SOHCAHTOA

SOHCAHTOA

A way of remembering how to compute the sine, cosine, and tangent of an angle.

SOH stands for Sine equals Opposite over Hypotenuse.

CAH stands for Cosine equals Adjacent over Hypotenuse.

TOA stands for Tangent equals Opposite over Adjacent.

Right triangle with angle θ at base. Sides labeled: hypotenuse, opposite side (vertical), adjacent side (horizontal). SOH: sin...

Example:

Find the values of sin θ, cos θ, and tan θ in the right triangle shown.

Right triangle with hypotenuse 5, vertical side 4, horizontal side 3, right angle at top-right, and angle θ at bottom-right.

Answer:

sin θ = 3/5 = 0.6

cosθ = 4/5 = 0.8

tanθ = 3/4 = 0.75

Right triangle with hypotenuse=5, opposite side=3, adjacent side=4, and angle θ at bottom right.

This triangle is oriented differently than the one shown in the SOHCAHTOA diagram, so make sure you know which sides are the opposite, adjacent, and hypotenuse.

Key Formula

\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}, \quad \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}, \quad \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}

Where:

$\theta$ = The acute angle in the right triangle you are examining
$\text{Opposite}$ = The side across from angle θ (it does not touch θ)
$\text{Adjacent}$ = The side next to angle θ that is not the hypotenuse
$\text{Hypotenuse}$ = The longest side of the right triangle, opposite the right angle

Worked Example

Problem: In a right triangle, the side opposite angle θ is 5, the side adjacent to angle θ is 12, and the hypotenuse is 13. Find sin θ, cos θ, and tan θ.

Step 1: Identify the three sides relative to angle θ. Opposite = 5, Adjacent = 12, Hypotenuse = 13.

Step 2: Apply SOH — Sine equals Opposite over Hypotenuse.

\sin\theta = \frac{5}{13} \approx 0.3846

Step 3: Apply CAH — Cosine equals Adjacent over Hypotenuse.

\cos\theta = \frac{12}{13} \approx 0.9231

Step 4: Apply TOA — Tangent equals Opposite over Adjacent.

\tan\theta = \frac{5}{12} \approx 0.4167

Answer: sin θ = 5/13 ≈ 0.385, cos θ = 12/13 ≈ 0.923, tan θ = 5/12 ≈ 0.417

Another Example

This example uses SOHCAHTOA to find an unknown side length rather than computing a ratio from known sides. It shows how to pick the correct ratio (SOH) based on the given and unknown quantities.

Problem: A ladder leans against a wall, forming a right triangle with the ground. The angle between the ladder and the ground is 40°. If the ladder (hypotenuse) is 10 metres long, how high up the wall does it reach?

Step 1: Draw the situation. The angle θ = 40° is between the ladder and the ground. The wall height is the side opposite the 40° angle, and the ladder is the hypotenuse.

Step 2: Choose the right ratio. You know the hypotenuse and want the opposite side, so use SOH: sin θ = Opposite / Hypotenuse.

\sin 40° = \frac{\text{height}}{10}

Step 3: Solve for the height by multiplying both sides by 10.

\text{height} = 10 \times \sin 40°

Step 4: Use a calculator: sin 40° ≈ 0.6428.

\text{height} \approx 10 \times 0.6428 = 6.428 \text{ m}

Answer: The ladder reaches approximately 6.43 metres up the wall.

Frequently Asked Questions

How do you remember which ratio is SOH, CAH, or TOA?

Read the mnemonic as three chunks: SOH, CAH, TOA. In each chunk, the first letter is the trig function (Sin, Cos, Tan), the second letter is the numerator (Opposite or Adjacent), and the third letter is the denominator (Hypotenuse or Adjacent). Some students remember the phrase 'Some Old Houses Can Always Hide Their Old Age' where the first letters match S-O-H-C-A-H-T-O-A.

Does SOHCAHTOA work for any triangle?

No. SOHCAHTOA applies only to right triangles, because the definitions of opposite, adjacent, and hypotenuse require a 90° angle. For non-right triangles, you need the Law of Sines or the Law of Cosines instead.

How do you decide which SOHCAHTOA ratio to use?

Look at what you know and what you need. If a problem involves the opposite side and the hypotenuse, use SOH (sine). If it involves the adjacent side and the hypotenuse, use CAH (cosine). If it involves the opposite and adjacent sides, use TOA (tangent). Pick the ratio that connects your known value to the unknown.

SOHCAHTOA (basic trig ratios) vs. Law of Sines / Law of Cosines

	SOHCAHTOA (basic trig ratios)	Law of Sines / Law of Cosines
Triangle type	Right triangles only	Any triangle (including non-right)
What you need	One acute angle and two sides of a right triangle	Various combinations of sides and angles in any triangle
Typical use	Finding a missing side or angle when a 90° angle is present	Solving triangles that have no right angle
Formulas	sin = O/H, cos = A/H, tan = O/A	a/sin A = b/sin B; c² = a² + b² − 2ab cos C

Why It Matters

SOHCAHTOA is one of the first tools you learn in trigonometry and appears constantly in geometry, physics, and engineering courses. Problems involving heights, distances, slopes, and angles of elevation almost always start with these three ratios. Mastering SOHCAHTOA also builds the foundation for the unit circle, inverse trig functions, and more advanced identities you will encounter later.

Common Mistakes

Mistake: Mixing up the opposite and adjacent sides when the triangle is rotated or drawn in an unusual orientation.

Correction: Always identify sides relative to the specific angle θ you are working with. The opposite side is directly across from θ, the adjacent side touches θ (and is not the hypotenuse), and the hypotenuse is always opposite the right angle.

Mistake: Using SOHCAHTOA on a triangle that has no right angle.

Correction: These ratios are defined only for right triangles. If no 90° angle is present, use the Law of Sines or Law of Cosines instead.

Related Terms

Sine — The 'SOH' ratio: opposite over hypotenuse
Cosine — The 'CAH' ratio: adjacent over hypotenuse
Tangent — The 'TOA' ratio: opposite over adjacent
Hypotenuse — Longest side; appears in SOH and CAH
Adjacent — Side next to the angle; used in CAH and TOA
Angle — θ determines which sides are opposite and adjacent
Side of a Polygon — Sides of the triangle used in the ratios