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Lesson Menu Five-Minute Check (over Chapter 11) CCSS Then/Now New Vocabulary Key Concept: Trigonometric Functions in Right Triangles Example 1: Evaluate Trigonometric Functions Example 2: Find Trigonometric Ratios Key Concept: Trigonometric Values for Special Angles Example 3: Find a Missing Side Length Example 4: Find a Missing Side Length Key Concept: Inverse Trigonometric Ratios Example 5: Find a Missing Angle Measure Example 6: Use Angles of Elevation and Depression

Over Chapter 11 5-Minute Check 1 A.Causation; a triangle must have a 90° angle to be a right triangle. B.Causation; a triangle’s angles must add to 180°. C.Correlation; a triangle must have a 90° angle to be a right triangle. D.Correlation; a triangle’s angles must add to 180°. When a triangle is a right triangle, one of its angles measures 90°. Does this show correlation or causation? Explain.

Over Chapter 11 5-Minute Check 2 From a box containing 8 blue pencils and 6 red pencils, 4 pencils are drawn and not replaced. What is the probability that all four pencils are the same color? A. B. C. D.

Over Chapter 11 5-Minute Check 3 A.accept B.reject Test the null hypothesis for H 0 = 82, h 1 > 82, n = 150, x = 83.1, and  = 2.1. _

Over Chapter 11 5-Minute Check 4 Jenny makes 60% of her foul shots. If she takes 5 shots in a game, what is the probability that she will make fewer than 4 foul shots? A. B. C. D.

CCSS Mathematical Practices 6 Attend to precision.

Then/Now You used the Pythagorean Theorem to find side lengths of right triangles. Find values of trigonometric functions for acute angles. Use trigonometric functions to find side lengths and angle measures of right triangles.

Vocabulary trigonometry trigonometric ratio trigonometric function sine cosine tangent cosecant secant cotangent reciprocal functions inverse sine inverse cosine inverse tangent angle of elevation angle of depression

Concept

Example 1 Evaluate Trigonometric Functions Find the values of the six trigonometric functions for angle G. Use opp = 24, adj = 32, and hyp = 40 to write each trigonometric ratio. For this triangle, the leg opposite  G is HF and the leg adjacent to  G is GH. The hypotenuse is GF.

Example 1 Evaluate Trigonometric Functions

Example 1 Evaluate Trigonometric Functions Answer:

Example 1 Find the value of the six trigonometric functions for angle A. A.B. C.D.

Example 2 Find Trigonometric Ratios In a right triangle,  A is acute and. Find the value of csc A. Step 1 Draw a right triangle and label one acute angle A. Since and, label the opposite leg 5 and the adjacent leg 3.

Example 2 Find Trigonometric Ratios Step 2 Use the Pythagorean Theorem to find c. a 2 + b 2 =c 2 Pythagorean Theorem =c 2 Replace a with 3 and b with 5. 34=c 2 Simplify. Take the square root of each side. Length cannot be negative.

Example 2 Find Trigonometric Ratios Now find csc A. Cosecant ratio Replace hyp with and opp with 5. Step 3 Answer:

Example 2 A. B. C. D.

Concept

Example 3 Find a Missing Side Length Use a trigonometric function to find the value of x. Round to the nearest tenth if necessary. The measure of the hypotenuse is 12. The side with the missing length is opposite the angle measuring 60 . The trigonometric function relating the opposite side of a right triangle and the hypotenuse is the sine function.

Example 3 Find a Missing Side Length Sine ratio Replace  with 60°, opp with x, and hyp with 12. Multiply each side by ≈ x Answer: Use a calculator. x =

Example 3 A. B. C. D. Write an equation involving sin, cos, or tan that can be used to find the value of x. Then solve the equation. Round to the nearest tenth.

Example 4 Find a Missing Side Length BUILDINGS To calculate the height of a building, Joel walked 200 feet from the base of the building and used an inclinometer to measure the angle from his eye to the top of the building. If Joel’s eye level is at 6 feet, what is the distance from the top of the building to Joel’s eye?

Example 4 Find a Missing Side Length Cosine function Use a calculator. Answer: The distance from the top of the building to Joel’s eye is about 827 feet. Replace  with 76°, adj with 200, and hyp with d. Solve for d.

Example 4 A.43 ft B.81 ft C.87 ft D.100 ft TREES To calculate the height of a tree in his front yard, Anand walked 50 feet from the base of the tree and used an inclinometer to measure the angle from his eye to the top of the tree, which was 62°. If Anand’s eye level is at 6 feet, about how tall is the tree?

Concept

Example 5 Find a Missing Angle Measure You know the measures of the sides. You need to find m  A. A. Find the measure of  A. Round to the nearest tenth if necessary. Inverse sine

Example 5 Find a Missing Angle Measure Answer: Therefore, m  A ≈ 32°. Use a calculator.

Example 5 Find a Missing Angle Measure Use the cosine function. B. Find the measure of  B. Round to the nearest tenth if necessary. Answer: Therefore, m  B ≈ 58º. Use a calculator. Inverse cosine

Example 5 A.m  A = 72º B.m  A = 80º C.m  A = 30º D.m  A = 55º A. Find the measure of  A.

Example 5 A.m  B = 18º B.m  B = 10º C.m  B = 60º D.m  B = 35º B. Find the measure of  B.

Example 6 Use Angles of Elevation and Depression A. GOLF A golfer is standing at the tee, looking up to the green on a hill. The tee is 36 yards lower than the green and the angle of elevation from the tee to the hole is 12°. From a camera in a blimp, the apparent distance between the golfer and the hole is the horizontal distance. Find the horizontal distance.

Example 6 Use Angles of Elevation and Depression Write an equation using a trigonometric function that involves the ratio of the vertical rise (side opposite the 12° angle) and the horizontal distance from the tee to the hole (adjacent). Multiply each side by x. Divide each side by tan 12°. Simplify. Answer: So, the horizontal distance from the tee to the green as seen from a camera in a blimp is about yards. x ≈ tan 

Example 6 Use Angles of Elevation and Depression B. ROLLER COASTER The hill of the roller coaster has an angle of descent, or an angle of depression, of 60°. Its vertical drop is 195 feet. From a blimp, the apparent distance traveled by the roller coaster is the horizontal distance from the top of the hill to the bottom. Find the horizontal distance.

Example 6 Use Angles of Elevation and Depression Multiply each side by x. Divide each side by tan 60°. Simplify. Answer: So, the horizontal distance of the hill is about feet. tan  x ≈ Write an equation using a trigonometric function that involves the ratio of the vertical drop (side opposite the 60° angle) and the horizontal distance traveled (adjacent).

Example 6 A.295 ft B.302 ft C.309 ft D.320 ft A. BASEBALL Mario hits a line drive home run from 3 feet in the air to a height of 125 feet, where it strikes a billboard in the outfield. If the angle of elevation of the hit was 22°, what is the horizontal distance from home plate to the billboard?

Example 6 A.34 ft B.49 ft C.73 ft D.85 ft B. KITES Angelina is flying a kite in the wind with a string with a length of 60 feet. If the angle of elevation of the kite string is 55°, then how high is the kite in the air?

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