Circle Identity (C)

It follows from the definition of cosine and sine as the coordinates of points on the unit circle that for any angle $\theta$,

$$(\cos(\theta))^2 + (\sin(\theta))^2 = 1.$$

This identity is called the Circle Identity. If the quadrant of an angle is known, this identity may be used to find the cosine of an angle if the sine is given, and vice versa.

Example: If $\theta$ is in the second quadrant and $\sin(\theta) = 5/13$, what is the cosine of $\theta$?

Solution: Since $\theta$ is in the second quadrant its cosine is negative. Also,

$$(\cos(\theta))^2 + (5/13)^2 = 1,$$

so $(\cos(\theta))^2 = 144/169 = (12/13)^2$. Therefore, $\cos(\theta) = -12/13$.

Exercises In each case find the indicated sine or cosine.

1.

$\theta$ in the first quadrant, $\sin(\theta) = 3/5$. Find $\cos(\theta)$.

2.

$\theta$ in the fourth quadrant, $\cos(\theta) = 3/5$. Find $\sin(\theta)$.

3.

$\theta$ in the third quadrant, $\cos(\theta) = -3/5$. Find $\sin(\theta)$.

4.

$\theta$ in the second quadrant, $\sin(\theta) = 1/5$. Find $\cos(\theta)$.

5.

$\theta$ in the first quadrant, $\sin(\theta) = 7/25$. Find $\cos(\theta)$.

6.

$\theta$ in the second quadrant, $\cos(\theta) = -24/25$. Find $\sin(\theta)$.

7.

$\theta$ in the third quadrant, $\cos(\theta) = -7/25$. Find $\sin(\theta)$.

8.

$\theta$ in the fourth quadrant, $\sin(\theta) = -1/25$. Find $\cos(\theta)$.