Trigonometry Practice

SOH-CAH-TOA Practice

Practice labeling right triangles, choosing sine, cosine, or tangent, and solving for missing sides and angles.

Start Practicing Review SOH-CAH-TOA

Try each problem first, then click Show solution to check your side labels, trig ratio, equation, and answer.

Keep These Ratios Nearby

SOH-CAH-TOA Essentials

SOH

\[ \sin(\theta) = \frac{\text{opposite}} {\text{hypotenuse}} \]

CAH

\[ \cos(\theta) = \frac{\text{adjacent}} {\text{hypotenuse}} \]

TOA

\[ \tan(\theta) = \frac{\text{opposite}} {\text{adjacent}} \]
Download Practice PDF Review SOH-CAH-TOA

Level 1

Label the Triangle & Choose the Ratio

Identify opposite, adjacent, and hypotenuse, then choose SOH, CAH, or TOA.

Problem 1

Label the Sides

Relative to \(\theta\), identify the opposite side, adjacent side, and hypotenuse.

A B C θ

Opposite:

Adjacent:

Hypotenuse:

Step-by-Step Solution

Start with the hypotenuse. It is always across from the \(90^\circ\) angle.

\[ \text{hypotenuse}=C \]

Relative to \(\theta\), side \(A\) is across from the angle, so it is opposite.

\[ \text{opposite}=A \]

Side \(B\) touches \(\theta\) and is not the hypotenuse, so it is adjacent.

Answer: \[ \boxed{ \begin{aligned} \text{opposite} &= A\\ \text{adjacent} &= B\\ \text{hypotenuse} &= C \end{aligned} } \]

Problem 2

Choose the Ratio

You know the opposite side and the hypotenuse. Which trig ratio should you use?

SOH
CAH
TOA

Step-by-Step Solution

We need the ratio that uses opposite and hypotenuse.

\[ \sin(\theta) = \frac{\text{opposite}} {\text{hypotenuse}} \]
Answer: \[ \boxed{\text{SOH}} \]

Problem 3

Choose the Ratio

You know the adjacent side and want to find the opposite side. Which trig ratio should you use?

SOH
CAH
TOA

Step-by-Step Solution

We need the ratio containing the opposite and adjacent sides.

\[ \tan(\theta) = \frac{\text{opposite}} {\text{adjacent}} \]
Answer: \[ \boxed{\text{TOA}} \]

Level 2

Find Missing Sides

Choose the correct trig ratio, write the equation, and solve for the missing side.

Problem 4

Sine

A right triangle has a \(38^\circ\) angle and a hypotenuse of \(14\). Find the side opposite the \(38^\circ\) angle.

x 14 38°

Step-by-Step Solution

We know the hypotenuse and want the opposite side, so use sine.

\[ \sin(38^\circ) = \frac{x}{14} \]

Multiply both sides by \(14\).

\[ x = 14\sin(38^\circ) \]

Evaluate.

\[ x \approx 8.62 \]
Answer: \[ \boxed{x\approx8.62} \]

Problem 5

Cosine

A right triangle has a \(52^\circ\) angle and a hypotenuse of \(18\). Find the side adjacent to the \(52^\circ\) angle.

x 18 52°

Step-by-Step Solution

We know the hypotenuse and want the adjacent side, so use cosine.

\[ \cos(52^\circ) = \frac{x}{18} \]

Multiply both sides by \(18\).

\[ x = 18\cos(52^\circ) \]

Evaluate.

\[ x \approx 11.08 \]
Answer: \[ \boxed{x\approx11.08} \]

Problem 6

Tangent

A right triangle has a \(31^\circ\) angle and an adjacent side of \(10\). Find the opposite side.

x 10 31°

Step-by-Step Solution

We know the adjacent side and want the opposite side, so use tangent.

\[ \tan(31^\circ) = \frac{x}{10} \]

Multiply both sides by \(10\).

\[ x = 10\tan(31^\circ) \]

Evaluate.

\[ x \approx 6.01 \]
Answer: \[ \boxed{x\approx6.01} \]

Level 3

Find Missing Angles

Set up the correct trig ratio, then use an inverse trig function to solve for the missing angle.

Calculator reminder: Make sure your calculator is in degree mode.

Problem 7

Inverse Sine

The side opposite \(\theta\) is \(8\), and the hypotenuse is \(15\). Find \(\theta\) to the nearest tenth of a degree.

8 15 θ

Step-by-Step Solution

We know the opposite side and the hypotenuse, so use sine.

\[ \sin(\theta) = \frac{8}{15} \]

Because the angle is unknown, use inverse sine.

\[ \theta = \sin^{-1} \left( \frac{8}{15} \right) \]

Evaluate in degree mode.

\[ \theta \approx 32.2^\circ \]
Answer: \[ \boxed{ \theta \approx 32.2^\circ } \]

Problem 8

Inverse Cosine

The side adjacent to \(\theta\) is \(12\), and the hypotenuse is \(17\). Find \(\theta\) to the nearest tenth of a degree.

12 17 θ

Step-by-Step Solution

We know the adjacent side and the hypotenuse, so use cosine.

\[ \cos(\theta) = \frac{12}{17} \]

Because the angle is unknown, use inverse cosine.

\[ \theta = \cos^{-1} \left( \frac{12}{17} \right) \]

Evaluate in degree mode.

\[ \theta \approx 45.1^\circ \]
Answer: \[ \boxed{ \theta \approx 45.1^\circ } \]

Problem 9

Inverse Tangent

The side opposite \(\theta\) is \(11\), and the adjacent side is \(16\). Find \(\theta\) to the nearest tenth of a degree.

11 16 θ

Step-by-Step Solution

We know the opposite and adjacent sides, so use tangent.

\[ \tan(\theta) = \frac{11}{16} \]

Because the angle is unknown, use inverse tangent.

\[ \theta = \tan^{-1} \left( \frac{11}{16} \right) \]

Evaluate in degree mode.

\[ \theta \approx 34.5^\circ \]
Answer: \[ \boxed{ \theta \approx 34.5^\circ } \]

Level 4

Mixed SOH-CAH-TOA Challenge

Decide which ratio to use without being told whether the problem needs sine, cosine, or tangent.

Problem 10

Mixed Challenge

A right triangle has a \(40^\circ\) angle and an opposite side of \(7\). Find the hypotenuse to the nearest hundredth.

7 x 40°

Step-by-Step Solution

We know the opposite side and want the hypotenuse, so use sine.

\[ \sin(40^\circ) = \frac{7}{x} \]

Multiply both sides by \(x\), then divide by \(\sin(40^\circ)\).

\[ x = \frac{7} {\sin(40^\circ)} \]

Evaluate.

\[ x \approx 10.89 \]
Answer: \[ \boxed{x\approx10.89} \]

Before You Finish

Use This Checklist Every Time

  1. Identify the reference angle.
  2. Label opposite, adjacent, and hypotenuse.
  3. Decide which two sides are involved.
  4. Choose SOH, CAH, or TOA.
  5. Use inverse trig if the missing value is an angle.

Continue Learning

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