Download 10-6 Study Guide and Intervention Trigonometric Ratios

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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.
A=
!"# !""!#$%& ∠" sine of ∠B =
!"# !""!#$%& ∠" sine of
cosine of
cosine of
tangent of
tangent of
!!"#$%&'(%
!!"#$%&'(
A=
!"# !"#!$%&' !" ∠!
B=
!"# !"#!$%&' !" ∠!
!"#$%&'()
A=
B=
!"#$%&'()
sin B =
!
!
!
!
cos B =
!
tan A =
!
tan B =
!"# !""!#$%& ∠!
!"# !"#!$%&' !" ∠! !"# !"#!$%&' !" ∠!
!
cos A =
!"# !""!#$%& ∠!
sin A =
!
!
!
!
!
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°
Chapter 10
37
Glencoe Algebra 1
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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
measure of the hypotenuse, use the cosine ratio.
cos 38° =
!"
Definition of cosine
!
cos 38° = 13
c=
A and are finding the
Multiply each side by 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
measure of the side opposite A, use the tangent ratio.
tan 38° =
!
!"
13 tan 38° = a
10.2 ≈ a
A and are finding the
Definition of tangent
Multiply each side by 13.
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.
Chapter 10
2.
3.
38
Glencoe Algebra 1