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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