How to Find a Missing Angle with Cosine

Step-by-step solution

1. Use the inverse cosine to solve for the angle.

   $$\theta = \cos^{-1}\left(\frac{1}{2}\right)$$

   Why: The inverse function answers the question: which angle has this cosine?

2. Read the angle from the unit circle.

   $$\cos 60^\circ = \frac{1}{2}$$

   Why: At 60 degrees the x-coordinate is one half.

3. Write the answer.

   $$\theta = 60^\circ$$

   Why: This is the exact angle for the given ratio.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

1. 1

   $\cos\theta = \frac{\sqrt{2}}{2}$

   Show answer

   $\theta = 45^\circ$

2. 2

   $\cos\theta = \frac{\sqrt{3}}{2}$

   Show answer

   $\theta = 30^\circ$

3. 3

   $\cos\theta = 1$

   Show answer

   $\theta = 0^\circ$

4. 4

   $\cos\theta = 0$

   Show answer

   $\theta = 90^\circ$

5. 5

   $\cos\theta = \frac{1}{2}$

   Show answer

   $\theta = 60^\circ$