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 received from the tribal elder consisted of many different right triangles. To decipher the map, they need to understand how to view a right triangle. A right triangle has two legs and a hypotenuse. However, if you look at the triangle from a specific angle, you can classify the legs according to their position. Take a look at ΔDEF from D. Side DF is the hypotenuse because it is from the right angle. Side EF is the leg because it is directly from D. Side DE is the leg because it is D.
But be careful! If you focus on a different angle, the names of the legs will change. Look at ΔDEF from the perspective of F now instead of D. Notice that the opposite and adjacent legs have changed because you are looking at the triangle from a different. Side DE is now the leg because it is directly F. Side EF is now the leg because it is F. Side DF is still the because the location of the right angle has not changed. You Try! Click on the box Learn [page 3] Sine, Cosine, and Tangent The trigonometric functions (also called circular functions) are functions of an angle. They are used to relate the angles of a right triangle to the lengths of the sides of a right triangle. Trigonometric functions are important in the study of right triangles. Here are the three basic trigonometric functions shown as ratios:
Learn [page 4] Similar Triangle and Trigonometric Functions There is a relationship that connects similar triangles together with the trigonometric ratios. Take a look at a video that explores this special relationship. Let's explore two nested similar triangles to see the relationship between the sides and their ratios. In the image to the right, two nested right triangles are shown with angle Θ marked by point C. The symbol Θ is the greek symbol for theta and is pronounced THAY-tuh. Small Triangle: vertical hypotenuse = 3 5 Large Triangle: vertical hypotenuse = 6 10 = 3 5 Now find the sin Θ for each right triangle. Notice the relationship between the first ratios you discovered and these two ratios. Small Triangle: sin θ = opp hyp = Large Triangle: sin θ = opp = = hyp
Notice how the sin Θ for the two similar triangles is the same. What do you think will happen if we find the cosine and tangent of Θ? Let's find out! Small Triangle: cos θ = adj hyp = tan θ = opp adj = Large Triangle: cos θ = adj = = hyp tan θ = opp adj = = The trigonometric ratios for triangles are the! Learn [page 5] Trigonometric Functions The chanting Carla heard in the woods (SOH-CAH-TOA) was referring to the trigonometric functions. There are many ways to remember the trigonometric functions and one way is a mnemonic device. A mnemonic device is a learning technique that aids information retention. If sine is equal to opposite over hypotenuse, we can use the first letters in sine (S), opposite (O), and hypotenuse (H) to start a mnemonic. You can assign words to the letters as well. While SOH-CAH-TOA is one way to remember, here's another: Some Old Horse Came A Hopping Through Our Apartment In the comic strip slides earlier in this lesson, Carla figured out what the (SOH-CAH-TOA) meant. Did you? Let's take a look at right triangle DEF from the perspective of F.
Sine The sine of F is represented by the length of the side over the length of the. sin F = opposite hypotenuse You can remember this by the acronym SOH (pronounced sō), which stands for - - Because DE is the opposite leg to F with a length of, and DF is the hypotenuse with a length of, the sine of F is equal to Cosine The cosine of F is represented by the length of the adjacent side over the length of the hypotenuse cos F = adjacent hypotenuse You can remember this by the acronym CAH (pronounced kă), which stands for - - Because EF is the adjacent leg to F with a length of, and DF is the hypotenuse with a length of, the cosine of F is equal to
Tangent The tangent of F is represented by the length of the opposite side over the length of the adjacent side. tan F = opposite adjacent You can remember this by the acronym TOA (pronounced tō-ah), which stands for - - Because DE is the opposite leg to F with a length of, and EF is the adjacent leg to F with a length of, the tangent of F is equal to You Try! Now see if you can identify the sine, cosine, and tangent of a given angle in a right triangle. For each problem, identify the sine, cosine, and tangent of P. Practice 1 sin < P = opposite hypotenuse = cos < P = adjacent hypotenuse = tan < P = opposite adjacent =
Practice Two sin < P = opposite hypotenuse = cos < P = adjacent hypotenuse = tan < P = opposite adjacent = Practice Three sin < P = opposite hypotenuse = cos < P = adjacent hypotenuse = tan < P = opposite adjacent =
Learn [page 6] Tangent and Slope There is something special about using the tangent ratio. Let's take another look at a right triangle graphed on the coordinate plane. The tangent of A is equivalent to the length of the opposite side over the length of the adjacent side. So, tan A = Let's recall how to find the slope of a line. If you have two points on a line, you can subtract the y values, subtract the x values, and then divide the differences to find the slope. In other words, you could use the formula m = y 2 y 1 x 2 x 1 Using this formula, you can find that the slope of AB is found by 6 0 7 0 = Because the rise of the line is the same as the opposite side length, and the run of the line is the same as the adjacent side length, the tangent of a given angle is equal to the slope of the hypotenuse! tan A = opposite rise = slope AB = = adjacent run Brian and Carla were so thankful to find the Trig Tribe! All that hard work trig-gered a mighty appetite in both of them, so they were very happy to get back home!
Learn [page 7] Complementary Angles There are other special relationships to explore with sine and cosine involving the complement of angles. Two angles are if their are. For example, a 45 angle would be complementary to another 45 angle. Also, a 50 angle would be complementary to another angle that measures. The angles do not need to be to each other. One angle can be on your paper and the other on the computer screen. As long as they add up to 90, they are. The following video will help you explore this special relationship.
Learn [page 8] Mixed Review You've learned a lot of information in this lesson! Take some time to practice some additional problems. Practice 1 Review the right triangle JKL. What trigonometric function does c d equal? There are two possible trigonometric functions c d could equal. Look at the triangle from KLJ. Side c is to KLJ, and side d is the. Because cos a = adjacent, cos a = c d. hypotenuse However, if you look at the triangle from LJK, side c represents the side instead of the adjacent side. Side d is still the. sin b = opposite hypotenuse = c d
Practice Two In right triangle JKL, if cos b = x y and tan b = 12, what is the value of sin b? x The best way to approach this problem is to draw a diagram and label it. Since cos b = x adjacent and cos b =, KL = and LJ = y hypotenuse Since tan b = 12 x opposite, and tan b =, it follows that JK = and KL = adjacent Once all sides have been labeled, determine the sine of the given angle. sin b = opposite hypotenuse =
Practice Three Use the image of the side view of a doorstop to answer the question that follows. If the height of the doorstop is 5 inches and the angle formed by its height and hypotenuse is 55, what is a trigonometric expression that represents the doorstop's length? The doorstop's length is the 55 angle, while its height of 5 inches is to the 55 angle. Therefore, use the ratio to find the expression. tan 55 = tan 55 = Multiply both sides of the equation by 5. = length The expression represents the length of the doorstop. Practice Four Complete the following statement: If the sin 30 = 1 2, then cos = 1 2 Remember that complementary angles have a special relationship with the trigonometric functions. The complement of 30 is the measure of the angle that, when added to the original angle, gives you a total of 90. This means the complement of 30 is 60. Complementary angles have congruent and values. If the sin 30 = 1 2, then cos = 1 2
Practice Five Use your calculator to find the cos 90. Now find the angle measure for sin in degrees that has the same value. The cos 90 = 0 Again, complementary angles have a special relationship with the trigonometric functions. The of 90 is the measure of the angle that, when added to the original angle, gives you a total of. This means the complement of 90 is 0. Complementary angles have congruent cosine and sine values: if the cos 90 = 0, then sin = 0. Click on the box GeOverview [page 9] Take a minute to refresh your memory a bit on what you just learned. The term trigonometry comes from a Greek word meaning. The of an angle within a right triangle is found by dividing the length of the side by the length of the. The of an angle within a right triangle is found by dividing the length of the side by the length of the. The of an angle within a right triangle is found by dividing the length of the side by the length of the side.
sin D = = cos D = = tan D = = Complementary angles have special relationships with the sine and cosine trigonometric functions: The of an angle has the same value as the of the complementary angle. The of an angle has the same value as the of the complementary angle. In the image to the right, two nested right triangles are shown with angle Θ marked by point C. Similar right triangles and their side ratios lead to the properties and definitions of the trigonometric ratios. Small Triangle: vertical hypotenuse = Large Triangle: vertical hypotenuse = =
Notice the relationship between the first ratios and sin Θ ratio. Small Triangle: sin θ = opp hyp = Large Triangle: sin θ = opp = = hyp Name period date HW: Geo 7.01 Algebra/Geometry Blend #1 Find the following trig values based on ABC. sin A = sin C = cos A = cos C = tan A = tan C = #2 Find the following trig values based on DEF. sin E = sin F = cos E = cos F = tan E = tan F =
#3 Find the following trig values based on XYZ. sin Y = sin Z = cos Y = cos Z = tan Y = tan Z = Fill in the blank to make each statement true. #4 If the sin 30 o = ½, then cos = ½ #5 If the cos 45 o = 2 2, then sin = 2 2 #6 If the cos 30 o = 3 2, then sin 60o = #7 If the cos 90 o = 0, then sin = 0 #8 If the sin 90 o = 1, then the cos 0 o =