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Sine and Cosine of
Complementary Angles
CK-12
Kaitlyn Spong
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Printed: October 9, 2014
AUTHORS
CK-12
Kaitlyn Spong
www.ck12.org
C HAPTER
Chapter 1. Sine and Cosine of Complementary Angles
1
Sine and Cosine of
Complementary Angles
Here you will explore how the sine and cosine of complementary angles are related.
∆ABC is a right triangle with m6 C = 90◦ and sin A = k. What is cos B?
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Guidance
Recall that the sine and cosine of angles are ratios of pairs of sides in right triangles.
• The sine of an angle in a right triangle is the ratio of the side opposite the angle to the hypotenuse.
• The cosine of an angle in a right triangle is the ratio of the side adjacent to the angle to the hypotenuse.
In the examples, you will explore how the sine and cosine of the angles in a right triangle are related.
Example A
Consider the right triangle below. Find the sine and cosine of angles A and B in terms of a, b, and c. What do you
notice?
Solution: sin A = ac , sin B = bc , cos A = bc , cos B = ac . Note that sin A = cos B and sin B = cos A.
Example B
Consider the triangle from Example A. How is 6 A related to 6 B?
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Solution: The sum of the measures of the three angles in a triangle is 180◦ . This means that m6 A + m6 B + m6 C =
180◦ . 6 C is a right angle so m6 C = 90◦ . Therefore, m6 A + m6 B = 90◦ . Angles A and B are complementary angles
because their sum is 90◦ .
In Example A you saw that sin A = cos B and sin B = cos A. This means that the sine and cosine of complementary
angles are equal.
Example C
Find 80◦ and cos 10◦ . Explain the result.
Solution: sin 80◦ ≈ 0.985 and cos 10◦ ≈ 0.985. sin 80◦ = cos 10◦ because 80◦ and 10◦ are complementary angle
measures. sin 80◦ and cos 10◦ are the ratios of the same sides of a right triangle, as shown below.
Concept Problem Revisited
∆ABC is a right triangle with m6 C = 90◦ and sin A = k. What is cos B?
6
A and 6 B are complementary because they are the two non-right angles of a right triangle. This means that sin A =
cos B and sin B = cos A. If sin A = k, then cos B = k as well.
Vocabulary
The tangent (tan) of an angle within a right triangle is the ratio of the length of the side opposite the angle to the
length of the side adjacent to the angle.
The sine (sin) of an angle within a right triangle is the ratio of the length of the side opposite the angle to the length
of the hypotenuse.
The cosine (cos) of an angle within a right triangle is the ratio of the length of the side adjacent to the angle to the
length of the hypotenuse.
The trigonometric ratios are sine, cosine, and tangent.
Trigonometry is the study of triangles.
θ, or “theta”, is a Greek letter. In geometry, it is often used as a variable to represent an unknown angle measure.
Two angles are complementary if the sum of their measures is 90◦ .
Guided Practice
1. If sin 30◦ = 21 , cos? = 21 .
2. Consider the right triangle below. Find tan A and tan B.
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Chapter 1. Sine and Cosine of Complementary Angles
3. In general, what is the relationship between the tangents of complementary angles?
Answers:
1. The sine and cosine of complementary angles are equal. 90◦ − 30◦ = 60◦ is complementary to 30◦ . Therefore,
cos 60◦ = 21 .
2. tan A =
a
b
and tan B = ab .
3. In general, the tangents of complementary angles are reciprocals.
Practice
1. How are the two non-right angles in a right triangle related? Explain.
2. How are the sine and cosine of complementary angles related? Explain.
3. How are the tangents of complementary angles related? Explain.
Let A and B be the two non-right angles in a right triangle.
4. If tan A = 12 , what is tan B?
7
10 , what is cos B?
cos A = 14 what is sin B?
sin A = 35 , cos? = 35 ?
5. If sin A =
6. If
7. If
sin A+cos B
.
2
9. If tan A = 32 what is tan B?
10. If tan B = 15 , what is tan A?
8. Simplify
Which angle is bigger, 6 A or 6 B?
Solve for θ.
11. cos 30◦ = sin θ
12. sin 75◦ = cos θ
13. cos 52◦ = sin θ
14. sin 18◦ = cos θ
15. cos 49◦ = sin θ
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References
1. . . CC BY-NC-SA
2. . . CC BY-NC-SA
3. . . CC BY-NC-SA
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