How to Solve a Right Triangle With an Angle & a Hypotenuse

The hypotenuse of a triangle is the side opposite the right angle.

Right triangles are the simplest triangles to solve because the right angle simplifies the law of sines to a single set of ratios. This set of ratios relates the sine, cosine and tangent functions to the lengths of the triangle's legs, in what is commonly known as the SOH-CAH-TOA rule (Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, and Tangent = Opposite / Adjacent). Use the sine and cosine ratios to solve the lengths of a right triangle when you know one angle and the hypotenuse.

Instructions

1. Add 90 to the known angle and subtract the result from 180 to get the measure of the unknown angle of the triangle. The sum of the angles of a triangle is always 180 degrees, so the measure of the unknown angle is the number obtained when subtracting the sum of the other two angles from 180. For example, if the known angle is 60 degrees, the third angle measure is 180 - (60 + 90) or 30 degrees.

2. Write a sine equation relating the known angle, the hypotenuse, and the side of the triangle opposite the known angle. For example, if the hypotenuse is 4 inches long, write the equation sin 60 = a/4. The variable a stands for the opposite leg in the formula sin x = opposite/hypotenuse.

3. Solve the equation from Step 2 by multiplying both sides by the denominator and computing the value of a. In the example, a = 4 sin 60, which is approximately 3.464. This is the length of the side opposite the known angle.

4. Write a cosine equation relating the known angle, the hypotenuse, and the side of the triangle adjacent to the known angle. In the above example, write the equation cos 60 = b/4. The variable b stands for the adjacent leg in the formula cos x = adjacent/hypotenuse.

5. Solve the equation from Step 4 by multiplying both sides by the denominator and computing the value of b. In the example, b = 4 cos 60, which equals 2. This is the length of the side adjacent to the known angle.

References

• Clark University Department of Mathematics: Right Triangles