Trigonometry

CHAPTER 8.4

TrigonometryThe word trigonometry comes from the Greek meaning “triangle measurement”.

Trigonometry uses the fact that the side lengths of similar triangles are always in the same ratio to find unknown sides and angles.

For example, when one of the angles in a right triangle is 30° the side opposite this angle is always half the length of the hypotenuse.

30°

8 cm?4 cm

6 cm12 cm

?30°

The sine ratioThe ratio of

the length of the opposite side

the length of the hypotenuse

is the sine ratio.

In trigonometry we use the Greek letter θ, theta, for the angle. The value of the sine ratio depends on the size of the angles in the triangle.

θ

OPPOSITE

HY

PO

TE

NU

SE

We say:

sin θ =opposite

hypotenuse

The sine ratio using a calculatorWhat is the value of sin 65°?

To find the value of sin 65° using a scientific calculator, start by making sure that your calculator is set to work in degrees.

Key in:

sin 6 5 =

Your calculator should display 0.906307787

This is 0.906 to the nearest thousandth.

The cosine ratioThe ratio of

the length of the adjacent side

the length of the hypotenuseis the cosine ratio.

The value of the cosine ratio depends on the size of the angles in the triangle.

θ

We say,

cos θ =adjacent

hypotenuse

A D J A C E N T

HY

PO

TE

NU

SE

The cosine ratio

It doesn’t matter how big the triangle is because all right triangles with an angle of 53° are similar.

The length of the opposite side divided by the length of the hypotenuse will always be the same value as long as the angle is the same.

In this triangle,

cos 53° =adjacent

hypotenuse

53°

6 cm

10 cm

=6

10

= 0.6

What is the value of cos 53°?

The cosine ratio using a calculatorWhat is the value of cos 30°?

To find the value of cos 30° using a scientific calculator, start by making sure that your calculator is set to work in degrees.

Key in:

cos 3 0 =

Your calculator should display 0.866025403

This is 0.866 to the nearest thousandth.

The tangent ratioThe ratio of the length of the opposite side

the length of the adjacent sideis the tangent ratio.

The value of the tangent ratio depends on the size of the angles in the triangle.

θ

OPPOSITE

We say,

tan θ =opposite

adjacent

A D J A C E N T

The tangent ratioWhat is the value of tan 71°?

It doesn’t matter how big the triangle is because all right triangles with an angle of 71° are similar.

The length of the opposite side divided by the length of the adjacent side will always be the same value as long as the angle is the same.

In this triangle,

tan 71° =opposite

adjacent71°

11.6 cm

4 cm

=11.6

4

= 2.9

Example 1Find Sine, Cosine, and Tangent Ratios

A. Express sin L as a fraction and as a decimal to the nearest hundredth.

Answer:

Example 1Find Sine, Cosine, and Tangent Ratios

B. Express cos L as a fraction and as a decimal to the nearest hundredth.

Answer:

Example 1Find Sine, Cosine, and Tangent Ratios

C. Express tan L as a fraction and as a decimal to the nearest hundredth.

Answer:

Example 1Find Sine, Cosine, and Tangent Ratios

D. Express sin N as a fraction and as a decimal to the nearest hundredth.

Answer:

Example 1Find Sine, Cosine, and Tangent Ratios

E. Express cos N as a fraction and as a decimal to the nearest hundredth.

Answer:

Example 1Find Sine, Cosine, and Tangent Ratios

F. Express tan N as a fraction and as a decimal to the nearest hundredth.

Answer:

Example 1A. Find sin A.

A.

B.

C.

D.

Example 1B. Find cos A.

A.

B.

C.

D.

Example 1C. Find tan A.

A.

B.

C.

D.

Example 1D. Find sin B.

A.

B.

C.

D.

Example 1E. Find cos B.

A.

B.

C.

D.

Example 1F. Find tan B.

A.

B.

C.

D.

Example 3 Estimate Measures Using Trigonometry

EXERCISING A fitness trainer sets the incline on a treadmill to 7°. The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor?

Example 3

A. 1 in.

B. 11 in.

C. 16 in.

D. 15 in.

CONSTRUCTION The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about 4.8°, about how high does the ramp rise off the ground to the nearest inch?

Try it! Find each length using trigonometry.

•Pg 570 #3A and 3B

Example 4Find Angle Measures Using Inverse Trigonometric Ratios

Use a calculator to find the measure of P to the nearest tenth.

Example 4

A. 44.1°

B. 48.3°

C. 55.4°

D. 57.2°

Use a calculator to find the measure of D to the nearest tenth.

Try it! Find each angle measure.

•Pg. 571 #4A and 4B

Example 5Solve a Right Triangle

Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree.

Example 5

A. mA = 36°, mB = 54°,

AB = 13.6

B. mA = 54°, mB = 36°,

AB = 13.6

C. mA = 36°, mB = 54°,

AB = 16.3

D. mA = 54°, mB = 36°,

AB = 16.3

Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

Example 1Angle of Elevation

CIRCUS ACTS At the circus, a person in the audience at ground level watches the high-wire routine. A 5-foot-6-inch tall acrobat is standing on a platform that is 25 feet off the ground. How far is the audience member from the base of the platform, if the angle of elevation from the audience member’s line of sight to the top of the acrobat is 27°?Make a drawing.

Example 2Angle of Depression

DISTANCE Maria is at the top of a cliff and sees a seal in the water. If the cliff is 40 feet above the water and the angle of depression is 52°, what is the horizontal distance from the seal to the cliff, to the nearest foot?Make a sketch of the situation.

Since are parallel, mBAC = mACD by the Alternate Interior Angles Theorem.

Example 1

A. 37°

B. 35°

C. 40°

D. 50°

DIVING At a diving competition, a 6-foot-tall diver stands atop the 32-foot platform. The front edge of the platform projects 5 feet beyond the ends of the pool. The pool itself is 50 feet in length. A camera is set up at the opposite end of the pool even with the pool’s edge. If the camera is angled so that its line of sight extends to the top of the diver’s head, what is the camera’s angle of elevation to the nearest degree?

Example 2

A. 19 ft

B. 20 ft

C. 44 ft

D. 58 ft

Luisa is in a hot air balloon 30 feet above the ground. She sees the landing spot at an angle of depression of 34. What is the horizontal distance between the hot air balloon and the landing spot to the nearest foot?