# 2.1.5: Sine and Cosine of Complementary Angles

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- 4216

Sine of an angle equals the cosine of its complement.

\(\Delta ABC\) is a right triangle with \(m\angle C=90^{\circ}\) and \(\sin A=k\). What is \(\cos B\)?

## Sine and **Cosine** of **Complementary Angles**

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 sidethe angle to the*opposite**hypotenuse.* - The
**cosine**of an angle in a right triangle is the ratio of the sideto the angle to the*adjacent*.*hypotenuse*

In the problems below, you will explore how the sine and cosine of the angles in a right triangle are related.

Let's take a look at a few example problems.

1. 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?

\(\sin A=\dfrac{a}{c}\), \(\sin B=\dfrac{b}{c}\), \(\cos A=\dfrac{b}{c}\), \(\cos B=\dfrac{a}{c}\) Note that \(\sin A=\cos B\) and \(\sin B=\cos A\).

2. Consider the triangle from the previous problem. How is \(\angle A\) related to \(\angle B\)?

The sum of the measures of the three angles in a triangle is \(180^{\circ}\). This means that \(m\angle A+m\angle B+m\angle C=180^{\circ}\). \(\angle C\) is a right angle so \(m\angle C=90^{\circ}\). Therefore, \(m\angle A+m\angle B=90^{\circ}\). Angles A and B are complementary angles because their sum is \(90^{\circ}\).

In #1 you saw that \(\sin A=\cos B\) and \(\sin B=\cos A\). This means that the sine and cosine of *complementary angles* are equal.

3. Find \(80^{\circ}\) and \(\cos 10^{\circ}\). Explain the result.

\(\sin 80^{\circ}\approx 0.985\) and \(\cos 10^{\circ}\approx 0.985\). \(\sin 80^{\circ}=\cos 10^{\circ}\) because \(80^{\circ}\) and \(10^{\circ}\) are complementary angle measures. \(\sin 80^{\circ}\) and \(\cos 10^{\circ}\) are the ratios of the same sides of a right triangle, as shown below.

Earlier, you were asked what is \(\cos B\).

\(\Delta ABC\) is a right triangle with \(m\angle C=90^{\circ}\) and \(\sin A=k\). What is \(\cos B\)?

**Solution**

\(\angle A\) and \(\angle B\) 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.

If \(\sin 30^{\circ}=\dfrac{1}{2}\), \(\cos\stackrel?{=}\dfrac{1}{2}\).

**Solution**

The sine and cosine of **complementary** angles are equal. \(90^{\circ}−30^{\circ}=60^{\circ}\) is complementary to \(30^{\circ}\). Therefore, \(\cos 60^{\circ}=\dfrac{1}{2}\).

Consider the right triangle below. Find \(\tan A\) and \(\tan B\).

**Solution**

\(\tan A=\dfrac{a}{b}\) and \(\tan B=\dfrac{b}{a}\).

In general, what is the relationship between the tangents of complementary angles?

**Solution**

In general, the tangents of complementary angles are reciprocals.

## Review

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 \(\tan A=\dfrac{1}{2}\), what is \(\tan B\)?

5. If \(\sin A=\dfrac{7}{10}\), what is \(\cos B\)?

6. If \(\cos A=\dfrac{1}{4}\) what is \(\sin B\)?

7. If \(\sin A=\dfrac{3}{5}\), \(\cos \stackrel?{=} \dfrac{3}{5}\)?

8. Simplify \(\dfrac{\sin A+\cos B}{2}\).

9. If \(\tan A=\dfrac{2}{3}\) what is \(\tan B\)?

10. If \(\tan B=\dfrac{1}{5}\), what is \(\tan A\)? Which angle is bigger, \(\angle A\) or \(\angle B\)?

Solve for \(\theta\).

11. \(\cos 30^{\circ}=\sin \theta\)

12. \(\sin 75^{\circ}=\cos \theta\)

13. \(\cos 52^{\circ}=\sin \theta\)

14. \(\sin 18^{\circ}=\cos \theta\)

15. \(\cos 49^{\circ}=\sin \theta\)

## Review (Answers)

To see the Review answers, open this PDF file and look for section 7.3.

## Vocabulary

Term | Definition |
---|---|

Complementary angles |
Complementary angles are a pair of angles with a sum of \(90^{\circ}\). |

cosine |
The cosine of an angle in a right triangle is a value found by dividing the length of the side adjacent the given angle by the length of the hypotenuse. |

sine |
The sine of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the hypotenuse. |