SOHCAHTOA Explained: Sine, Cosine, and Tangent for Right Triangles

SOHCAHTOA is the mnemonic that unlocks all of right-triangle trigonometry. The six letters stand for the definitions of the three primary trigonometric ratios:

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

This guide explains what each ratio means geometrically, walks through how to identify “opposite” and “adjacent” sides relative to a given angle, and works through enough examples that you can solve any right-triangle trig problem confidently.

The setup: a right triangle, and one of its acute angles

Trig ratios always start with a right triangle (one 90° angle) plus a choice of which acute angle (one of the other two, less than 90°) you want to focus on. Call that focus angle θ (theta).

Relative to θ, the three sides of the triangle have names:

• Hypotenuse: the longest side, opposite the right angle. (Same for both acute angles.)
• Opposite: the side across from the angle θ. Not touching θ at either end.
• Adjacent: the side that touches θ (but is not the hypotenuse).

If you switch focus to the other acute angle, “opposite” and “adjacent” swap. The hypotenuse stays the same.

The three ratios

For the chosen angle θ:

sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent

The values these ratios produce depend ONLY on the angle θ, not on the size of the triangle. Two triangles with the same θ but different scales have the same sin/cos/tan values for that angle. This is what makes trig ratios universal — they let you convert between angles and side ratios in any right triangle.

Worked Example 1 — Finding a side from an angle

A right triangle has a hypotenuse of 10 and one acute angle of 30°. Find the side opposite the 30° angle.

Use SOH (sine):

sin(30°) = opposite / 10
0.5 = opposite / 10
opposite = 5

The opposite side is 5.

(We know sin(30°) = 0.5 exactly because the 30° angle in a 30-60-90 triangle has opposite/hypotenuse = 1/2.)

Worked Example 2 — Finding an angle from sides

A right triangle has opposite = 4 and adjacent = 3 relative to the angle we want.

Use TOA (tangent):

tan(θ) = 4 / 3 ≈ 1.333
θ = arctan(1.333) ≈ 53.13°

This is a famous triangle: the 3-4-5 right triangle. Its non-right angles are approximately 36.87° (opposite the 3-side) and 53.13° (opposite the 4-side).

Choosing which ratio to use

The choice depends on which sides and angles are involved:

Memorizing this table is overkill. The faster habit: identify which two of {opposite, adjacent, hypotenuse} appear in the problem, then pick the ratio that uses exactly those two.

The three exact-value angles to memorize

For 30°, 45°, and 60°, the sin/cos/tan values are exact and worth memorizing:

These exact values come directly from the 30-60-90 and 45-45-90 special right triangles. The “Special Right Triangles Calculator” page derives them in detail.

Inverse trig: getting an angle back from a ratio

If you know a sin/cos/tan value and want to recover the angle, use the inverse functions:

• arcsin (also written sin⁻¹): input a ratio, output an angle. For example, arcsin(0.5) = 30°.
• arccos (cos⁻¹): arccos(0.5) = 60°.
• arctan (tan⁻¹): arctan(1) = 45°.

On a calculator, these are usually labeled sin⁻¹, cos⁻¹, tan⁻¹ (often the SHIFT + sin / cos / tan combination). Make sure your calculator is in degree mode for right-triangle work — radian mode gives different numerical answers for the same input.

Common mistakes

• Using sin/cos/tan with the wrong angle. The “opposite” and “adjacent” labels depend on which angle you’ve picked. Confusing them is the #1 SOHCAHTOA error.
• Wrong calculator mode. sin(30°) ≈ 0.5 if calculator is in degree mode. In radian mode, sin(30) ≈ −0.988 — totally different. Always check mode first.
• Writing sin⁻¹(x) as 1/sin(x). sin⁻¹ is the inverse function (arcsin), not the reciprocal. The reciprocal of sin is csc (cosecant): csc(x) = 1/sin(x).
• Trying to use SOHCAHTOA on non-right triangles. These ratios are defined for right triangles. For general triangles, use the Law of Sines or Law of Cosines instead.

Beyond the right triangle

SOHCAHTOA defines sin/cos/tan only for acute angles in a right triangle. The unit-circle definition generalizes these functions to all real numbers, including negative angles and angles greater than 90°. But for nearly all introductory geometry and trig homework, SOHCAHTOA is the foundation.

Try it yourself

The Triangle Solver applies SOHCAHTOA, the Law of Sines, and the Law of Cosines automatically. Input any three values (with at least one side) and it derives the rest with full step-by-step working. For specifically SOHCAHTOA practice problems, the Pythagorean Theorem Calculator handles right-triangle setups and the Special Right Triangles Calculator works with the 30-60-90 and 45-45-90 exact-value triangles where SOHCAHTOA simplifies dramatically.

FAQ

How do I remember which is sine and which is cosine? Some students remember “sine = opposite” by noticing that “sine” and “opposite” both have “o” + “i” patterns. Others just use the SOHCAHTOA mnemonic directly. Whatever sticks for you is fine.

What does sin(90°) = 1 mean physically? When the focus angle is 90°, the “opposite” side would be the hypotenuse itself — so opposite/hypotenuse = 1. The right angle’s sine is 1. Similarly cos(90°) = 0 because the “adjacent” side has shrunk to zero length.

Why do all six “tri” functions exist (sin, cos, tan, csc, sec, cot)? The last three are reciprocals: csc = 1/sin, sec = 1/cos, cot = 1/tan. They are less common in introductory work but appear in calculus and advanced trig identities.