NAME
DATE
10-6
PERIOD
Study Guide and Intervention
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 hypotenuse leg opposite ∠B sine of ∠B = − hypotenuse
a sin A = − c
sine of ∠A = −
leg adjacent to ∠A hypotenuse leg adjacent to ∠B cosine of ∠B = −− hypotenuse
a cos B = − c
tangent of ∠A = −−
leg opposite ∠A leg adjacent to ∠A
a tan A = −
leg opposite ∠B tangent of ∠B = −− leg adjacent to ∠B
b tan B = − a
b cos A = − c
cosine of ∠A = −−
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example
"
b sin B = − c
c
b
a
$
b
#
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 2 2 2 a + 8 = 10 b = 8 and c = 10 2 a + 64 = 100 Simplify. 2 a = 36 Subtract 64 from each side. a=6 Take the positive square root of each side.
# 10
a
$
"
8
Step 2 Use the side lengths to write the trigonometric ratios. opp 6 3 sin A = − = − =− hyp
10
adj 8 4 cos A = − = − =−
5
hyp
10
5
opp adj
6 3 tan A = − = − =− 8
4
Exercises Find the values of the three trigonometric ratios for angle A. 1. " 8
2. "
3
$
3.
#
17
$
7
15 8 sin A = − , cos A = − , 17 15 tan A = − 8
"
5
#
17
#
3 4 4 sin A = − , cos A = − , tan A = − 5 5 3
24
$
7 24 sin A = − , cos A = − , 25 7 tan A = − 24
25
Use a calculator to find the value of each trigonometric ratio to the nearest ten-thousandth. 4. sin 40° 0.6428 Chapter 10
5. cos 25° 0.9063
37
6. tan 85° 11.4301 Glencoe Algebra 1
Lesson 10-6
Trigonometric Ratios
NAME
DATE
10-6
Study Guide and Intervention
PERIOD
(continued)
Trigonometric Ratios Use Trigonometric Ratios
When you find all of the unknown measures of the sides and angles of a right triangle, you are solving the triangle. You can find the missing measures of a right triangle if you know the measure of two sides of the triangle, or the measure of one side and the measure of one acute angle. Example
Solve the right triangle. Round each side length to the nearest tenth.
Step 1 Find the measure of ∠B. The sum of the measures of the angles in a triangle is 180. 180° − (90° + 38°) = 52° The measure of ∠B is 52°. −− Step 2 Find the measure of AB. Because you are given the measure of the side adjacent to ∠ A and are finding the measure of the hypotenuse, use the cosine ratio. 13 cos 38° = − c
c cos 38° = 13
# c
a
38° 13
"
$
Definition of cosine Multiply each side by c.
13 c=− Divide each side by cos 38°. cos 38° −− So the measure of AB is about 16.5. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
−−− Step 3 Find the measure of BC. Because you are given the measure of the side adjacent to ∠ A and are finding the measure of the side opposite ∠ A, use the tangent ratio. a tan 38° = −
Definition of tangent
13
13 tan 38° = a
Multiply each side by 13.
10.2 ≈ a
Use a calculator.
−−− So the measure of BC is about 10.2.
Exercises Solve each right triangle. Round each side length to the nearest tenth. $
1. b
"
#
2. a
30° 9
#
"
∠B = 60°, AC ≈ 7.8, BC = 4.5 Chapter 10
56°
44°
c
8
b
$
∠A = 46°, AC ≈ 7.7, AB ≈ 11.1 38
3. " b
$
c
16
#
∠B = 34°, AC ≈ 19.3, AB ≈ 10.8 Glencoe Algebra 1