How to Use the Mnemonic 'SOHCAHTOA' in Trigonometry

A right-angled triangle, also known as a right triangle, has one angle that is exactly 90 degrees. The side opposite this right angle is the longest side, called the hypotenuse.

The other two sides are referred to as the opposite side and the adjacent side, depending on their relationship to the angle being measured or referenced.

Mathematicians need a way to solve for and relate the angles of a triangle to the length of its sides. To do this, they use the fundamental math functions called trigonometric functions, which have applications across science, engineering and everyday life.

Defined based on the ratios of the side lengths in a right-angled triangle, the trigonometric functions are:

These ratios depend on the angle θ, an acute angle (less than 90 degrees) in the triangle.

What Does SOHCAHTOA Stand for?

Here’s how the mnemonic device SOHCAHTOA helps mathematicians and math students remember the trigonometric functions and ratios.

Mathematicians have to calculate the unknown side lengths or angles in a right triangle all the time. To do this, they apply the trigonometric functions.

For example, if you know the value of angle θ, you can find the two sides of a right-angled triangle.

Trigonometric Function Example

Suppose you have a right triangle with:

You want to find the length of the opposite side b.

The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side, so:

You know that:

So, using the tangent function looks like this:

You know from trigonometric tables or by using a calculator that:

So:

Now, to find b:

The length of the opposite side b is approximately 2.885 units.

If you know the lengths of two sides of a triangle, you can find the angle θ using the inverse trigonometric functions:

Inverse Trigonometric Function Example

Suppose you have a right triangle with:

You want to find the angle θ.

The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side, so:

To find the angle θ, you can use one of the inverse trigonometric functions:

Using a calculator, you can then determine:

In other words, the angle θ is approximately 30.96 degrees, making it also an acute angle.

Another essential concept in trigonometry is the Pythagorean theorem, which Pythagoras, a philosopher in ancient Greece devised. It states:

where a and b are the lengths of the legs (opposite and adjacent sides), and c is the hypotenuse.

The Pythagorean theorem is useful when you know the two sides of a right triangle and are trying to find a missing side.

Pythagorean Theorem Calculator

To solve for missing sides, you can rearrange the trigonometric equations. For example:

Using a calculator, you can compute the sine, cosine or tangent of the angle θ, and then multiply by the known side length to find the unknown side.

Example Calculation

Consider a right-angled triangle where the hypotenuse is 10 units, and the angle θ is 30 degrees. To find the opposite side, you would use the sine function:

Since sin⁡30° =0.5:

Trig identities are equations involving trigonometric functions that are true for every value of the variable. They include fundamental identities like this basic Pythagorean identity:

Other common identities include angle sum and difference identities, double angle identities, half-angle identities, reciprocal identities and co-function identities. These identities are useful when you're trying to simplify trigonometric expressions and solving equations.

It's easy to view trigonometry as just a theoretical concept, but that's far from the case. It actually has practical applications across many different fields, including engineering, physics and astronomy, and even in everyday problem-solving scenarios like navigation and construction.

Engineers use trigonometric ratios to analyze forces in structures like bridges and buildings, or to design mechanical systems in engines. With these ratios, physicists can determine the trajectory, range and height of objects in projectile motion, and astronomers can calculate the necessary angles for precisely aligning their telescopes to specific celestial objects or events.

We created this article in conjunction with AI technology, then made sure it was fact-checked and edited by a HowStuffWorks editor.