How to Find a Missing Side with Sine

Step-by-step solution

1. Choose the ratio with opposite and hypotenuse.

   $$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$$

   Why: The sine ratio links the side opposite the angle to the hypotenuse.

2. Substitute the angle and the hypotenuse.

   $$\sin 30^\circ = \frac{x}{12}$$

   Why: The unknown side is opposite the 30 degree angle.

3. Use the exact value of sin 30 degrees.

   $$\frac{1}{2} = \frac{x}{12}$$

   Why: sin 30 degrees is exactly one half.

4. Multiply both sides by 12.

   $$x = 6$$

   Why: The unknown side is 6 units long.

Practice problems

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

1. 1

   $30^\circ,\ \text{hypotenuse } 20$

   Show answer

   $10$

2. 2

   $45^\circ,\ \text{hypotenuse } 10$

   Show answer

   $5\sqrt{2}$

3. 3

   $60^\circ,\ \text{hypotenuse } 8$

   Show answer

   $4\sqrt{3}$

4. 4

   $30^\circ,\ \text{hypotenuse } 5$

   Show answer

   $2.5$

5. 5

   $45^\circ,\ \text{hypotenuse } 6$

   Show answer

   $3\sqrt{2}$