Find a Missing Right-Triangle Side

Find a Missing Side in a Right Triangle

When one side and one acute angle of a right triangle are known, trigonometric ratios help find the missing side. The key idea is to relate the side you know to the side you want using the angle.

1. Identify the sides

• Hypotenuse: the longest side, opposite the right angle.
• Opposite: across from the given acute angle.
• Adjacent: next to the given acute angle, but not the hypotenuse.

2. Choose the correct ratio

Use the ratio that matches the sides involved:

• (\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}})
• (\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}})
• (\tan \theta = \frac{\text{opposite}}{\text{adjacent}})

Set up an equation using the known side and angle. Then solve for the missing side.

3. Solve and simplify

If the missing side is multiplied by a trig value, divide to isolate it. If the side is in the denominator, multiply both sides first. Keep exact values when possible, and simplify radicals or fractions in the final answer.

4. Check your result

Your answer should make sense for the triangle: the hypotenuse must be the longest side, and the side lengths should be positive. A quick estimate can help confirm the size is reasonable.

Example check

If you find the hypotenuse, it should be larger than either leg. If you find a leg, compare it with the given side and angle to see whether the result fits the triangle’s shape.