## How to Find Missing Sides and Angles of a Right Triangle

In a right triangle, when one side and an angle are given we can find a missing side.

### Trigonometric Ratios

The following trigonometric ratios can be used to determine the length of the missing side in a right triangle.

$$Sin \ θ \ = \ \frac{Opposite \ Side}{Hypotenuse}$$

$$Cos \ θ \ = \ \frac{Adjacent \ Side}{Hypotenuse}$$

$$tan \ θ \ = \ \frac{Opposite \ Side}{Adjacent \ Side}$$

$$Cot \ θ \ = \ \frac{Adjacent \ Side }{Opposite \ Side}$$

$$Cosec \ θ \ = \ \frac{Hypotenuse}{Opposite \ Side}$$

$$Sec \ θ \ = \ \frac{Hypotenuse}{Adjacent \ Side \ to}$$

### Example

Find the value of $$x$$.

Solution

• Hypotenuse Side $$= \ AC$$
• Opposite Side $$= \ AB \ = \ x$$
• Adjacent Side $$= \ BC \ = \ 7$$
• $$θ \ = \ 30$$

The trigonometric ratio $$tan$$ involves the adjacent and opposing sides.

$$tan(θ) \ = \ \frac{Opposite \ Side}{Adjacent \ Side} \ = \ \frac{AB}{BC}$$ $$⇒ \ \frac{\sqrt{3}}{3} \ = \ \frac{x}{7} \ ⇒ \ x \ = \ \frac{7\sqrt{3}}{3} \ ≈ \ 4.04$$

So, $$4.04$$ is the measurement of the missing side.

### Exercises for Finding the Missing Sides and Angles of a Right Triangle

1) Find the answer: $$x \ = \ ?$$

2) Find the answer: $$x \ = \ ?$$

3) Find the answer: $$x \ = \ ?$$

4) Find the answer: $$x \ = \ ?$$

5) Find the answer: $$x \ = \ ?$$

6) Find the answer: $$sin \ x \ = \ ? \ , \ cos \ x \ = \ ?$$

7) Find the answer: $$cos \ x \ = \ ? \ , \ tan \ x \ = \ ?$$

8) Find the answer: $$cot \ x \ = \ ? \ , \ sin \ x \ = \ ?$$

9) Find the answer: $$x \ = \ ?$$

10) Find the answer: $$x \ = \ ?$$

1) Find the answer: $$x \ = \ ?$$

$$\color{red}{tan \ 45° \ = \ 1 \ = \ \frac{AB}{BC} \ = \ \frac{x}{7} \ ⇒ \ x \ = \ \frac{\sqrt{2}}{2} \times 7 \ = \ \frac{7\sqrt{2}}{2}}$$

2) Find the answer: $$x \ = \ ?$$

$$\color{red}{tan \ x \ = \ \frac{AB}{BC} \ = \ \frac{5}{10} \ = \ \frac{1}{2} \ ⇒ \ x \ ≈ \ 26.565°}$$

3) Find the answer: $$x \ = \ ?$$

$$\color{red}{tan \ 60° \ = \ \sqrt{3} \ = \ \frac{AB}{BC} \ = \ \frac{6}{x} \ ⇒ \ x \ = \ \frac{6}{\sqrt{3}} \ = \ 2\sqrt{3}}$$

4) Find the answer: $$x \ = \ ?$$

$$\color{red}{tan \ 60° \ = \ \sqrt{3} \ = \ \frac{BC}{AB} \ = \ \frac{x}{11} \ ⇒ \ x \ = \ \sqrt{3} \times 11 \ = \ 11\sqrt{3}}$$

5) Find the answer: $$x \ = \ ?$$

$$\color{red}{tan \ x \ = \ \frac{BC}{AB} \ = \ \frac{4\sqrt{3}}{4} \ = \ \sqrt{3} \ ⇒ \ x \ = \ 60°}$$

6) Find the answer: $$sin \ x \ = \ ? \ , \ cos \ x \ = \ ?$$

$$\color{red}{AC \ = \ \sqrt{3^2 \ + \ 4^2} \ = \ 5}$$
$$\color{red}{sin \ x \ = \ \frac{BC}{AC} \ = \ \frac{4}{5}}$$
$$\color{red}{cos \ x \ = \ \frac{AB}{AC} \ = \ \frac{3}{5}}$$

7) Find the answer: $$cos \ x \ = \ ? \ , \ tan \ x \ = \ ?$$

$$\color{red}{BC \ = \ \sqrt{10^2 \ - \ 8^2} \ = \ \sqrt{36} \ = \ 6}$$
$$\color{red}{cos \ x \ = \ \frac{BC}{AC} \ = \ \frac{6}{10} \ = \ \frac{3}{5}}$$
$$\color{red}{tan \ x \ = \ \frac{AB}{AC} \ = \ \frac{8}{6} \ = \ \frac{4}{3}}$$

8) Find the answer: $$cot \ x \ = \ ? \ , \ sin \ x \ = \ ?$$

$$\color{red}{AB \ = \ \sqrt{15^2 \ - \ 12^2} \ = \ \sqrt{81} \ = \ 9}$$
$$\color{red}{cot \ x \ = \ \frac{BC}{AB} \ = \ \frac{12}{9} \ = \ \frac{4}{3}}$$
$$\color{red}{sin \ x \ = \ \frac{AB}{AC} \ = \ \frac{9}{15} \ = \ \frac{3}{5}}$$

9) Find the answer: $$x \ = \ ?$$

$$\color{red}{cos \ 45° \ = \ \frac{\sqrt{2}}{2} \ = \ \frac{BC}{AC} \ = \ \frac{9}{x} \ ⇒ \ x \ = \ 9 \times \frac{2}{\sqrt{2}} \ = \ \frac{\sqrt{2}}{9}}$$

10) Find the answer: $$x \ = \ ?$$

$$\color{red}{cos \ 49° \ ≈ \ 0.656 \ = \ \frac{BC}{AC} \ = \ \frac{9}{x} \ ⇒ \ x \ = \ 9 \times \frac{1}{0.656} \ ≈ \ 13.719}$$

## Missing Sides and Angles of a Right Triangle Practice Quiz

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