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.