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Lesson 10-8 Trigonometric Ratios
Trigonometry – the study of relationships among angles and sides of a triangle
Trigonometry Ratio – is a ratio that compares the side lengths of two sides of a triangle
Sine Ratio – the ratio of the leg opposite to the hypotenuse in relationship to an acute angle in a
right triangle (SOH)
Cosine Ratio - the ratio of the leg adjacent to the hypotenuse in relationship to an acute angle in
a right triangle (CAH)
Tangent Ratio - the ratio of the leg opposite to the leg adjacent in relationship to an acute angle
in a right triangle (TOA)
Example 1 Find Sine, Cosine, and Tangent Ratios
Find the values of the three trigonometric ratios for angle A.
Step 1 Use the Pythagorean Theorem to find BC.
a2 + b2 = c2
a2 + 242 = 262
a + 576 = 676
a2 = 100
a = 10
Step 2
Pythagorean Theorem
b = 24 and c = 26
Simplify.
Subtract 576 from each side.
Take the square root of each side.
Use the side lengths to write the trigonometric ratios.
opp 10 5
adj 24 12
sin A =
cos A =




hyp 26 13
hyp 26 13
B
26
A
tan A =
a
24
C
opp 10 5


adj 24 12
Example 2 Use a Calculator to Evaluate Expressions
Use a calculator to find sin 75 to the nearest ten-thousandth.
KEYSTROKES:
SIN 75
)
ENTER 0.9659258263
Round to the nearest ten-thousandth, sin 75°  0.9659.
Example 3 Solve a Triangle
Solve the right triangle. Round each side length to the nearest tenth.
Step 1 Find the measure of  N. The sum of the
measures of the angles in a triangle is 180.
180° – (90° + 35°) = 55°
The measure of  N is 55°.
21
M
N
35
L
Step 2
Find the measure of LM . Since you are given the measure of the side opposite L and
are finding the measure of the side adjacent to L, use the tangent ratio.
21
tan 35° = n
Definition of tangent
n tan 35° = 21
Multiply each side by n.
21
n=
Divide each side by tan 35.
tan 35
n  30.0
Use a calculator.
So, the measure of LM is about 30.0.
Step 3
Find the measure of LN . Since you are given the measure of the side opposite L and
are finding the measure of the hypotenuse, use the sine ratio.
21
sin 35° = m
Definition of sine
m sin 35° = 21
Multiply each side by m.
21
m=
Divide each side by sin 35.
sin 35
m  36.6
Use a calculator.
So, the measure of LN is about 36.6.
Real-World Example 4 Find a Missing Side Length
DECORATING A plant shelf 8 inches wide makes a 56 angle
with the brace b. Approximately how many inches long is the
brace?
8 in.
You need to find the hypotenuse. You are given the side adjacent to the
56° angle. Use the cosine ratio.
8
cos 56° = b
Definition of cosine
b  cos 56° = 8
Multiply each side by b.
8
b=
Divide each side by cos 56.
cos 56
b  14.3
Use a calculator.
56
b
So, the brace is about 14.3 inches long.
Inverse Trigonometric Functions – the inverse of the trigonmetric ratios to find the acute angle in a
right triangle
Inverse Sine - if sin A=x then A=sin-1x
Inverse Cosine - if cos A=x then A=cos-1x
Inverse Sine - if tan A=x then A=tan-1x
Example 5 Find a Missing Angle Measure
Find mT to the nearest degree.
You know the measure of the side opposite to T and the
measure of the hypotenuse. Use the sine ratio.
15
sin T =
Definition of sine
23
Use a calculator and the [COS-1] function to find the measure
of the angle.
KEYSTROKES:
So, mT  40.
2ND [SIN-1] 15

23 )
S
T
15
23
R
ENTER 40.70570683