Five-Minute Check (over Chapter 11)CCSSThen/NowNew VocabularyKey Concept: Trigonometric Functions in Right TrianglesExample 1: Evaluate Trigonometric FunctionsExample 2: Find Trigonometric RatiosKey Concept: Trigonometric Values for Special AnglesExample 3: Find a Missing Side LengthExample 4: Find a Missing Side LengthKey Concept: Inverse Trigonometric RatiosExample 5: Find a Missing Angle MeasureExample 6: Use Angles of Elevation and Depression

Over Chapter 11

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

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

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

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

A. accept

B. reject

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

_

Over Chapter 11

A. accept

B. reject

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

_

Over Chapter 11

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.

Over Chapter 11

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.

Mathematical Practices6 Attend to precision.

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.

• 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

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.

Evaluate Trigonometric Functions

Evaluate Trigonometric Functions

Answer:

Evaluate Trigonometric Functions

Answer:

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

A. B.

C. D.

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

A. B.

C. D.

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.

Find Trigonometric Ratios

Step 2Use the Pythagorean Theorem to find c.

a2 + b2 = c2 Pythagorean Theorem32 + 52 = c2 Replace a with 3 and

b with 5.34 = c2 Simplify.

Take the square root of each side. Length cannot be

negative.

Find Trigonometric Ratios

Now find csc A.

Cosecant ratio

Replace hyp withand opp with 5.

Step 3

Answer:

Find Trigonometric Ratios

Now find csc A.

Cosecant ratio

Replace hyp withand opp with 5.

Step 3

Answer:

A.

B.

C.

D.

A.

B.

C.

D.

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.

Find a Missing Side Length

Sine ratio

Replace with 60°, opp with x, and hyp with 12.

Multiply each side by 12.

10.4 ≈ x

Answer:

Use a calculator.

Find a Missing Side Length

Sine ratio

Replace with 60°, opp with x, and hyp with 12.

Multiply each side by 12.

10.4 ≈ x

Answer:

Use a calculator.

x =

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.

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.

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?

Find a Missing Side Length

Cosine function

Use a calculator.

Answer:

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.

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

Solve for d.

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?

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?

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

Find a Missing Angle Measure

Answer:

Use a calculator.

Find a Missing Angle Measure

Answer: Therefore, mA ≈ 32°.

Use a calculator.

Find a Missing Angle Measure

Use the cosine function.

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

Answer:

Use a calculator.

Inverse cosine

Find a Missing Angle Measure

Use the cosine function.

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

Answer: Therefore, mB ≈ 58º.

Use a calculator.

Inverse cosine

A. mA = 72º

B. mA = 80º

C. mA = 30º

D. mA = 55º

A. Find the measure of A.

A. mA = 72º

B. mA = 80º

C. mA = 30º

D. mA = 55º

A. Find the measure of A.

A. mB = 18º

B. mB = 10º

C. mB = 60º

D. mB = 35º

B. Find the measure of B.

A. mB = 18º

B. mB = 10º

C. mB = 60º

D. mB = 35º

B. Find the measure of B.

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.

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:

x ≈ 169.4

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.Answer: So, the horizontal distance from the tee to the

green as seen from a camera in a blimp is about 169.4 yards.

x ≈ 169.4

tan

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.

Use Angles of Elevation and Depression

Multiply each side by x.

Divide each side by tan 60°.

Simplify.Answer:

tan

x ≈ 112.6

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).

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

112.6 feet.

tan

x ≈ 112.6

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).

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?

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?

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?

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?