# SohCahToa

SohCahToa is an acronym that serves as a mnemonic for recalling the right triangle definitions of the three trigonometric functions: sine, cosine, and tangent.

## Sides of a right triangle

The three sides of a triangle are named: adjacent, opposite, hypotenuse. They correspond to the following sides of a right triangle:

- Adjacent - the adjacent side of a right triangle is the side next to the given angle that is not the hypotenuse.
- Opposite - the opposite side of a right triangle is the side opposite the given angle.
- Hypotenuse - the hypotenuse of a right triangle is the longest side of the right triangle and always lies across from the right angle of the triangle.

The right triangle below shows the adjacent (a), opposite (o), and hypotenuse (h) relative to a given angle θ.

## SohCahToa formula

The mnemonic SohCahToa is used to help us remember the formulas for solving a right triangle using the trigonometric functions sine, cosine, and tangent. Each set of 3 letters gives us 1 right triangle formula for each of the 3 trigonometric functions:

Soh:

Cah:

Toa:

## When to use SohCahToa

To use SohCahToa, we need to know certain measures of a right triangle:

- We can use SohCahToa if we know two sides of a right triangle.
- We can use SohCahToa if we know one side and one angle other than the right angle.

Given the first case where we know two sides, it is also possible to find the third side of the triangle using the Pythagorean theorem, a^{2} + b^{2} = c^{2}. In the case of a right triangle we can replace the variables a, b, and c with o, a, and h (o and a can be substituted for either a or b).

## How to use SohCahToa

To use SohCahToa, given that the appropriate measures of the right triangle are known (as described above), plug the known information into the appropriate SohCahToa formula. For example, if we know the angle, θ, and the opposite side of the triangle, we would use sine to find the hypotenuse, since . Since we know two of the values, we can use algebra to find the hypotenuse. Once we know the hypotenuse, we can then similarly use Cah to find the adjacent side. Below are some SohCahToa examples.

Example:

Find sin(θ), cos(θ), and tan(θ) for the right triangle below.

First, find the hypotenuse, h, using the Pythagorean Theorem, where a = 6 and o = 10.

Now,

Example:

Find the hypotenuse and adjacent side for the right triangle below.

We'll find h first.

h = 8

We'll use tangent to find a next.

You could also use the cosine function to find a since you have the value for c. You could even use the Pythagorean Theorem to find a since you know the other two sides of the triangle.