NAME ____________________________________________ DATE _____________________________ PERIOD _____________
Chapter 10 37 Glencoe Algebra 1
10-6 Study Guide and Intervention 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.
sine of �A = !"# !""!#$%& ∠" !!"#$%&'(%
sine of ∠B = !"# !""!#$%& ∠" !!"#$%&'(
sin A = !!
sin B = !!
cosine of �A = !"# !"#!$%&' !" ∠!!"#$%&'()
cosine of �B = !"# !"#!$%&' !" ∠!!"#$%&'()
cos A = !!
cos B = !!
tangent of �A = !"# !""!#$%& ∠!!"# !"#!$%&' !" ∠!
tangent of �B = !"# !""!#$%& ∠!!"# !"#!$%&' !" ∠!
tan A = !!
tan B = !!
Example: Find the values of the three trigonometric ratios for angle A .
Step 1 Use the Pythagorean Theorem to find BC.
𝑎! + 𝑏! = 𝑐! Pythagorean Theorem
𝑎! + 8! = 10! b = 8 and c = 10
𝑎! + 64 = 100 Simplify.
𝑎! = 36 Subtract 64 from each side.
a = 6 Take the positive square root of each side.
Step 2 Use the side lengths to write the trigonometric ratios.
sin A = !""!"#
= !!"
= !! cos A = !"#
!"# = !
!" = !
! tan A = !""
!"# = !
! = !
!
Exercises Find the values of the three trigonometric ratios for angle A.
1. 2. 3.
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°
NAME ____________________________________________ DATE _____________________________ PERIOD _____________
Chapter 10 38 Glencoe Algebra 1
10-6 Study Guide and Intervention (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.
cos 38° = !"!
Definition of cosine
cos 38° = 13 Multiply each side by c.
c = !"!"# !"°
Divide each side by cos 38°.
So the measure of 𝐴𝐵 is about 16.5. 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.
tan 38° = !!"
Definition of tangent
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. 2. 3.
