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Splash Screen. 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

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### Splash Screen

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

When a triangle is a right triangle, one of its angles measures 90°. Does this show correlation or causation? Explain.

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°.

### 5-Minute Check 1

A.

B.

C.

D.

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?

### 5-Minute Check 2

Test the null hypothesis for H0 = 82, h1 > 82, n = 150, x = 83.1, and  = 2.1.

_

A.accept

B.reject

### 5-Minute Check 3

A.

B.

C.

D.

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?

### 5-Minute Check 4

Mathematical Practices

6 Attend to precision.

### CCSS

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.

### Then/Now

• 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

For this triangle, the leg opposite G is HF and the leg adjacent to G is GH. The hypotenuse is GF.

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.

### Example 1

Evaluate Trigonometric Functions

### Example 1

Evaluate Trigonometric Functions

### Example 1

A.B.

C.D.

Find the value of the six trigonometric functions for angle A.

### Example 1

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.

Find Trigonometric Ratios

### Example 2

Find Trigonometric Ratios

Step 2

Use the Pythagorean Theorem to find c.

a2 + b2=c2Pythagorean Theorem

32 + 52=c2Replace a with 3 and b with 5.

34=c2Simplify.

Take the square root of each side. Length cannot be negative.

### Example 2

Replace hyp withand opp with 5.

Find Trigonometric Ratios

Step 3

Now find csc A.

Cosecant ratio

A.

B.

C.

D.

### Concept

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

x =

Find a Missing Side Length

Sine ratio

Replace  with 60°, oppwith x, and hyp with 12.

Multiply each side by 12.

10.4 ≈ x

Use a calculator.

### 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 3

Find a Missing Side Length

BUILDINGSTo 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

Replace  with 76°, adj with 200, and hyp with d.

Solve for d.

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.

### Example 4

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?

A.43 ft

B.81 ft

C.87 ft

D.100 ft

### Concept

Find a Missing Angle Measure

A. Find the measure of A. Round to the nearest tenth if necessary.

You know the measures of the sides. You need to find mA.

Inverse sine

### Example 5

Find a Missing Angle Measure

Use a calculator.

Answer: Therefore, mA≈ 32°.

### Example 5

Inverse cosine

Use a calculator.

Find a Missing Angle Measure

B. Find the measure of B. Round to the nearest tenth if necessary.

Use the cosine function.

Answer: Therefore, mB ≈ 58º.

### Example 5

A. Find the measure of A.

A.mA = 72º

B.mA = 80º

C.mA = 30º

D.mA = 55º

### Example 5

B. Find the measure of B.

A.mB = 18º

B.mB = 10º

C.mB = 60º

D.mB = 35º

### Example 5

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

tan 

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.

x≈ 169.4

Answer: So, the horizontal distance from the tee to the green as seen from a camera in a blimp is about 169.4 yards.

### 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

tan 

Use Angles of Elevation and Depression

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).

Multiply each side by x.

Divide each side by tan 60°.

x≈ 112.6

Simplify.

Answer: So, the horizontal distance of the hill is about 112.6 feet.

### Example 6

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?

A.295 ft

B.302 ft

C.309 ft

D.320 ft

### Example 6

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?

A.34 ft

B.49 ft

C.73 ft

D.85 ft