What does SOH-CAH-TOA stand for?

SOH-CAH-TOA is a mnemonic: \sin(\theta) = \frac{\text{opp}}{\text{hyp}}, \cos(\theta) = \frac{\text{adj}}{\text{hyp}}, \tan(\theta) = \frac{\text{opp}}{\text{adj}}. It helps you remember which sides pair with each trig function for a given acute angle in a right triangle.

When should I use tangent instead of sine or cosine?

Use tangent when you know (or want to find) only the opposite and adjacent sides, but not the hypotenuse. Tangent is the best choice when the hypotenuse is not given, not needed, or would make the problem more complicated.

Do these ratios work for all angles, or just 0°–90°?

In the context of right triangles and SOH-CAH-TOA, these ratios apply only to the acute angles (greater than 0° and less than 90°). For other angles, you would use the unit circle or inverse trig functions.

Can I use these ratios if I only know two sides and no angles?

Not directly; you would need to use Inverse Trigonometric Functions like \sin^{-1}, \cos^{-1}, or \tan^{-1} to find the angle first, then use SOH-CAH-TOA to find the missing side.

Sine Cosine Tangent

Find missing sides or angles in right triangles using sine, cosine, and tangent ratios—essential for SAT and ACT geometry.

\sin\theta = \frac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \frac{\text{opp}}{\text{adj}}

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What each symbol means

What Sine Cosine Tangent takes

Symbol	Meaning
$\theta$	The acute angle (between 0° and 90°, not including the right angle) in a right triangle that you use to determine which sides are opposite and adjacent; using the wrong angle will swap these roles.

When to use it

You need to find a missing side or angle in a right triangle when you know one acute angle and one side.

Level

Usually taught in: Geometry · Appears on: SAT, ACT

Worked examples

1. Find the opposite side using sine

Problem

In a right triangle, one angle is 30° and the hypotenuse is 20 cm. Find the side opposite the 30° angle.

$\sin(30^\circ) = \frac{\text{opp}}{20}$
Set up the sine ratio for the 30° angle, where opposite is unknown and hypotenuse is 20 cm.
$\text{opp} = 20 \cdot \sin(30^\circ)$
Isolate the opposite side by multiplying both sides by 20.
$\text{opp} = 20 \cdot 0.5 = 10 \text{ cm}$
Since $\sin(30^\circ) = 0.5$ , multiply to get 10 cm.

Answer: $\text{opp} = 10 \text{ cm}$

Sine directly relates the opposite side and hypotenuse, so when you know the hypotenuse and angle, you multiply to find the opposite. This works cleanly here because $\sin(30^\circ) = 0.5$ .

2. Find the hypotenuse using sine

Problem

A right triangle has an angle of 50° and the side opposite to it is 15 m. Find the hypotenuse.

$\sin(50^\circ) = \frac{15}{\text{hyp}}$
Set up the sine ratio for the 50° angle, where 15 m is the opposite side and hyp is the unknown hypotenuse.
$\text{hyp} = \frac{15}{\sin(50^\circ)}$
Isolate hyp by dividing both sides by $\sin(50^\circ)$ , which rearranges the fraction.
$\text{hyp} = \frac{15}{0.766} \approx 19.6 \text{ m}$
Substitute $\sin(50^\circ) \approx 0.766$ and divide to get approximately 19.6 m.

Answer: $\text{hyp} \approx 19.6 \text{ m}$

When the opposite side is known and the hypotenuse is unknown, rearrange the sine ratio to divide the opposite by sine. This example requires division, which is more complex than Example 1, and involves decimal approximation.

3. Find the height of a tent pole using sine

Problem

For the school bake sale, a tent rope is 10 feet long, tied from the top of a pole to the ground at a 30° angle. How tall is the pole?

$\sin(30^\circ) = \frac{\text{height}}{10}$
The rope forms the hypotenuse (10 feet), and the pole height is the opposite side to the 30° angle at ground level.
$\text{height} = 10 \cdot \sin(30^\circ)$
Isolate height by multiplying both sides by 10.
$\text{height} = 10 \cdot 0.5 = 5 \text{ feet}$
Since $\sin(30^\circ) = 0.5$ , the pole is 5 feet tall.

Answer: $\text{height} = 5 \text{ feet}$

Real-world problems often involve heights or lengths you cannot measure directly, which is why SOH-CAH-TOA is so useful. Here, you translate the rope and angle into an equation, then solve for the pole height.

Common mistakes

Where Sine Cosine Tangent usually goes wrong

The mistake	Why it is wrong	The fix
$\sin(35^\circ) = \frac{\text{adj}}{\text{hyp}}$	This reverses the ratio for sine; sine pairs opposite (not adjacent) with hypotenuse.	Use $\sin(35^\circ) = \frac{\text{opp}}{\text{hyp}}$ instead; remember SOH (Sine = Opposite over Hypotenuse).
Using $\sin(90^\circ) = \frac{\text{opp}}{\text{hyp}}$ to find a side length in a right triangle	The 90° angle is the right angle of the triangle, not one of the acute angles SOH-CAH-TOA is designed for; $\sin(90^\circ) = 1$ , which doesn't help you find sides.	Always use one of the acute angles given in the problem, not the right angle.
Setting up $\sin(35^\circ) = \frac{\text{opp}}{10}$ , then solving $\text{opp} = \frac{\sin(35^\circ)}{10}$	The algebra is backwards; you need to multiply by 10, not divide, to isolate the opposite side.	Multiply both sides by 10: $\text{opp} = 10 \cdot \sin(35^\circ)$ .

Tips and when to use something else

SOH-CAH-TOA works for right triangles only; check that you have a right angle before using these ratios.
If you have two sides but no angle, use Inverse Trigonometric Functions (like $\sin^{-1}$ ) to find the angle first.
For non-right triangles, use the Law of Sines or Law of Cosines instead of SOH-CAH-TOA.
Always label sides relative to your chosen angle: opposite is across from it, adjacent is next to it, and hypotenuse is always the longest side opposite the right angle.

Related formulas

Pythagorean IdentityThe Pythagorean Identity states that sine squared plus cosine squared always equals one, helping you simplify trigonometric expressions and solve equations.Law of SinesLaw of Sines connects side lengths to their opposite angles in a triangle, used to find unknown sides or angles when an angle-side pair is known.Law of CosinesFinds unknown sides or angles in any triangle when you know two sides and the angle between them, or all three sides.Inverse Trigonometric FunctionsInverse trigonometric functions find angles when you know their sine, cosine, or tangent values; use them to solve equations where an angle is unknown.Unit CircleThe unit circle is a fundamental tool that shows how angles map to sine and cosine coordinates: the point (cos θ, sin θ) always lies on x² + y² = 1.

Frequently asked questions

What does SOH-CAH-TOA stand for?: SOH-CAH-TOA is a mnemonic: $\sin(\theta) = \frac{\text{opp}}{\text{hyp}}$ , $\cos(\theta) = \frac{\text{adj}}{\text{hyp}}$ , $\tan(\theta) = \frac{\text{opp}}{\text{adj}}$ . It helps you remember which sides pair with each trig function for a given acute angle in a right triangle.
When should I use tangent instead of sine or cosine?: Use tangent when you know (or want to find) only the opposite and adjacent sides, but not the hypotenuse. Tangent is the best choice when the hypotenuse is not given, not needed, or would make the problem more complicated.
Do these ratios work for all angles, or just 0°–90°?: In the context of right triangles and SOH-CAH-TOA, these ratios apply only to the acute angles (greater than 0° and less than 90°). For other angles, you would use the unit circle or inverse trig functions.
Can I use these ratios if I only know two sides and no angles?: Not directly; you would need to use Inverse Trigonometric Functions like $\sin^{-1}$ , $\cos^{-1}$ , or $\tan^{-1}$ to find the angle first, then use SOH-CAH-TOA to find the missing side.

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Reviewed 2026-09-18