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NAME _____________________________________________ DATE ____________________________ PERIOD _____________
12-1 Study Guide and Intervention
Trigonometric Functions in Right Triangles
Trigonometric Functions for Acute Angles Trigonometry is the study of relationships among the angles and sides
of a right triangle. A trigonometric function has a rule given by a trigonometric ratio, which is a ratio that compares the
side lengths of a right triangle.
Trigonometric Functions
in Right Triangles
If θ is the measure of an acute angle of a right triangle, opp is the measure of the leg
opposite θ, adj is the measure of the leg adjacent to θ, and hyp is the measure of the
hypotenuse, then the following are true.
sin θ =
csc θ =
opp
cos θ =
hyp
hyp
sec θ =
opp
hyp
hyp
tan θ =
opp
cot θ =
opp
𝟑
Example: In a right triangle, ∠ B is acute and cos B = 𝟕. Find the value of tan B.
Step 1 Draw a right triangle and label one acute angle B. Label the adjacent side 3 and the hypotenuse 7.
Step 2 Use the Pythagorean Theorem to find b.
𝑎2 + 𝑏 2 = 𝑐 2
Pythagorean Theorem
32 + 𝑏 2 = 72
a = 3 and c = 7
9 + 𝑏 2 = 49
Simplify.
2
Subtract 9 from each side.
b = √40 or 2√10
Take the positive square root of each side.
𝑏 = 40
Step 3 Find tan B.
tan B =
opp
Tangent function
tan B =
2√10
3
Replace opp with 2√10 and adj with 3.
Exercises
Find the values of the six trigonometric functions for angle θ.
1.
2.
3.
In a right triangle, ∠ A and ∠ B are acute.
7
4. If tan A = 12 , what is cos A?
Chapter 12
1
5. If cos A = 2 , what is tan A?
5
3
6. If sin B = 8 , what is tan B?
Glencoe Algebra 2
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
12-1 Study Guide and Intervention (continued)
Trigonometric Functions in Right Triangles
Use Trigonometric Functions You can use trigonometric functions to find missing side lengths and missing angle
measures of right triangles. You can find the measure of the missing angle by using the inverse of sine, cosine, or tangent.
Example: Find the measure of ∠ C. Round to the nearest tenth if necessary.
You know the measure of the side opposite ∠ C and the measure of the hypotenuse. Use the sine function.
opp
sin C = hyp
8
sin C = 10
sin−1
8
10
= m∠ C
53.1° ≈ m∠ C
Sine function
Replace opp with 8 and hyp with 10.
Inverse sine
Use a calculator.
Exercises
Use a trigonometric function to find each value of x. Round to the nearest tenth if necessary.
1.
2.
3.
4.
5.
6.
Find x. Round to the nearest tenth if necessary.
7.
8.
Chapter 12
9.
6
Glencoe Algebra 2