Solving Right Triangles Given certain measures in a right triangle, we often want to find the other angle and side measures. This is called solving the triangle. All of the following examples will use degree measure, so set the mode on your calculator to degree.

c 7 in a  Example 1: Solve the given triangle: First, solve for θ. Since the sum of the two acute angles in a right triangle is 90°, we have … So θ is 28°.

c 7 in a Now solve for c, which is the hypotenuse. The side measured 7 in. is the side adjacent to the given 62° angle. The trigonometric ratio using adjacent and hypotenuse is cosine.

c 7 in a The value of c is approximately inches.

c 7 in a Last, solve for a, which is the side opposite the 62° angle. The side measured 7 in. is the side adjacent to the given 62° angle. The trigonometric ratio using opposite and adjacent is tangent.

c 7 in a The value of a is approximately inches.

84.7 cm 62.3 cm b  Example 2: Solve the given triangle: First, solve for θ. Since the hypotenuse and the side adjacent to θ are given, use cosine.

84.7 cm 62.3 cm b So θ is approximately 42.6°.

Now solve for α cm 62.3 cm b Since the sum of the two acute angles in a right triangle is 90°, we have … So α is approximately 47.4°

Last, solve for b, which is the side opposite θ. The side measured 62.3 in. is the side adjacent to θ The trigonometric ratio using opposite and adjacent is tangent cm 62.3 cm b

The value of b is approximately 57.3 cm cm 62.3 cm b Note that we could have used the Pythagorean Theorem to solve for side b.