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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

5-Minute Check 1

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 triangles

angles must add to 180. C. Correlation a

triangle must have a 90 angle to be a right

triangle. D. Correlation a triangles angles

must add to 180.

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?

5-Minute Check 3

A. accept B. reject

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?

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.

Example 1

Evaluate Trigonometric Functions

Example 1

Evaluate Trigonometric Functions

Example 1

Find the value of the six trigonometric functions

for angle A.

Example 2

Find Trigonometric Ratios

Example 2

Find Trigonometric Ratios

Step 2 Use the Pythagorean Theorem to find c.

a2 b2 c2 Pythagorean Theorem 32

52 c2 Replace a with 3 and b with

5. 34 c2 Simplify.

Take the square root of each side. Length

cannot be negative.

Example 2

Find Trigonometric Ratios

Step 3

Now find csc A.

Cosecant ratio

Example 2

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 12.

10.4 x

Use a calculator.

Answer

Example 3

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

Joels eye level is at 6 feet, what is the

distance from the top of the building to Joels

eye?

Example 4

Find a Missing Side Length

Cosine function

Use a calculator.

Answer The distance from the top of the

building to Joels 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 Anands 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

Example 5

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

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 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.

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

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.

Simplify.

x 112.6

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

End of the Lesson