SOHCAHTOA Explained: Sine, Cosine and Tangent in Right Triangles

SOHCAHTOA is a memory aid for the three basic trigonometric ratios in a right triangle: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent. With it you can find any missing side or angle once you know one side and one acute angle, or two sides. For example, in a right triangle with a 35° angle and a 12-foot hypotenuse, the side opposite the angle is 12 × sin 35° ≈ 6.88 feet.

Naming the sides

Everything depends on labeling the sides relative to the angle you are using, called θ (theta).

• Hypotenuse: the longest side, opposite the right angle. It never changes.
• Opposite: the side directly across from θ.
• Adjacent: the side next to θ that is not the hypotenuse.

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

The three ratios

These ratios depend only on the angle, not on the size of the triangle. Every right triangle with a 35° angle has sin 35° ≈ 0.5736, whether it is drawn on paper or spans a canyon. That is what makes trigonometry useful for measuring things you cannot reach.

Choosing the right ratio

Pick the ratio that contains exactly the two sides involved: the one you have and the one you need.

Finding a missing side

When the unknown is on top

Angle 35°, hypotenuse 12. Find the opposite and adjacent sides.

Check with the Pythagorean theorem: 6.88² + 9.83² ≈ 47.3 + 96.6 ≈ 144 = 12² ✓.

When the unknown is on the bottom

Angle 35°, opposite side 6.88. Find the hypotenuse.

When the unknown is in the denominator, you divide instead of multiply. Writing out the ratio first, then rearranging, prevents mixing the two up.

Finding a missing angle

Use the inverse functions, written sin⁻¹, cos⁻¹ and tan⁻¹ (or arcsin, arccos, arctan). They answer “which angle has this ratio?”

The other acute angle is 90° − 29.05° = 60.95°, because a triangle’s angles sum to 180°. The inverse trig functions calculator returns angles in degrees or radians.

Real-world examples

Height of a tree

You stand 50 feet from a tree and measure a 32° angle of elevation to its top. Your eyes are 5.5 feet above the ground.

Forgetting to add eye height is the classic mistake in this problem.

A ladder

A 20-foot ladder leans at 75° to the ground. It reaches 20 × sin 75° ≈ 19.32 ft up the wall, and its foot sits 20 × cos 75° ≈ 5.18 ft out. That matches the common safety guideline of about a 4-to-1 height-to-base ratio, which corresponds to roughly 75°.

Angle of depression

From the top of a 40-foot lookout, you sight a boat at an 8° angle of depression, measured downward from horizontal. The horizontal line and the vertical drop form a right triangle with the 8° angle at your eye, so the boat is 40 ÷ tan 8° ≈ 285 feet from the base. Because the horizontal line at the top is parallel to the ground, the angle of depression from above equals the angle of elevation from the boat, which is often the easier way to draw the triangle.

A kite

With 100 feet of string at a 40° angle, a kite flies about 100 × sin 40° ≈ 64.3 feet above your hand, ignoring sag in the string.

Special angles worth memorizing

Two right triangles have exact, easy ratios: the 45-45-90 triangle (sides 1, 1, √2) and the 30-60-90 triangle (sides 1, √3, 2).

Notice that sin 30° = cos 60°: the sine of an angle equals the cosine of its complement. That is where the “co” in cosine comes from. The full table for every whole degree is on the trigonometric ratios table page, and the unit circle calculator extends these values beyond 90°.

The reciprocal functions

Three more functions are the reciprocals of the first three:

Most calculators have no csc, sec or cot keys. Compute them as reciprocals: cot 35° = 1 ÷ tan 35° ≈ 1.428. Do not use tan⁻¹ for this; that is the inverse function, which returns an angle.

Avoiding calculator errors

• Degree vs radian mode. sin 30 is 0.5 in degree mode and about −0.988 in radian mode. If an answer looks wildly wrong, check the mode first. For converting between the two, see how to convert degrees to radians.
• Sine and cosine can never exceed 1 in a right triangle, because the hypotenuse is the longest side. A ratio above 1 means the sides are mislabeled.
• Round at the end. Keep full precision through intermediate steps.

The trig functions calculator evaluates all six functions in either mode, and the right triangle calculator solves the whole triangle from any two values. For finding a side from the other two without any angle, use the Pythagorean theorem calculator or read how to use the Pythagorean theorem.