MAT1793-ABED_1: Solving Right Triangles

Constructing Knowledge

You now have several tools available to determine unknown angle measures and side lengths in a right triangle. These tools include the sine, cosine, and tangent ratios, the Pythagorean theorem, and the triangle angle sum. You can use all of these tools to solve right triangles.

Solving a Right Triangle

Determining all unknown side lengths and angle measures in a right triangle

Process for solving a right triangle:

When solving a right triangle do the simplest math first. In many cases, this means using the triangle angle sum (angle measures in a triangle add to 180°).

To calculate side lengths when only one side length is given the primary trigonometric ratios of sine, cosine or tangent can be used.

If you know two side lengths the Pythagorean Theorem can be used to calculate the third side length.

There are many ways of solving a right triangle. As long as you correctly apply the Pythagorean theorem, the triangle angle sum, and the primary trigonometric ratios, you may solve the triangle in whichever order makes the most sense to you, using whichever strategies are appropriate.

Multimedia

A video describing how to solve a right triangle is provided.

EXAMPLE 1

Solve the following triangle. Express angle measures to the nearest degree and lengths to the nearest tenth.

Solution

Step 1: Use a trigonometric ratio to determine one of the unknown angle measures from the two known side lengths.

Angle p will be solved for first. The tangent ratio will be used as the lengths opposite and adjacent to angle p are known.

\(\begin{align} \text{tan}\,p&=\frac{\text{opp}}{\text{adj}} \\ \\ \text{tan}\,p&=\frac{8}{14} \\ \\ p&=\text{tan}^{-1}\left(\frac{8}{14}\right) \\ \\ p&=30° \end{align}\)

Step 2: Solve for the third angle

The two acute angles in a right triangle add to 90° (All three angles sum to 180° and the right angle is 90°).

\(\begin{align} p+q&=90° \\ \\ 30°+q&=90° \\ \\ 30°+q-30°&=90°-30° \\ \\ q&=60° \end{align}\)

Step 3: Solve for the third side

Use the Pythagorean theorem to determine the length of the hypotenuse from the two known side lengths.

\(\begin{align} a^2+b^2&=c^2 \\ \\ 8^2+14^2&=c^2 \\ \\ 64+196&=c^2 \\ \\ 260&=c^2 \\ \\ \sqrt{260}&=\sqrt{c^2} \\ \\ 16.1&=c \\ \end{align}\)

Step 4: Label the diagram with all side lengths and angle measures.