Trigonometry Guide

SOH-CAH-TOA Explained

Learn how sine, cosine, and tangent connect the sides and angles of a right triangle.

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Right Triangle Trigonometry

What Is SOH-CAH-TOA?

SOH-CAH-TOA is a memory tool for the three basic trigonometric ratios: sine, cosine, and tangent.

Each ratio compares two sides of a right triangle. Once you know which sides are involved, you can choose the correct trig function and solve for a missing side or angle.

The Three Ratios

SOH — CAH — TOA

SOH

Sine = Opposite ÷ Hypotenuse

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

Cosine = Adjacent ÷ Hypotenuse

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

Tangent = Opposite ÷ Adjacent

\[ \tan(\theta) = \frac{\text{opposite}} {\text{adjacent}} \]

Before Using SOH-CAH-TOA

Know the Three Side Names

Before choosing sine, cosine, or tangent, identify the hypotenuse, opposite side, and adjacent side.

Hypotenuse

The hypotenuse is the side directly across from the \(90^\circ\) angle.

It is always the longest side of a right triangle.

Opposite

The opposite side is directly across from the reference angle \(\theta\).

Its location depends on which acute angle you are using.

Adjacent

The adjacent side touches the reference angle \(\theta\), but it is not the hypotenuse.

Think: next to the angle.

Start With the Hypotenuse

The hypotenuse is the easiest side to identify because it is always opposite the right angle. After finding it, use \(\theta\) to decide which of the remaining sides is opposite and which is adjacent.

Visual Model

Label the Triangle Before You Calculate

The side names opposite and adjacent are always based on the reference angle \(\theta\).

opposite adjacent hypotenuse θ 90°
With \(\theta\) at the lower-right corner, the vertical side is opposite, the bottom side is adjacent, and the slanted side is the hypotenuse.

Opposite and Adjacent Can Switch

If the reference angle changes, the opposite and adjacent sides can switch names. The hypotenuse never changes.

Choose the Right Ratio

Which One Do You Use?

Look at the side you know and the side you are trying to find. Then choose the trig ratio that contains both of them.

Use Sine

Opposite + Hypotenuse

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

If the problem involves the opposite side and the hypotenuse, use SOH.

Use Cosine

Adjacent + Hypotenuse

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

If the problem involves the adjacent side and the hypotenuse, use CAH.

Use Tangent

Opposite + Adjacent

\[ \tan(\theta) = \frac{\text{opposite}} {\text{adjacent}} \]

If the problem involves the opposite side and the adjacent side, use TOA.

A Simple Decision Process

Use These Four Steps

1

Find the reference angle

Identify the acute angle the problem is using.

2

Label the sides

Mark opposite, adjacent, and hypotenuse.

3

Identify known and unknown

Decide which side is given and which side you need to find.

4

Pick SOH, CAH, or TOA

Choose the ratio containing the two sides involved.

Worked Examples

Finding Missing Sides

Once you identify the correct ratio, substitute the known values and solve the equation.

Example 1

Use Sine to Find the Opposite Side

SOH

A right triangle has a \(35^\circ\) angle and a hypotenuse of \(12\). Find the side opposite the \(35^\circ\) angle.

x 12 35°
  1. Identify the sides.

    We know the hypotenuse and want the opposite side.

  2. Choose SOH.
    \[ \sin(35^\circ) = \frac{x}{12} \]
  3. Multiply by \(12\).
    \[ x = 12\sin(35^\circ) \]
  4. Evaluate.
    \[ x \approx 6.88 \]
Answer \[ \boxed{x\approx6.88} \]

Example 2

Use Cosine to Find the Adjacent Side

CAH

A right triangle has a \(42^\circ\) angle and a hypotenuse of \(15\). Find the side adjacent to the \(42^\circ\) angle.

x 15 42°
  1. Identify the sides.

    We know the hypotenuse and want the adjacent side.

  2. Choose CAH.
    \[ \cos(42^\circ) = \frac{x}{15} \]
  3. Multiply by \(15\).
    \[ x = 15\cos(42^\circ) \]
  4. Evaluate.
    \[ x \approx 11.15 \]
Answer \[ \boxed{x\approx11.15} \]

Example 3

Use Tangent to Find the Opposite Side

TOA

A right triangle has a \(28^\circ\) angle and an adjacent side of \(9\). Find the opposite side.

x 9 28°
  1. Identify the sides.

    We know the adjacent side and want the opposite side.

  2. Choose TOA.
    \[ \tan(28^\circ) = \frac{x}{9} \]
  3. Multiply by \(9\).
    \[ x = 9\tan(28^\circ) \]
  4. Evaluate.
    \[ x \approx 4.79 \]
Answer \[ \boxed{x\approx4.79} \]

Finding Angles

Use Inverse Trig to Find a Missing Angle

When the missing value is an angle, set up SOH, CAH, or TOA just like before. Then use the inverse trig function to solve for \(\theta\).

Inverse Sine

\[ \theta = \sin^{-1} \left( \frac{\text{opposite}} {\text{hypotenuse}} \right) \]

Inverse Cosine

\[ \theta = \cos^{-1} \left( \frac{\text{adjacent}} {\text{hypotenuse}} \right) \]

Inverse Tangent

\[ \theta = \tan^{-1} \left( \frac{\text{opposite}} {\text{adjacent}} \right) \]

Check Your Calculator Mode

For these right-triangle problems, make sure your calculator is in degree mode before evaluating an inverse trig function.

Example 4

Use Inverse Sine

SOH

The side opposite \(\theta\) is \(7\), and the hypotenuse is \(13\). Find \(\theta\).

7 13 θ
  1. Identify the sides.

    We have the opposite side and the hypotenuse.

  2. Choose SOH.
    \[ \sin(\theta) = \frac{7}{13} \]
  3. Use inverse sine.
    \[ \theta = \sin^{-1} \left( \frac{7}{13} \right) \]
  4. Evaluate.
    \[ \theta \approx 32.6^\circ \]
Answer \[ \boxed{ \theta \approx 32.6^\circ } \]

Example 5

Use Inverse Cosine

CAH

The side adjacent to \(\theta\) is \(8\), and the hypotenuse is \(11\). Find \(\theta\).

8 11 θ
  1. Identify the sides.

    We have the adjacent side and the hypotenuse.

  2. Choose CAH.
    \[ \cos(\theta) = \frac{8}{11} \]
  3. Use inverse cosine.
    \[ \theta = \cos^{-1} \left( \frac{8}{11} \right) \]
  4. Evaluate.
    \[ \theta \approx 43.3^\circ \]
Answer \[ \boxed{ \theta \approx 43.3^\circ } \]

Example 6

Use Inverse Tangent

TOA

The side opposite \(\theta\) is \(9\), and the adjacent side is \(12\). Find \(\theta\).

9 12 θ
  1. Identify the sides.

    We have the opposite and adjacent sides.

  2. Choose TOA.
    \[ \tan(\theta) = \frac{9}{12} \]
  3. Use inverse tangent.
    \[ \theta = \tan^{-1} \left( \frac{9}{12} \right) \]
  4. Evaluate.
    \[ \theta \approx 36.9^\circ \]
Answer \[ \boxed{ \theta \approx 36.9^\circ } \]

Check Your Reasoning

Common SOH-CAH-TOA Mistakes

Most mistakes come from labeling the triangle incorrectly or choosing the wrong ratio.

Mixing Up Opposite and Adjacent

Opposite and adjacent depend on the reference angle. Always label the triangle from the angle you are actually using.

Misidentifying the Hypotenuse

The hypotenuse is always opposite the \(90^\circ\) angle. It does not depend on the reference angle.

Choosing the Wrong Ratio

Use the known side and the side you are trying to find. Choose the ratio that contains both.

Using the Wrong Calculator Mode

Most right-triangle problems use degree measure. Check that your calculator is in degree mode before evaluating trig or inverse trig.

Quick Review

SOH-CAH-TOA at a Glance

SOH

Sine

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

CAH

Cosine

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

TOA

Tangent

\[ \tan(\theta) = \frac{\text{opposite}} {\text{adjacent}} \]

Finding a side? Use the regular trig function. Finding an angle? Use the inverse trig function.

Ready to Practice?

Try SOH-CAH-TOA Problems

Practice labeling sides, choosing the correct trig ratio, finding missing sides, and finding missing angles.

Practice SOH-CAH-TOA Review Special Right Triangles

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