AP Physics 1 Summer Packet Review of Trigonometry used in Physics
|
|
- Cleopatra Stevens
- 5 years ago
- Views:
Transcription
1 AP Physics 1 Summer Packet Review of Trigonometry used in Physics For some of you this material will seem pretty familiar and you will complete it quickly. For others, you may not have had much or any of this material yet in your math classes. In either case, the concepts covered in this packet are some of the biggest stumbling blocks to progress in a physics class and it is important that we have as much of an understanding of it as possible to begin the year. Please review this materially thoroughly and take it seriously if you do, it will make your life in AP Physics 1 much easier. Please note, in addition to specific exercises, there are some places in the notes of this material where you need to fill in blanks don t just skip these! You will need to print this document and write your answers on it. It is to be turned in on the first day of class. This packet is designed to guide you through it without any need for additional resources. However, if you have questions you can reach me by (jmarshall@sahs1.org) or my cell phone ( ). If I do not answer a phone call directly, leave a message so I know it was a student. Marshall 1. Introduction Consider the following situation An airplane flies at a speed of 100 m/s toward the east (on an x- y graph this would be in a direction of 0 degrees). At the same time a strong wind is blowing toward the north at 20 m/s (on an x- y graph this would be in a direction of 90 degrees). This situation could be represented on a graph by considering the plane to be at the origin. The speed of the plane toward the east could be represented by arrow toward the east with a length of 100 units, while the speed of the wind could be represented by an arrow toward the north with a length of 20 units: Hopefully it makes sense that because of the wind, some of the speed of the wind would be added to the speed of the airplane and to a person observing the plane from the ground, the actual speed of the plane would be a little more than 100 m/s and the direction of the plane would no longer be directly east, but a direction that is a little north of east. But, how much is the speed of the plane changed by the wind? Is all of the wind speed added to the plane speed, or is only part of the wind speed added because the direction of plane and the wind are different? What is the actual direction the plane is now travelling in? Looking at the above diagram, we will find out that if we draw a line from the tip of each arrow in the direction of the other arrow to the point where the lines intersect, we will create a parallelogram (see diagram on next page): 1
2 Further, if we draw a line from the origin to the point where these two lines intersect, the length of this line will represent the actual speed of the plane: Additionally, the angle between the x- axis (the horizontal direction or east ) and this new line will represent the direction the plane is actually now flying: Is there some way we can actually determine how long this new line is and what the angle above the x axis it is? Notice that the line we drew from the tip of the plane speed arrow is the same length as the wind speed arrow. If we move the actual wind speed arrow to this location on the diagram we can see that the two arrows and line we originally drew form a right triangle: Is there some way that you know of that you can now find the length of the 2
3 unknown line? Of course there is! You should remember that we call the side of a right triangle that is opposite the right angle, the hypotenuse. The Pythagorean theorem tells us that for a right triangle, the square of the length of the hypotenuse (which we call side c ) is equal to the sum of the squares of the other sides of the triangle (sides a and b ): c 2 = a 2 + b 2 Or, in this case: (actual speed) 2 = ( plane speed) 2 + (wind speed) 2 If we substitute real values in this gives us: (actual speed) 2 = (100 m / s) 2 + (20 m / s) 2 If we take the square root of both sides of the equation we get: (actual speed) 2 = (100 m / s) 2 + (20 m / s) 2 or actual speed = (100 m / s) 2 + (20 m / s) 2 = 102 m / s So, with the wind blowing as stated, the plane is actually moving 102 m/s that s the easy part. You will use similar Pythagorean theorem calculations dozens (if not hundreds) of time throughout the AP Physics 1 course get used to it! However, we do not want to just know the actual speed of the plane we also want to know the direction the plane is now flying because of the wind that is, what is the angle between the original direction of the of the plane (originally the plane was flying along the x axis, or east, or at 0 degrees), and the new direction of the plane. Being able to determine this requires a basic knowledge or trigonometry, which some of you have not yet had in math classes. Even if you have, the review will be really helpful. 3
4 2. Lets define some terminology. As we have already stated: hypotenuse the hypotenuse of a right triangle is the side of the triangle opposite the right angle: There are two ways to identify the hypotenuse of a right triangle. 1) It will always be the longest side of the right triangle 2) Identify the right angle of the triangle it will be the one which is Then look across the triangle the hypotenuse will be the side of the triangle that is across from the right angle. Until you become thoroughly familiar with these concepts and they become intuitive (automatic) to you, whenever you have a right triangle I would get in the habit of immediately labeling the hypotenuse of a right triangle with a small letter h. There are still two other sides to the right triangle and we need a way to label or reference them. How we do this will depend on which angle (other than the right angle) we are talking about. In physics, we usually use the small Greek letter, θ (theta), to reference an angle. For the triangle we are talking about, we could label the two remaining angles, θ 1 and θ 2 : The angle we were concerned about finding for the direction of the airplane was θ 1. With reference to this angle, we can now label the sides. The side of the triangle that is next to the angle is called the adjacent side, because it is adjacent to the angle (yes, the hypotenuse is also adjacent to the angle, but we are already calling it the hypotenuse). The side of the triangle that is on the side opposite the angle is called the opposite side. As with the hypotenuse, whenever you have a right triangle I 4
5 would get in the habit of immediately labeling the adjacent side of a right triangle with a small letter a, and the opposite side with a small letter o: Recognize again, what you label as adjacent and opposite will depend on the angle you are interested in. We are interested in angle θ 1, but if we were interested in angle θ 2 instead, the adjacent and opposite sides would be switched: On the diagram of our airplane problem shown below, label the angle we wanted to find as θ, and also label the hypotenuse (h), adjacent side (a), and opposite side (o). 5
6 3. For any right triangle, for a given angle, the ratios of the lengths of the sides of the triangle will always be the same (huh?) the meaning of sine, cosine and tangent. As you become more and more experienced with physics involving right triangles, you will become aware that there are five features or pieces of information we want access to regarding the triangle. We already know that one of the angles is a 90 degree angle. To completely characterize the triangle we will want to know the number of degrees of the other two angles, and the lengths of each of the sides. You will find out, if you don t know this already, that if we have a right triangle, and are given information that provides us with (a) one of the non- right angles and the length of one of the sides, or (b) the length of two sides, we will be able to determine the other three features. The ability to do this depends on the concept that for any right triangle, for a given angle (that is not the right angle), the ratios of lengths of the sides will always be the same. Lets see if we can use our airplane triangle to make sense of this. Below is our triangle notice I have labeled the angle we are interested in as θ, and I have labeled the hypotenuse (h), the adjacent side (a), and the opposite side (o). Notice also the opposite side has a value of 20 and the adjacent side has a value of 100. What is the ratio of the value of the opposite side compared to the value of the adjacent side? You can see that it is o = 20 = 0.2. Stated another way, the length of a 100 the opposite side is 1/5 of the length of the adjacent side. What if we were to make this triangle five times bigger (see diagram on next page)? You can see from simply inspecting the two triangles that while the lengths of all the sides are multiplied by 5, all of the angles are 6
7 exactly the same in both triangles. Also, and most importantly, the ratio of the length of the opposite side to the length of the adjacent side remains the same. o = 20 = 100 = 0.2 a Because all of the angles of the right triangle have remained the same, both the smaller and larger triangle are stated to be similar they have the same shape. This means that even though the lengths of the sides have increased, the ratios of the lengths of each side must be the same. That is, even though we increased the value of the length of opposite side from 20 to 100, the ratio of the length of the opposite side to the adjacent side must still be the same. The opposite side must still have a length that is 1/5 of the length of the adjacent side. Therefore, the length of the adjacent side must now have a value of 500. Based on this reasoning, it should make sense that for this particular angle, θ, the ratio of the length of the opposite side to the length of the adjacent side, that is, o, will always be the same, no matter what the specific lengths a are. For this angle of θ, if the opposite side has a length of 2, the adjacent side must have a length of 10. If the opposite side has a length of 7, the adjacent side must have a length of 35. If the opposite side has a length of 15, the adjacent side must have a length of 75. If the opposite side has a length of 2.5, the adjacent side must have a length of 12.5 and so on. For this angle of θ, the ratio o will always equal 0.2. a 7
8 It should now further make sense that if the ratio of o is some value other a than 0.2, the angle θ would also be different. For example, if the opposite side had a value of 5 and the adjacent side had a value of 12, the ratio o a would be 5 =.416. As this ratio is larger than in the previous case, the angle 12 θ is also greater. Finally, it should make sense that when the ratio of the length of the opposite side to the length of adjacent side ( o ) is a certain value, we can measure a the angle that corresponds to this ratio and write that value down somewhere in a reference book, or that we can further write down the angles that correspond to all of the possible ratios of opposite side to adjacent side. But, it s going to get really annoying to keep saying ratio of the length of the opposite side to the length of the adjacent side, so we are going actually give this ratio a special name tangent. Lets not forget, we actually have three sides, so we can also have the ratio of the length of the opposite side to the length of the hypotenuse, or the ratio of the length of the adjacent side to the length of the hypotenuse. Of course, we give these ratios special names, too. sine (abbreviated sin) ratio of the length of the opposite side to the length of the hypotenuse: sin = o h cosine (abbreviated cos) ratio of the length of the adjacent side to the length of the hypotenuse: cos = a h tangent (abbreviated tan) ratio of the length of the opposite side to the length of the adjacent side: tan = o a 8
9 These ratios are also called trig functions (short for trigonometry). The point is, in a right triangle, for a given number of degrees for an angle θ, these ratios of the lengths of sides will always be the same no matter what the actual values are, and we have reference books that tell us what those ratios will be for any number of degrees. Conversely, if we have the ratio, we can find out what the corresponding number of degrees must be. Lets use an old fashioned illustration of this. Go to the following link on the internet: This page shows the reference I am talking about a trig table. In our first example we said that the ratio of the length of the opposite side to the length of the adjacent side was 20 =.2. Again, the ratio of 100 the opposite to the adjacent side is called the tangent o a = tan. So, look under the Tan column in this table and find the values.1944 and.2126 (.2 is between these two values). Notice that in the Deg column, the corresponding values for degrees are 11 and 12. That is, if the ratio of opposite to adjacent is.2, the corresponding angle must be somewhere between 11 and 12 degrees. 9
10 We also had a right triangle where the ratio was 5/12 or.416. Using the same trig table, between what two values must the angle θ be for this ratio? There are two ways to view this information. We can either know the angle θ and find out what the ratios sin, cos, and tan are. Or, we can know what the ratio is, and find out what the angle is that corresponds to it. However, we use calculators now, not trig tables (I had to use a trig table in high school because we did not have calculators). If you know the angle is a certain number of degrees, to find either sin, cos or tan on your calculator, simply hit the sin, cos, or tan button. This will print the name of the ratio on the screen and give you a left parenthesis. sin( cos( tan( Then type in the number of degrees and close the parentheses (always make a habit of closing parentheses). For now, make sure your calculator is set to degrees and not radians. When you find the sin of a certain angle, make sure you understand, you are finding the ratio of the length of the opposite side of a triangle to the length of the hypotenuse. (1) If the angle is small (<30 degrees), the opposite side will be short compared to the hypotenuse and the sin will be less than 0.5 (2) At an angle of 30 degrees, the opposite side will be one- half the length of the hypotenuse, and so the sin will be 0.5. (3) At an angle greater than 30 degrees, the length of the opposite side will approach the length of the hypotenuse and the sin will be greater than 0.5. For a right triangle, the length of the opposite side can never by greater than the hypotenuse so the sin can never be greater than 1. 10
11 When you find the cos of a certain angle, make sure you understand, you are finding the ratio of the length of the adjacent side of a triangle to the length of the hypotenuse. (1) If the angle is great (>60 degrees), the adjacent side will be short compared to the hypotenuse and the cos will be less than 1. (2) At an angle of 60 degrees, the adjacent side will be one- half the length of the hypotenuse, and so the cos will be 0.5. (3) At an angle less than 60 degrees, the length of the adjacent side will approach the length of the hypotenuse and the cos will be greater than 0.5. For a right triangle, the length of the adjacent side can never by greater than the hypotenuse so the cos can never be greater than 1. When you find the tan of a certain angle, make sure you understand, you are finding the ratio of the length of the opposite side of a triangle to the length of the adjacent side. (1) If the angle is great (>45 degrees), the opposite side will be longer than the adjacent side and the tan will be greater than 1. (2) At an angle of 45 degrees, the adjacent side will equal the length of the opposite side, so the tan will be 1. (3) At an angle less than 45 degrees, the length of the adjacent side will greater than the opposite side and the tan will be less than 1. 11
12 What if we are given the ratio (that is, the sin, cos, or tan) and want to know the angle. The process of finding the angle given the ratio is called finding the inverse. If you are given the sin and want to know the angle, you find the inverse of the sin (designated sin - 1 ) (this is also known as the arcsin ). If you are given the cos and want to know the angle, you find the inverse of the cos (designated cos - 1 ) (this is also known as the arccos). If you are given the tan and want to know the angle, you find the inverse of the tan (designated tan - 1 ) (this is also known as the arctan). To accomplish this on the calculator, hit the 2 nd button, and then the ratio you want to use (sin, cos, or tan). On the screen will then be printed the name of the ratio, raised to the negative 1 power (this means inverse of ), followed by a left parenthesis. sin - 1 ( cos - 1 ( tan - 1 ( Then type in the value of the ratio and close the parentheses. The answer will then be the number of degrees. Lets test this. In our above example we said that when the ratio of the opposite side to the adjacent side was 0.2, the corresponding angle was between 11 and 12 degrees according to the trig table. So, find the inverse tangent of 0.2 on your calculator. You should get , confirming what our trig table said. The relationship between the ratios of the different sides of a right triangle and the corresponding angles is so important that we have a mnemonic to help us remember the names of the ratios and the corresponding sides: SOH- CAH- TOA (pronounced sew- caw- toe- a) 12
13 Lets make sure you have a good understanding of how to find values for sin, cos and tan given an angle, and how to find angles given a value for sin, cos and tan. Using the trig table ( given the following angles, which are not exact whole number angles, state the values between which the actual sin, cos and tan must be (notice that the table headings for values of trig functions between 0 and 45 degrees are at the top of the table, while the table headings for values of trig functions between 45 and 90 degrees are listed at the bottom of the table). (round to the nearest thousandth). I have done the first one for you Angle sin is between cos is between tan is between and and and and and and and and and and and and and and and Now, using your calculator, find the exact values of the trig functions for these same angles (round to nearest thousandth). Angle sin cos tan
14 Given the following values for sin, use the trig table ( trig/tables.htm) to determine the angles between which the actual angle must be. Then use your calculator to determine the exact value (round to nearest hundredth of a degree) Value of sin angle is between exact value (nearest hundredth).6092 and (think this means that the length of opposite side is.6092 times the length of the hypotenuse what does that mean for the size of the angle don t.0035 and (think this means that the length of opposite side is.0035 times the length of the hypotenuse what does that mean for the size of the angle don t.1987 and (think this means that the length of opposite side is.1987 times the length of the hypotenuse what does that mean for the size of the angle don t.8888 and (think this means that the length of opposite side is.8888 times the length of the hypotenuse what does that mean for the size of the angle don t.3443 and (think this means that the length of opposite side is.3443 times the length of the hypotenuse what does that mean for the size of the angle don t Given the following values for cos, use the trig table( trig/tables.htm) to determine the angles between which the actual angle must be. Then use your calculator to determine the exact value (round to nearest hundredth of a degree) Value of cos angle is between exact value (nearest hundredth).6092 and (think this means that the length of adjacent side is.6092 times the length of the hypotenuse what does that mean for the size of the angle don t.0035 and (think this means that the length of adjacent side is.0035 times the length of the hypotenuse what does that mean for the size of the angle don t.1987 and (think this means that the length of adjacent side is.1987 times the length of the hypotenuse what does that mean for the size of the angle don t 14
15 .8888 and (think this means that the length of adjacent side is.8888 times the length of the hypotenuse what does that mean for the size of the angle don t.3443 and (think this means that the length of adjacent side is.3443 times the length of the hypotenuse what does that mean for the size of the angle don t Here s something interesting you may have already noticed it add up the values of the exact angles for each value of sin and cos as follows: angle for sin - 1 (.6092) + angle for cos - 1 (.6092) = angle for sin - 1 (.0035) + angle for cos - 1 (.0035) = angle for sin - 1 (.1987) + angle for cos - 1 (.1987) = angle for sin - 1 (.8888) + angle for cos - 1 (.8888) = angle for sin - 1 (.3443) + angle for cos - 1 (.3443) = sin and cos are said to be reciprocal functions this means that the sin of one angle will equal the cos of the complement of that angle (remember that two complementary angles add up to 90 degrees) so, sin(35 0 ) = cos(55 0 ). This means that if you know the sin of an angle, you also automatically know the cos of its complement. Given the following values for tan, use the trig table( trig/tables.htm) to determine the angles between which the actual angle must be. Then use your calculator to determine the exact value (round to nearest hundredth of a degree) Value of tan angle is between exact value (nearest hundredth).0178 and (think this means that the length of opposite side is.0178 times the length of the adjacent side what does that mean for the size of the angle don t and (think this means that the length of opposite side is times the length of the adjacent side what does that mean for the size of the angle don t and (think this means that the length of opposite side is times the length of the adjacent side what does that mean for the size of the angle don t.8888 and 15
16 (think this means that the length of opposite side is.8888 times the length of the adjacent side what does that mean for the size of the angle don t and (think this means that the length of opposite side is times the length of the adjacent side what does that mean for the size of the angle don t 16
17 4. How to use sin, cos, and tan (trig functions) to find unknown sides/angles of right triangles. In the introductory problem we saw that in solving some types of physics problems, we might have a critical need to be able to determine the value of unknown lengths of sides and angles of a right triangle. Underlying this skill is the understanding that for a specific angle in a right triangle, the ratio of the length of any of the sides to one of the other sides is a constant value, no matter what the actual lengths are. For a particular angle, we can either look up these ratios (sin, cos, and tan) on a trig table, or we can find them using a calculator. Knowing these ratios for a particular angle, if you know the length of one of the sides you can determine the length of the other side. Or, if you know the length of two sides, you can determine the angle associated with those two sides. The following exercises will guide you through and give you practice in the strategy we need to follow to develop these necessary skills. - - Given the right triangle illustrated to the right - - Given the angle designated θ 1. Label the sides of the triangle as: h for hypotenuse a for adjacent o for opposite θ 2. The sin of θ is defined as the ratio of the length of the side to the length of the. 3. The cos of θ is defined as the ratio of the length of the side to the length of the. 4. The tan of θ is defined as the ratio of the length of the side to the length of the side. (Recognize that if we had called the other non- right angle θ, the sides that we called opposite and adjacent would be reversed, although the hypotenuse would still be the same side.) 17
18 5. We commonly use the abbreviations listed in question 1 for the hypotenuse, adjacent and opposite sides of a right triangle as they relate the angle designated as θ. Using those abbreviations, complete the following equations: sin θ = cos θ = tan θ = 6. Remember that we have created a mnemonic to help us remember these ratios Write out that mnemonic: 7. For the following right triangles (starting on the next page), you have been given all three sides of the triangle. For example, for triangle 1 you have been given that one side has a value of 14.7, another has a value of 3.31, and the last has a value of Get used to the idea that we will commonly label sides of a triangle with variables that mean something to us and their corresponding value. In the first triangle, V, VA, and VB are referring to velocity, velocity A and velocity B. It will not only be important for you to recognize the hypotenuse, adjacent side and opposite side of a right triangle, but that these sides correspond to specific values in a physics problem and you can substitute these values in for h, o, and a to help you solve problems. Recognizing this will allow you determine which trig functions you need to use to solve problems. For each triangle, for each angle listed (θ 1 and θ 2): (a) Write out the general formula for the trig function asked for using the standard abbreviations, h for hypotenuse, o for opposite, and a for adjacent (b) Then write that formula in terms of the labels on the triangle; (c) Then plug the correct values in and solve Recognize that what the adjacent and opposite sides will be will depend on whether it is θ1 or θ2. One of the functions for each angle of the first triangle has been done for you as an example of what I want. Remember also that, as ratios of lengths of sides of a right triangle, the final values will have no units they are simply ratios. For the example I have completed, the sin of angle θ1 is simply a ratio telling us that side VA is.225 times as long as side V. Recognize also that there will only be one angle that can give us that specific ratio. Round all values to the thousandths place. 18
19 Triangle 1 For angleθ 1 sinθ 1 = o h = V A V = =.225 cos θ 1 tan θ 1 For angleθ 2 sinθ 1 = o h = V B V = =.973 cos θ 2 tan θ 2 Triangle 2 19
20 For angleθ 1 sin θ 1 cos θ 1 tan θ 1 For angleθ 2 sin θ 2 cos θ 2 tan θ 2 Triangle 3 20
21 For angleθ 1 sin θ 1 cos θ 1 tan θ 1 For angleθ 2 sin θ 2 cos θ 2 tan θ 2 8. Now lets use our knowledge to find missing pieces of information. If we are given an angle and one side of a right triangle, we can always find the the lengths of the other two sides (if we have a calculator or a trig table). The basic strategy is this (this will seem confusing at first but we will go through several examples step by step): - - It is always a good idea to label a right triangle with h, o, and a. - - Ask the question, related to the given angle, what side of the triangle have I been given, the hypotenuse, the adjacent side or the opposite side? - - Then ask the question, related to the given angle, what side of the triangle do I want, the hypotenuse, the adjacent side or the opposite side? - - Then ask the question, because I can remember which trig functions are related to which sides because I know the mnemonic, SOH CAH TOA, which trig function contains both the side I have been given and the side I want? 21
22 - - Then write down the basic formula for that trig function (as you did in problem 5). - - Then substitute in the labels on the diagram of the triangle for the abbreviations of the sides in the formula. - - Then recognize that to get the length of the side you want, isolate that variable on one side of the equation. (We will develop the habit of not ever substituting values into a formula until the variable you want has been isolated on one side of the equation) - - Then plug in the values and solve. Using this strategy, find the missing pieces of information in the following problems the first one has been done for you. In each successive problem, I have decreased the amount of guidance. (PLEASE NOTE I am not asking you to write down these questions they are questions you will ask yourself in your brain as you develop the ability use this skill). Triangle 1 Given side V = 23.9 m/s and θ2 = 71 o, find the value of side VB. Label the triangle with h, o, and a. Related to angle θ2, what side have you been given? Hypotenuse Related to angle θ2, what side do you want? Opposite What trig function contains both of these sides? Sin What is the formula for this trig function? sinθ 2 = o h Substitute labels from the diagram into this formula: sinθ 2 = o h = V B V Isolate the desired value on one side of the equation: We want the length of side VB this means that before we plug values into the formula, we isolate VB on one side of the formula. 22
23 In this particular circumstance, in order to get VB by itself on one side of the equation, we need to multiply both sides by the denominator, V. (V )(sinθ 2 ) = o h = (V ) V B V (I like the isolated variable on the left) V sinθ 2 = V B V B = V sinθ 2 Plug in given values and solve: V B = (23.9 m / s)(sin71 0 ) = 22.6 m / s Triangle 2 Given side DA = 414 m/s and θ1 = 32 o, find the value of side DB. Label the triangle with h, o, and a. Related to angle θ1, what side have you been given? Related to angle θ1, what side do you want? What trig function contains both of these sides? What is the formula for this trig function? Substitute labels from the diagram into this formula: Isolate desired value on one side of the equation: Plug in given values and solve: 23
24 Triangle 3 VA =? VB = 191 km/h θ2 θ1 V =? Given side VB = 191 km/h and θ2 = 74 o, find the value of side VA. Note even though I have left out some of the guiding questions you should still be thinking them in your mind to get to the answer of the questions I have left in. What trig function contains both the given side and the side you want? Write the formula for this function using the labels from the given triangle: Isolate the desired value on one side of the equation: Plug in given values and solve: Triangle 4 Given side VA =7.45 m/s and θ 1 = 27 0, find the value of side V. You re on your own for this one just follow the strategy and it should be no problem. You need to show me the formula you are using to solve the problem, how you isolate the desired variable, how you plug the correct values into the formula, and the answer. 24
25 9. Now lets reverse the process. Previous problems have shown us (a) that if we know the lengths of the sides of a right triangle, we can find ratio of the length of the opposite side compared to the hypotenuse (sin), the ratio of the length of the adjacent side compared to the hypotenuse (cos), and the ratio of the length of the opposite side compared to the adjacent side (tan) (b) that if we know the length of a side and an angle, we can find the lengths of the other two sides. We have also stated that for any non- right angle of a right triangle, whether it s 1.7 o, 59.6 o, 74.3 o, or whatever the angle, for that angle, there is one and only one sin, cos and tan that corresponds to that angle. Therefore, for a given value of sin, cos, and tan, there is one and only one angle that can correspond with that value. Therefore, if I have a calculator or a trig table, and I know a value for sin, cos or tan, or I can calculate one of these values using the lengths of sides, I can find the angle by finding the inverse of that function, denoted: sin - 1, cos - 1, and tan - 1. The basic strategy is this given two sides of a right triangle, or, given information that will allow me to calculate the length of two sides of a right triangle: - - Label the triangle h, o, and a. - - Ask the question, related to the angle I want to find, what two sides of the right triangle do I have, the opposite and hypotenuse, the adjacent and hypotenuse, or the opposite and adjacent? - - Then ask the question, what trig function contains both of these sides? - - Then write down the standard formula for that trig function, remembering to include the θ symbol after the name of the function. - - Then move the name of the function to the other side of the equation, in front of the ratio for that function, and place a superscript - 1 after the name of the function, so that just the θ symbol is on one side of the equation, and the inverse designation for the trig function, and its ratio are on the other side. - - Then substitute in the labels on the diagram of the triangle for the abbreviations of the sides in the formula. - - Then plug in the values given and solve. In the following problems, find the requested angle the first one has been done for you. The remaining three I have given you no additional help on. 25
26 Triangle 1 Find angleθ 1 In relationship to angle θ1, what two sides have you been given? opposite and hypotenuse What trig function contains both of these sides? sin What is the formula for this trig function? sinθ 1 = o h Write the inverse form of this formula: θ 1 = sin 1 o h o Substitute in labels from the triangle: θ 1 = sin 1 h = V A sin 1 V Plug in values: θ 1 = sin = Triangle 2 Find angleθ 2 (on your own) Triangle 3 26
27 Find angleθ 1 (on your own) Triangle 4 Find angleθ 2 (on your own) VA = 7.45 m/s θ2 V = 16.4 m/s VB =? θ1 27
Algebra/Geometry Blend Unit #7: Right Triangles and Trigonometry Lesson 1: Solving Right Triangles. Introduction. [page 1]
Algebra/Geometry Blend Unit #7: Right Triangles and Trigonometry Lesson 1: Solving Right Triangles Name Period Date Introduction [page 1] Learn [page 2] Pieces of a Right Triangle The map Brian and Carla
More informationLearning Goal: I can explain when to use the Sine, Cosine and Tangent ratios and use the functions to determine the missing side or angle.
MFM2P Trigonometry Checklist 1 Goals for this unit: I can solve problems involving right triangles using the primary trig ratios and the Pythagorean Theorem. U1L4 The Pythagorean Theorem Learning Goal:
More informationApplication of Geometric Mean
Section 8-1: Geometric Means SOL: None Objective: Find the geometric mean between two numbers Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse
More informationBASICS OF TRIGONOMETRY
Mathematics Revision Guides Basics of Trigonometry Page 1 of 9 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Foundation Tier BASICS OF TRIGONOMETRY Version: 1. Date: 09-10-015 Mathematics Revision
More informationLesson 30, page 1 of 9. Glencoe Geometry Chapter 8.3. Trigonometric Ratios
Lesson 30 Lesson 30, page 1 of 9 Glencoe Geometry Chapter 8.3 Trigonometric Ratios Today we look at three special ratios of right triangles. The word Trigonometry is derived from two Greek words meaning
More informationMORE TRIGONOMETRY
MORE TRIGONOMETRY 5.1.1 5.1.3 We net introduce two more trigonometric ratios: sine and cosine. Both of them are used with acute angles of right triangles, just as the tangent ratio is. Using the diagram
More informationUnit 2: Right Triangle Trigonometry RIGHT TRIANGLE RELATIONSHIPS
Unit 2: Right Triangle Trigonometry This unit investigates the properties of right triangles. The trigonometric ratios sine, cosine, and tangent along with the Pythagorean Theorem are used to solve right
More informationChapter 7. Right Triangles and Trigonometry
Chapter 7 Right Triangles and Trigonometry 4 16 25 100 144 LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor 8 20 32 = = = 4 *2 = = = 75 = = 40 = = 7.1 Apply the Pythagorean Theorem Objective:
More information8-1. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary
8-1 The Pythagorean Theorem and Its Converse Vocabulary Review 1. Write the square and the positive square root of each number. Number 9 Square Positive Square Root 1 4 1 16 Vocabulary Builder leg (noun)
More informationWeek 11, Lesson 1 1. Warm Up 2. Notes Sine, Cosine, Tangent 3. ICA Triangles
Week 11, Lesson 1 1. Warm Up 2. Notes Sine, Cosine, Tangent 3. ICA Triangles HOW CAN WE FIND THE SIDE LENGTHS OF RIGHT TRIANGLES? Essential Question Essential Question Essential Question Essential Question
More information8-1. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary
8-1 he Pythagorean heorem and Its Converse Vocabulary Review 1. Write the square and the positive square root of each number. Number Square Positive Square Root 9 81 3 1 4 1 16 1 2 Vocabulary Builder leg
More informationTrig Functions Learning Outcomes. Solve problems about trig functions in right-angled triangles. Solve problems using Pythagoras theorem.
1 Trig Functions Learning Outcomes Solve problems about trig functions in right-angled triangles. Solve problems using Pythagoras theorem. Opposite Adjacent 2 Use Trig Functions (Right-Angled Triangles)
More information1 What is Trigonometry? Finding a side Finding a side (harder) Finding an angle Opposite Hypotenuse.
Trigonometry (9) Contents 1 What is Trigonometry? 1 1.1 Finding a side................................... 2 1.2 Finding a side (harder).............................. 2 1.3 Finding an angle.................................
More informationGeom- Chpt. 8 Algebra Review Before the Chapter
Geom- Chpt. 8 Algebra Review Before the Chapter Solving Quadratics- Using factoring and the Quadratic Formula Solve: 1. 2n 2 + 3n - 2 = 0 2. (3y + 2) (y + 3) = y + 14 3. x 2 13x = 32 1 Working with Radicals-
More informationTrig Functions Learning Outcomes. Solve problems about trig functions in right-angled triangles. Solve problems using Pythagoras theorem.
1 Trig Functions Learning Outcomes Solve problems about trig functions in right-angled triangles. Solve problems using Pythagoras theorem. Opposite Adjacent 2 Use Trig Functions (Right-Angled Triangles)
More informationUnit 3 Trigonometry. 3.1 Use Trigonometry to Find Lengths
Topic : Goal : Unit 3 Trigonometry trigonometry I can use the primary trig ratios to find the lengths of sides in a right triangle 3.1 Use Trigonometry to Find Lengths In any right triangle, we name the
More informationSimilar Right Triangles
MATH 1204 UNIT 5: GEOMETRY AND TRIGONOMETRY Assumed Prior Knowledge Similar Right Triangles Recall that a Right Triangle is a triangle containing one 90 and two acute angles. Right triangles will be similar
More informationEQ: SRT.8 How do I use trig to find missing side lengths of right triangles?
EQ: SRT.8 How do I use trig to find missing side lengths of right triangles? Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question Essential
More informationModule 13 Trigonometry (Today you need your notes)
Module 13 Trigonometry (Today you need your notes) Question to ponder: If you are flying a kite, you know the length of the string, and you know the angle that the string is making with the ground, can
More informationPut in simplest radical form. (No decimals)
Put in simplest radical form. (No decimals) 1. 2. 3. 4. 5. 6. 5 7. 4 8. 6 9. 5 10. 9 11. -3 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 3 28. 1 Geometry Chapter 8 - Right Triangles
More informationUNIT 2 RIGHT TRIANGLE TRIGONOMETRY Lesson 2: Applying Trigonometric Ratios Instruction
Prerequisite Skills This lesson requires the use of the following skills: defining and calculating sine, cosine, and tangent setting up and solving problems using the Pythagorean Theorem identifying the
More informationRiverboat and Airplane Vectors
Grade Homework Riverboat and Airplane Vectors It all depends on your point of view It s all relative On occasion objects move within a medium that is moving with respect to an observer. In such instances,
More informationThe statements of the Law of Cosines
MSLC Workshop Series: Math 1149 and 1150 Law of Sines & Law of Cosines Workshop There are four tools that you have at your disposal for finding the length of each side and the measure of each angle of
More informationFunctions - Trigonometry
10. Functions - Trigonometry There are si special functions that describe the relationship between the sides of a right triangle and the angles of the triangle. We will discuss three of the functions here.
More informationStudent Instruction Sheet: Unit 4, Lesson 4. Solving Problems Using Trigonometric Ratios and the Pythagorean Theorem
Student Instruction Sheet: Unit 4, Lesson 4 Suggested Time: 75 minutes Solving Problems Using Trigonometric Ratios and the Pythagorean Theorem What s important in this lesson: In this lesson, you will
More informationRight is Special 1: Triangles on a Grid
Each student in your group should have a different equilateral triangle. Complete the following steps: Using the centimeter grid paper, determine the length of the side of the triangle. Write the measure
More information77.1 Apply the Pythagorean Theorem
Right Triangles and Trigonometry 77.1 Apply the Pythagorean Theorem 7.2 Use the Converse of the Pythagorean Theorem 7.3 Use Similar Right Triangles 7.4 Special Right Triangles 7.5 Apply the Tangent Ratio
More informationParking Lot HW? Joke of the Day: What do you get when you combine a flat, infinite geometric figure with a beef patty?
Parking Lot HW? Joke of the Day: What do you get when you combine a flat, infinite geometric figure with a beef patty? a plane burger Agenda 1 23 hw? Finish Special Right Triangles L8 3 Trig Ratios HW:
More informationWelcome to Trigonometry!
Welcome to Trigonometry! Right Triangle Trigonometry: The study of the relationship between the sides and the angles of right triangles. Why is this important? I wonder how tall this cake is... 55 0 3
More information8.7 Extension: Laws of Sines and Cosines
www.ck12.org Chapter 8. Right Triangle Trigonometry 8.7 Extension: Laws of Sines and Cosines Learning Objectives Identify and use the Law of Sines and Cosines. In this chapter, we have only applied the
More informationSimplifying Radical Expressions and the Distance Formula
1 RD. Simplifying Radical Expressions and the Distance Formula In the previous section, we simplified some radical expressions by replacing radical signs with rational exponents, applying the rules of
More informationMath Section 4.1 Special Triangles
Math 1330 - Section 4.1 Special Triangles In this section, we ll work with some special triangles before moving on to defining the six trigonometric functions. Two special triangles are 30 60 90 triangles
More informationReview on Right Triangles
Review on Right Triangles Identify a Right Triangle Example 1. Is each triangle a right triangle? Explain. a) a triangle has side lengths b) a triangle has side lengths of 9 cm, 12 cm, and 15 cm of 5 cm,7
More information8.3 Trigonometric Ratios-Tangent. Geometry Mr. Peebles Spring 2013
8.3 Trigonometric Ratios-Tangent Geometry Mr. Peebles Spring 2013 Bell Ringer 3 5 Bell Ringer a. 3 5 3 5 = 3 5 5 5 Multiply the numerator and denominator by 5 so the denominator becomes a whole number.
More informationA life not lived for others is not a life worth living. Albert Einstein
life not lived for others is not a life worth living. lbert Einstein Sides adjacent to the right angle are legs Side opposite (across) from the right angle is the hypotenuse. Hypotenuse Leg cute ngles
More informationChapter 8: Right Triangles (page 284)
hapter 8: Right Triangles (page 284) 8-1: Similarity in Right Triangles (page 285) If a, b, and x are positive numbers and a : x = x : b, then x is the between a and b. Notice that x is both in the proportion.
More informationUse SOH CAH TOA to memorize the three main trigonometric functions.
Use SOH CAH TOA to memorize the three main trigonometric functions. Content Objective Content Objective Content Objective Content Objective Content Objective Content Objective Content Objective Content
More information84 Geometric Mean (PAAP and HLLP)
84 Geometric Mean (PAAP and HLLP) Recall from chapter 7 when we introduced the Geometric Mean of two numbers. Ex 1: Find the geometric mean of 8 and 96.ÿ,. dÿ,... : J In a right triangle, an altitude darn
More informationSin, Cos, and Tan Revealed
Sin, Cos, and Tan Revealed Reference Did you ever wonder what those keys on your calculator that say sin, cos, and tan are all about? Well, here s where you find out. You ve seen that whenever two right
More information1. A right triangle has legs of 8 centimeters and 13 centimeters. Solve the triangle completely.
9.7 Warmup 1. A right triangle has legs of 8 centimeters and 13 centimeters. Solve the triangle completely. 2. A right triangle has a leg length of 7 in. and a hypotenuse length of 14 in. Solve the triangle
More informationEQ: How do I use trigonometry to find missing side lengths of right triangles?
EQ: How do I use trigonometry to find missing side lengths of right triangles? Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question Essential
More informationAP Physics B Summer Homework (Show work)
#1 NAME: AP Physics B Summer Homework (Show work) #2 Fill in the radian conversion of each angle and the trigonometric value at each angle on the chart. Degree 0 o 30 o 45 o 60 o 90 o 180 o 270 o 360 o
More informationOVERVIEW Similarity Leads to Trigonometry G.SRT.6
OVERVIEW Similarity Leads to Trigonometry G.SRT.6 G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric
More informationThe study of the measurement of triangles is called Trigonometry.
Math 10 Workplace & Apprenticeship 7.2 The Sine Ratio Day 1 Plumbers often use a formula to determine the lengths of pipes that have to be fitted around objects. Some common terms are offset, run, and
More information*Definition of Cosine
Vetors - Unit 3.3A - Problem 3.5A 3 49 A right triangle s hypotenuse is of length. (a) What is the length of the side adjaent to the angle? (b) What is the length of the side opposite to the angle? ()
More information11.4 Apply the Pythagorean
11.4 Apply the Pythagorean Theorem and its Converse Goal p and its converse. Your Notes VOCABULARY Hypotenuse Legs of a right triangle Pythagorean theorem THE PYTHAGOREAN THEOREM Words If a triangle is
More informationStudent Outcomes. Lesson Notes. Classwork. Discussion (20 minutes)
Student Outcomes Students explain a proof of the converse of the Pythagorean Theorem. Students apply the theorem and its converse to solve problems. Lesson Notes Students had their first experience with
More informationTRAINING LAB BLOOD AS EVIDENCE BLOOD DROPS FALLING AT AN ANGLE NAME
TRAINING LAB BLOOD AS EVIDENCE BLOOD DROPS FALLING AT AN ANGLE NAME Background: You just completed studying the behavior of passive blood drops that drip straight down from a wound, but not all blood drops
More informationMath 20-3 Admission Exam Study Guide Notes about the admission exam:
Math 20-3 Admission Exam Study Guide Notes about the admission exam: To write the exam, no appointment is necessary; drop-in to MC221 (Testing) and ask for the 20-3 exam. You ll be given a form to take
More informationUnit 2 Day 4 Notes Law of Sines
AFM Unit 2 Day 4 Notes Law of Sines Name Date Introduction: When you see the triangle below on the left and someone asks you to find the value of x, you immediately know how to proceed. You call upon your
More informationRight-angled triangles and trigonometry
Right-angled triangles and trigonometry 5 syllabusref Strand: Applied geometry eferenceence Core topic: Elements of applied geometry In this cha 5A 5B 5C 5D 5E 5F chapter Pythagoras theorem Shadow sticks
More informationApplying Trigonometry: Angles of Depression and Elevation
Applying Trigonometry: Angles of Depression and Elevation An angle of elevation is the angle formed by a horizontal line and the line of sight to a point above. In the diagram, 1 is the angle of elevation.
More informationI can add vectors together. IMPORTANT VOCABULARY
Pre-AP Geometry Chapter 9 Test Review Standards/Goals: G.SRT.7./ H.1.b.: I can find the sine, cosine and tangent ratios of acute angles given the side lengths of right triangles. G.SRT.8/ H.1.c.: I can
More informationNote! In this lab when you measure, round all measurements to the nearest meter!
Distance and Displacement Lab Note! In this lab when you measure, round all measurements to the nearest meter! 1. Place a piece of tape where you will begin your walk outside. This tape marks the origin.
More informationThe Battleship North Carolina s Fire Control
The Battleship North Carolina s Fire Control Objectives: 1. Students will see the application of trigonometry that the Mark 14 gun sight used with the 20mm guns aboard the NC Battleship. (Geometry SCOS:
More informationBesides the reported poor performance of the candidates there were a number of mistakes observed on the assessment tool itself outlined as follows:
MATHEMATICS (309/1) REPORT The 2013 Mathematics (309/1) paper was of average standard. The paper covered a wide range of the syllabus. It was neither gender bias nor culture bias. It did not have language
More informationLesson 3: Using the Pythagorean Theorem. The Pythagorean Theorem only applies to triangles. The Pythagorean Theorem + = Example 1
Lesson 3: Using the Pythagorean Theorem The Pythagorean Theorem only applies to triangles. The Pythagorean Theorem + = Example 1 A sailboat leaves dock and travels 6 mi due east. Then it turns 90 degrees
More informationParallel Lines Cut by a Transversal
Name Date Class 11-1 Parallel Lines Cut by a Transversal Parallel Lines Parallel Lines Cut by a Transversal A line that crosses parallel lines is a transversal. Parallel lines never meet. Eight angles
More informationTrigonometric Functions
Trigonometric Functions (Chapters 6 & 7, 10.1, 10.2) E. Law of Sines/Cosines May 21-12:26 AM May 22-9:52 AM 1 degree measure May 22-9:52 AM Measuring in Degrees (360 degrees) is the angle obtained when
More informationThe Pythagorean Theorem Diamond in the Rough
The Pythagorean Theorem SUGGESTED LEARNING STRATEGIES: Shared Reading, Activating Prior Knowledge, Visualization, Interactive Word Wall Cameron is a catcher trying out for the school baseball team. He
More informationIn previous examples of trigonometry we were limited to right triangles. Now let's see how trig works in oblique (not right) triangles.
The law of sines. In previous examples of trigonometry we were limited to right triangles. Now let's see how trig works in oblique (not right) triangles. You may recall from Plane Geometry that if you
More informationApplications of trigonometry
Applications of trigonometry This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,
More informationLEARNING OBJECTIVES. Overview of Lesson. guided practice Teacher: anticipates, monitors, selects, sequences, and connects student work
D Rate, Lesson 1, Conversions (r. 2018) RATE Conversions Common Core Standard N.Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units
More informationCH 21 THE PYTHAGOREAN THEOREM
121 CH 21 THE PYTHAGOREAN THEOREM The Right Triangle A n angle of 90 is called a right angle, and when two things meet at a right angle, we say they are perpendicular. For example, the angle between a
More informationRelated Rates - Classwork
Related Rates - Classwork Earlier in the year, we used the basic definition of calculus as the mathematics of change. We defined words that meant change: increasing, decreasing, growing, shrinking, etc.
More informationName Date PD. Pythagorean Theorem
Name Date PD Pythagorean Theorem Vocabulary: Hypotenuse the side across from the right angle, it will be the longest side Legs are the sides adjacent to the right angle His theorem states: a b c In any
More informationMarch Madness Basketball Tournament
March Madness Basketball Tournament Math Project COMMON Core Aligned Decimals, Fractions, Percents, Probability, Rates, Algebra, Word Problems, and more! To Use: -Print out all the worksheets. -Introduce
More informationTrigonometry. terminal ray
terminal ray y Trigonometry Trigonometry is the study of triangles the relationship etween their sides and angles. Oddly enough our study of triangles egins with a irle. r 1 θ osθ P(x,y) s rθ sinθ x initial
More informationWrite these equations in your notes if they re not already there. You will want them for Exam 1 & the Final.
Tuesday January 30 Assignment 3: Due Friday, 11:59pm.like every Friday Pre-Class Assignment: 15min before class like every class Office Hours: Wed. 10-11am, 204 EAL Help Room: Wed. & Thurs. 6-9pm, here
More information5.8 The Pythagorean Theorem
5.8. THE PYTHAGOREAN THEOREM 437 5.8 The Pythagorean Theorem Pythagoras was a Greek mathematician and philosopher, born on the island of Samos (ca. 582 BC). He founded a number of schools, one in particular
More informationChapter. Similar Triangles. Copyright Cengage Learning. All rights reserved.
Chapter 5 Similar Triangles Copyright Cengage Learning. All rights reserved. 5.4 The Pythagorean Theorem Copyright Cengage Learning. All rights reserved. The Pythagorean Theorem The following theorem will
More informationMath 3 Plane Geometry Review Special Triangles
Name: 1 Date: Math 3 Plane Geometry Review Special Triangles Special right triangles. When using the Pythagorean theorem, we often get answers with square roots or long decimals. There are a few special
More informationAreas of Parallelograms and Triangles 7-1
Areas of Parallelograms and Triangles 7-1 Parallelogram A parallelogram is a quadrilateral where the opposite sides are congruent and parallel. A rectangle is a type of parallelogram, but we often see
More informationLast First Date Per SETTLE LAB: Speed AND Velocity (pp for help) SPEED. Variables. Variables
DISTANCE Last First Date Per SETTLE LAB: Speed AND Velocity (pp108-111 for help) Pre-Activity NOTES 1. What is speed? SPEED 5-4 - 3-2 - 1 2. What is the formula used to calculate average speed? 3. Calculate
More informationMarch Madness Basketball Tournament
March Madness Basketball Tournament Math Project COMMON Core Aligned Decimals, Fractions, Percents, Probability, Rates, Algebra, Word Problems, and more! To Use: -Print out all the worksheets. -Introduce
More informationStudent Resource / Program Workbook INTEGERS
INTEGERS Integers are whole numbers. They can be positive, negative or zero. They cannot be decimals or most fractions. Let us look at some examples: Examples of integers: +4 0 9-302 Careful! This is a
More informationCH 34 MORE PYTHAGOREAN THEOREM AND RECTANGLES
CH 34 MORE PYTHAGOREAN THEOREM AND RECTANGLES 317 Recalling The Pythagorean Theorem a 2 + b 2 = c 2 a c 90 b The 90 angle is called the right angle of the right triangle. The other two angles of the right
More informationChapter 10. Right Triangles
Chapter 10 Right Triangles If we looked at enough right triangles and experimented a little, we might eventually begin to notice some relationships developing. For instance, if I were to construct squares
More informationLesson 21: Special Relationships within Right Triangles Dividing into Two Similar Sub-Triangles
: Special Relationships within Right Triangles Dividing into Two Similar Sub-Triangles Learning Targets I can state that the altitude of a right triangle from the vertex of the right angle to the hypotenuse
More informationMotion in 1 Dimension
A.P. Physics 1 LCHS A. Rice Unit 1 Displacement, Velocity, & Acceleration: Motion in 1 Dimension In-Class Example Problems and Lecture Notes 1. Freddy the cat started at the 3 meter position. He then walked
More informationVectors in the City Learning Task
Vectors in the City Learning Task Amy is spending some time in a city that is laid out in square blocks. The blocks make it very easy to get around so most directions are given in terms of the number of
More informationDate Lesson Assignment Did it grade Friday Feb.24
PAP Pre-Calculus Lesson Plans Unit Sem 2 3 rd term Johnston (C4) and Noonan (C6) February 24 th to March 9 th 202 - Vectors Date Lesson Assignment Did it grade Friday Feb.24 Law of Sines/Cosines, Area
More informationMathematics. Leaving Certificate Examination Paper 1 Higher Level Friday 10 th June Afternoon 2:00 4:30
2016. M29 Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination 2016 Mathematics Paper 1 Higher Level Friday 10 th June Afternoon 2:00 4:30 300 marks Examination number
More informationUnit 4. Triangle Relationships. Oct 3 8:20 AM. Oct 3 8:21 AM. Oct 3 8:26 AM. Oct 3 8:28 AM. Oct 3 8:27 AM. Oct 3 8:27 AM
Unit 4 Triangle Relationships 4.1 -- Classifying Triangles triangle -a figure formed by three segments joining three noncollinear points Classification of triangles: by sides by angles Oct 3 8:20 AM Oct
More information2.6 Related Rates Worksheet Calculus AB. dy /dt!when!x=8
Two Rates That Are Related(1-7) In exercises 1-2, assume that x and y are both differentiable functions of t and find the required dy /dt and dx /dt. Equation Find Given 1. dx /dt = 10 y = x (a) dy /dt
More informationWhere are you right now? How fast are you moving? To answer these questions precisely, you
4.1 Position, Speed, and Velocity Where are you right now? How fast are you moving? To answer these questions precisely, you need to use the concepts of position, speed, and velocity. These ideas apply
More informationWorksheet 1.1 Kinematics in 1D
Worksheet 1.1 Kinematics in 1D Solve all problems on your own paper showing all work! 1. A tourist averaged 82 km/h for a 6.5 h trip in her Volkswagen. How far did she go? 2. Change these speeds so that
More informationSolving Quadratic Equations (FAL)
Objective: Students will be able to (SWBAT) solve quadratic equations with real coefficient that have complex solutions, in order to (IOT) make sense of a real life situation and interpret the results
More informationMATERIALS: softball, stopwatch, measuring tape, calculator, writing utensil, data table.
1 PROJECTILE LAB: (SOFTBALL) Name: Partner s Names: Date: PreAP Physics LAB Weight = 1 PURPOSE: To calculate the speed of a softball projectile and its launch angle by measuring only the time and distance
More informationSHOT ON GOAL. Name: Football scoring a goal and trigonometry Ian Edwards Luther College Teachers Teaching with Technology
SHOT ON GOAL Name: Football scoring a goal and trigonometry 2006 Ian Edwards Luther College Teachers Teaching with Technology Shot on Goal Trigonometry page 2 THE TASKS You are an assistant coach with
More informationVectors. Wind is blowing 15 m/s East. What is the magnitude of the wind s velocity? What is the direction?
Physics R Scalar: Vector: Vectors Date: Examples of scalars and vectors: Scalars Vectors Wind is blowing 15 m/s East. What is the magnitude of the wind s velocity? What is the direction? Magnitude: Direction:
More information9.3 Altitude-on-Hypotenuse Theorems
9.3 Altitude-on-Hypotenuse Theorems Objectives: 1. To find the geometric mean of two numbers. 2. To find missing lengths of similar right triangles that result when an altitude is drawn to the hypotenuse
More informationTwo-Dimensional Motion and Vectors
Science Objectives Students will measure and describe one- and two-dimensional position, displacement, speed, velocity, and acceleration over time. Students will graphically calculate the resultant of
More informationLesson 14: Modeling Relationships with a Line
Exploratory Activity: Line of Best Fit Revisited 1. Use the link http://illuminations.nctm.org/activity.aspx?id=4186 to explore how the line of best fit changes depending on your data set. A. Enter any
More informationSection 4.2 Objectives
Section 4. Objectives Determine whether the slope of a graphed line is positive, negative, 0, or undefined. Determine the slope of a line given its graph. Calculate the slope of a line given the ordered
More informationThe Rule of Right-Angles: Exploring the term Angle before Depth
The Rule of Right-Angles: Exploring the term Angle before Depth I m a firm believer in playing an efficient game. Goaltenders can increase their efficiency in almost every category of their game, from
More informationSection 8: Right Triangles
The following Mathematics Florida Standards will be covered in this section: MAFS.912.G-CO.2.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition
More informationAlgebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task 3.1.2
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic. Pythagorean Theorem; Task 3.. TASK 3..: 30-60 RIGHT TRIANGLES Solutions. Shown here is a 30-60 right triangle that has one leg of length and
More informationTHE BEHAVIOR OF GASES
14 THE BEHAVIOR OF GASES SECTION 14.1 PROPERTIES OF GASES (pages 413 417) This section uses kinetic theory to explain the properties of gases. This section also explains how gas pressure is affected by
More informationSpecial Right Triangles
GEOMETRY Special Right Triangles OBJECTIVE #: G.SRT.C.8 OBJECTIVE Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. *(Modeling Standard) BIG IDEA (Why is
More information