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 feet Or maybe more relevant... I wonder how tall this tree is! 100 m 55 o
If we have a right triangle with acute angle A: Opposite means: Across from Adjacent means: Next to A hypotenuse will never be a leg!
Opposite A Hypotenuse Adjacent to A Hypotenuse Opposite A Adjacent to A Opposite A Adjacent to A Hypotenuse
Hypotenuse Adjacent to A Opposite A Adjacent to A Opposite A Hypotenuse
Opposite A Adjacent to A Hypotenuse tangent - The ratio between the length of opposite side to that of the adjacent side.
Ex. 1 What is the tangent of angle A? Consider this triangle with angle A approximately 31 o. B 3 A 31 o 5 C What is the tangent of angle A?
There is something really cool about right triangles... If the angle is about 31 o, then the tangent (ratio between opposite and adjacent legs) will in fact ALWAYS be about 3 to 5. 31 o 3 5 31 o 10 6 31 o 9 15 sine the ratio between the length of opposite side to that of the hypotenuse
Ex. 2 What is the sine of angle A? cosine The ratio between the length of adjacent side to that of the hypotenuse.
Ex. 3 What is the cosine of angle A? To remember which Trig Ratio is which...
1. What is the cosine of angle E? 2. What is the sine of angle E?
3. What is the tangent of angle E? 4. What is the tangent of B?
5. What is the tangent of A? Find the trigonometric value, rounding to four decimal places. sin 52 0 When using trigonometry, it is VERY IMPORTANT that your calculator be in DEGREE MODE! Go to MODE, then down to where it says radian/degree and highlight degree.
Find the trigonometric value, rounding to four decimal places. cos 78 0 Find the trigonometric value, rounding to four decimal places. tan 72 0
Last chapter, we learned that if we knew the ratio, we could set up a proportion to figure out a missing side in a triangle. 4 x 3 5 This chapter, we're not given similar triangles, but we can find out what the ratio is supposed to be by using our knowledge about trigonometry! 28 o x 15 Look at what sides you have and determine if you need sin, cos, or tan. Then set up the proportion and solve.
Ex. 1 Find the missing side length. x 40 o 7 Ex. 2 Find the missing side length. x 40 o 7
Ex. 3 Find the missing side length. 40 o x 7 Ex. 4 Find the missing side length. 50 o x 7
Ex. 5 Find the missing side length. a 15 40 o Ex. 6 Find the missing side length. 6 30 o x
Ex. 7 Find the missing side length. 25 b 75 o Inverse trig functions: Sin 1, Cos 1, Tan 1 Inverses are used to find the degree measure of an angle. (Sin, Cos and Tan find the side lengths.) An inverse function will negate ("undo") a trig function.
Ex. 1 Find θ sin θ =.5567 Round to one decimal place. Ex.2 Find θ tan θ = 1.5355 Round to one decimal place. Goal: Find the measure of angle A. B C What is the tangent of angle A? Then use tan 1 to find the measure of the angle.
Ex. 4 What is the measure of angle B? B 13 5 12 Ex. 5 What is the measure of angle A? 17 15 A 8
Ex. 6 What is the measure of B angle B? 41 9 Ex. 7 What is the measure of angle C? 15 C 17
Ex. 8 Draw a triangle that would match the trig sentence. Then, solve for θ cos θ = 5 12 Ex. 9 Draw a triangle that fits the trig sentence below. Then, find the value of x. Round to one decimal tan 35 ο = x 14
Ex. 10 Draw a triangle that fits the trig sentence below. Then, find the value of θ. sin θ ο = 23 43 Ex. 11 Draw a triangle that fits the trig sentence below. Then, find the value of θ sin θ ο = 17 13
Ex. 12 Draw a triangle that fits the trig sentence below. Then, find the value of 0. tan θ ο = 17 14 13.2 Word Problems with Right Triangle Trigonometry
Our situation: A girl standing at the bottom of a hill is waving up to her friend who is roasting marshmallows at a campfire on the top of the hill. If the angle of elevation is 27 o and the girl is 200 feet from the bottom of the hill, approximately how tall is the hill? Angle of Depression Horizontal Angle of Elevation Horizontal Hill Formal Definitions: Angle of Elevation: If you are looking up, it is the angle from the horizontal UP to the line of sight. Angle of Depression: If you are looking down, it is the angle from the horizontal DOWN to the line of sight.
What conjecture can you make about the angles of elevation and depression? Think about parallel lines... Angle of Depression Angle of Elevation The angle of elevation from a sailboat to the top of a 121 foot lighthouse on the shore is 16 degrees. To the nearest foot, how far is the sailboat from the shore? Step 1: Draw a picture Step 2: Decide sin, cos or tan Step 3: Solve
Ex. 2) A helicopter is hovering over a landing pad 100 meters from where you are standing. The helicopter s angle of elevation with the ground is 12 o. What is the altitude of the helicopter? Ex. 3) Ray, a lighthouse operator sits in the top of his lighthouse 25 meters above sea level. He sights a sailboat in the distance. The angle of depression of the sighting is 10 o. How far is the boat from the base of the lighthouse? Give your answer to the nearest 10 meters.
Ex. 4) You are flying a kite and let out 240 feet of string. The kite s angle of elevation with the ground is 40 0. If the string is stretched straight, how high is the kite above the ground? Homework: 13.1 Page 647#1-5, 7-17 odd, 18-26 13.2 Page 650 1, 2, 3-17 odd, 18-23