How to Find Missing Sides and Angles of a Right Triangle

Learn how to find the missing sides or angles of a right triangle when one length and one angle is provided.

Step by step guide to finding missing sides and angles of a Right Triangle

• By using Sine, Cosine or Tangent, we can find an unknown side in a right triangle when we have one length, and one angle (apart from the right angle).
• Adjacent, Opposite and Hypotenuse, in a right triangle is shown below.
• Recall the three main trigonometric functions:
SOH – CAH – TOA, sin $$θ=\frac{opposite}{hypotenuse}$$, Cos $$θ=\frac{adjacent}{hypotenuse}$$, tan $$⁡θ=\frac{opposite}{adjacent}$$

Missing Sides and Angles of a Right Triangle – Example 1:

Find AC in the following triangle. Round answers to the nearest tenth.

Solution:

sin $$(θ=\frac{opposite}{hypotenuse}$$. sine $$45^\circ=\frac{AC}{8}→8 ×sin 45^\circ=AC$$,
now use a calculator to find sin $$45^\circ$$. sin $$40^\circ=\frac{\sqrt{2}}{2}→AC \cong 0.70710$$

Missing Sides and Angles of a Right Triangle – Example 2:

Find AC in the following triangle. Round answers to the nearest tenth.

Solution:

sine $$θ=\frac{opposite}{hypotenuse}$$. sine $$40^\circ=\frac{AC}{6}→6 ×$$ sine $$40^\circ=AC$$,
now use a calculator to find sine $$40^\circ$$. sine $$40^\circ\cong 0.642→AC \cong 3.9$$

Exercises for Finding Missing Sides and Angles of a Right Triangle

Find the missing side. Round answers to the nearest tenth.

• $$\color{blue}{31.4}$$
• $$\color{blue}{7.0}$$
• $$\color{blue}{16.2}$$
• $$\color{blue}{31.1}$$

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