SECTION 8.4

TRIGONOMETRY

The word trigonometry comes from two greek terms, trigon, meaning triangle, and metron, meaning measure. a trigonometric ratio is a ratio of the lengths of two sides of a right triangle.

By AA Similarity, a right triangle with a given acute angle is similar to every other right triangle with the same acute angle measure. So trigonometric ratios are constant for a given angle measure.

Example 1:

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

opposite legsin

hypotenuse

12or 0.32

37

L

MN

LN

Example 1:

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

adjacent legcos

hypotenuse

35or 0.95

37

L

LM

LN

Example 1:

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

opposite legtan

adjacent leg

12or 0.34

35

L

MN

LM

Example 1:

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

opposite legsin

hypotenuse

35or 0.95

37

N

LM

LN

Example 1:

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

adjacent legcos

hypotenuse

12or 0.32

37

N

MN

LN

Example 1:

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

opposite legtan

adjacent leg

35or 2.92

12

N

LM

MN

Example 2:

a) Use a special right triangle to express the cosine of 60° as a fraction and as a decimal to the nearest hundredth.

Special right triangles can be used to find the sine, cosine, and tangent of 30°, 45° and 60° angles.

adjacentcos60 Definition of cosine ratio

hypotenuse

Substitution21

or 0.50 Simplify2

x

x

Example 2:

b) Use a special right triangle to express the tangent of 60° as a fraction and as a decimal to the nearest hundredth.

oppositetan 60 Definition of tangent ratio

adjacent

3Substitution

3or 1.73 Simplify

1

x

x

Example 3: 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?

Let y be the height of the treadmill from the floor in inches. The length of the treadmill is 5 feet, or 60 inches.

leg oppositesin 7 sin

60 hypotenuse

60sin 7 Multiply each side by 60

Use a calculator to find .

KEYSTROKES: 60 SIN 7 ENTER 7.312160604

The treadmill is about 7.3 inches high.

y

y

y

Example 4: 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?

Let y be the height of the ramp from the floor in feet. The length of the ramp is 15 feet.

leg oppositetan 4.8 tan

15 leg adjacent

15tan 4.8 Multiply each side by 15

Use a calculator to find .

KEYSTROKES: 15 TAN 4.8 ENTER

The ramp is about 15 feet high.

y

y

y

y

If you know the sine, cosine, or tangent of an acute angle, you can use a calculator to find the measure of the angle, which is the inverse of the trigonometric ratio.

Example 5:

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

The measures given are those of the leg adjacent to P and the hypotenuse, so write the equation using the cosine ratio.

1

13 adjcos cos

19 hyp

13 13If cos = , then cos . Use a calculator.

19 19

KEYSTROKES:2ND [COS](13 19) ENTER 46.82644889

So, the measure of is approximately 46.8 .

P P

P m P

P

Example 5:

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

The measures given are those of the leg opposite to D and the hypotenuse, so write the equation using the sine ratio.

1

16 oppsin sin

23 hyp

16 16If sin = , then sin . Use a calculator.

23 23

KEYSTROKES:2ND [SIN](16 23) ENTER 44.07920985

So, the measure of is approximately 44.1 .

D P

D m D

D

Example 6: Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree.

a)

1

Find by using a tangent ratio.

4 opptan tan

7 adj

4tan Definition of inverse tangent

729.7448813 Use a calculator

So, the measure of is about 30 .

m A

A A

m A

m A

A

Find mB using complementary angles.

mB ≈ 60° Subtract 30 from each side.

So, the measure of B is about 60 .

30° + mB ≈ 90° mA ≈ 30

mA + mB = 90° Definition of complementary angles

Find AB by using the Pythagorean Theorem.

(AC)2 + (BC)2 = (AB)2 Pythagorean Theorem

72 + 42 = (AB)2 Substitution

65 = (AB)2 Simplify.

Take the positive square root of each side.

8.06 ≈ AB Use a calculator.

65 AB

Example 6: Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree.

b)

1

Find by using a tangent ratio.

11 opptan tan

8 adj

11tan Definition of inverse tangent

853.97262661 Use a calculator

So, the measure of is about 54 .

m A

A A

m A

m A

A

Find mB using complementary angles.

mB ≈ 36° Subtract 54 from each side.

So, the measure of B is about 36 .

54° + mB ≈ 90° mA ≈ 54

mA + mB = 90° Definition of complementary angles

Find AB by using the Pythagorean Theorem.

(AC)2 + (BC)2 = (AB)2 Pythagorean Theorem

82 + 112 = (AB)2 Substitution

185 = (AB)2 Simplify.

Take the positive square root of each side.

13.6 ≈ AB Use a calculator.

185 AB