Trigonometric Ratios

Trigonometric Ratios Trigonometry is the study of relationships of the angles and the
sides of a right triangle. The three most common trigonometric ratios are the sine, cosine,
and tangent.
leg opposite ∠A a
sine of ∠A = − sin A = −
c
hypotenuse
leg opposite ∠B b "
sine of ∠B = − sin B = −
c
hypotenuse
leg adjacent to ∠A b
cosine of ∠A = −− cos A = −
c
hypotenuse c
b
leg adjacent to ∠B a
cosine of ∠B = −− cos B = −
c
hypotenuse

leg opposite ∠A a
tangent of ∠A = −− tan A = − a
leg adjacent to ∠A b $ #
leg opposite ∠B b
tangent of ∠B = −− tan B = −
a
leg adjacent to ∠B

Example Find the values of the three trigonometric ratios for angle A .

Step 1 Use the Pythagorean Theorem to find BC. #

a2 + b2 = c2 Pythagorean Theorem
10
a2 + 82 = 102 b = 8 and c = 10 a
a2 + 64 = 100 Simplify.

a = 36
2
Subtract 64 from both sides. $ 8 "
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

a=6 Take the square root of each side.

Step 2 Use the side lengths to write the trigonometric ratios.

opp 6 3 8 4 adj 6 3 opp
sin A = − = − =− cos A = − = − =− tan A = − = − =−
hyp 10 5 hyp 10 5 adj 8 4

Exercises
Find the values of the three trigonometric ratios for angle A.

1. " 2. " 3 $ 3. #
17
8 7
"
$ # 5 24 $

Use a calculator to find the value of each trigonometric ratio to the nearest
ten-thousandth.

4. sin 40° 5. cos 25° 6. tan 85°

Chapter 10 149 Glencoe Algebra 1

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