SOH CAH TOA

SOH CAH TOA is the mnemonic that helps you remember the three fundamental trigonometric ratios. Each group of three letters tells you which sides of a right triangle to divide:

• SOH — Sine = Opposite / Hypotenuse
• CAH — Cosine = Adjacent / Hypotenuse
• TOA — Tangent = Opposite / Adjacent

These three ratios are the foundation of all trigonometry. Every other concept — the unit circle, trig identities, the law of sines — builds on them.

Identifying the Sides

Before you can use any trig ratio, you need to identify the three sides of the right triangle relative to your chosen angle $\theta$.

• Opposite (O) — the side directly across from the angle $\theta$
• Adjacent (A) — the side next to $\theta$ that is not the hypotenuse
• Hypotenuse (H) — the longest side, always across from the right angle

Remember: if you change which angle you are working with, the opposite and adjacent sides swap. The hypotenuse never changes.

The Three Ratios Defined

Notice that sine and cosine always involve the hypotenuse, while tangent compares the two legs directly.

There is also a useful connection between the three:

$\tan\theta = \frac{\sin\theta}{\cos\theta}$

This is because $\frac{O/H}{A/H} = \frac{O}{A}$, which is exactly the tangent.

How to Choose the Right Ratio

When solving a problem, follow these steps:

1. Label the sides — identify which side is opposite, adjacent, and hypotenuse relative to the angle in the problem.
2. Determine what you know and what you need — you will always know two of the three values (angle and one side, or two sides).
3. Pick the ratio that connects your known and unknown values:
   • Know O and H (or need one of them)? Use sine.
   • Know A and H (or need one of them)? Use cosine.
   • Know O and A (or need one of them)? Use tangent.

Worked Examples

Example 1: Finding Sine, Cosine, and Tangent from a Triangle

A right triangle has an angle $\theta$, an opposite side of 5, an adjacent side of 12, and a hypotenuse of 13. Find all three trig ratios.

$\sin\theta = \frac{O}{H} = \frac{5}{13} \approx 0.385$

$\cos\theta = \frac{A}{H} = \frac{12}{13} \approx 0.923$

$\tan\theta = \frac{O}{A} = \frac{5}{12} \approx 0.417$

Verification: $\frac{\sin\theta}{\cos\theta} = \frac{5/13}{12/13} = \frac{5}{12} = \tan\theta$ . Correct.

Example 2: A Ladder Against a Wall

A 20-foot ladder leans against a wall at a 65-degree angle with the ground. How high up the wall does it reach?

The ladder is the hypotenuse ($H = 20$). The height up the wall is opposite the 65-degree angle. We know the angle and the hypotenuse, and we need the opposite — use sine.

$\sin(65°) = \frac{O}{20}$

$O = 20 \times \sin(65°) \approx 20 \times 0.9063 = 18.126 \approx 18.13 \text{ ft}$

Answer: The ladder reaches approximately 18.13 feet up the wall.

Example 3: Finding an Angle from Two Sides

A ramp rises 3 feet over a horizontal distance of 8 feet. What angle does the ramp make with the ground?

The rise (3) is opposite the angle; the run (8) is adjacent. We know O and A — use tangent.

$\tan\theta = \frac{3}{8} = 0.375$

$\theta = \tan^{-1}(0.375) \approx 20.6°$

Answer: The ramp makes approximately a 20.6-degree angle with the ground.

Common Mistakes

1. Mixing up opposite and adjacent. Always identify sides relative to the angle you are using, not relative to the right angle.
2. Using the wrong ratio. If your problem involves the hypotenuse and the opposite side, you need sine — not tangent.
3. Calculator in the wrong mode. Make sure your calculator is set to degrees (not radians) when working in degrees.
4. Forgetting that trig ratios are just fractions. Sine of 30 degrees is just a number (0.5). There is nothing mysterious about it.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• SOH CAH TOA is the mnemonic for the three trig ratios: Sine = O/H, Cosine = A/H, Tangent = O/A
• Always label sides relative to the angle you are working with — opposite, adjacent, and hypotenuse
• Choose your ratio based on which two of the three sides (O, A, H) are involved in your problem
• Tangent equals sine divided by cosine: $\tan\theta = \frac{\sin\theta}{\cos\theta}$
• Make sure your calculator is in the correct angle mode (degrees vs. radians)

Return to Trigonometry for more topics in this section.