Basic trigonometry

There are three trigonometric functions we’ll focus on here: sine, cosine, and tangent. These functions relate the side lengths of a right triangle. That means the definitions in this lesson apply only to right triangles.

SOH CAH TOA

The most common way to remember how to use these three trigonometric functions is the mnemonic SOH CAH TOA. In this mnemonic, S (SOH) stands for sine, C (CAH) stands for cosine, and T (TOA) stands for tangent. Each part of SOH CAH TOA tells you both which function you’re using and which sides to compare.

Because we only use these functions with right triangles, we’ll refer to the triangle below as we define each function:

In the diagram, the sides are labeled opposite, adjacent, and hypotenuse. These match the remaining letters in SOH CAH TOA (O, A, and H).

In each case, you’ll write an equation in this form:

• function $=$ (first side) divided by (second side)

We use the symbol $\theta$ (theta) to represent the angle.

Sine

Start with SOH. Replacing the letters with words gives Sine Opposite Hypotenuse. That tells you:

$$\sin (\theta )=\text{opposite}/\text{hypotenuse}$$

To find $\sin (\theta )$, substitute the side lengths into the formula. For example, if the opposite side is 4 and the hypotenuse is 8:

$$\sin (\theta )=4/8=1/2$$

Cosine

Next is CAH, which stands for Cosine Adjacent Hypotenuse:

$$\cos (\theta )=\text{adjacent}/\text{hypotenuse}$$

For example, if the adjacent side is $6$ and the hypotenuse is $8$:

$$\cos (\theta )=6/8=3/4$$

Tangent

Finally, TOA stands for Tangent Opposite Adjacent:

$$\tan (\theta )=\text{opposite}/\text{adjacent}$$

For example, if the opposite side is $4$ and the adjacent side is $6$:

$$\tan (\theta )=4/6=2/3$$

Theta ($\theta$)

It’s important to know where $\theta$ is in the triangle because the labels opposite and adjacent depend on which angle you’re using.

For example, if you use the top-right angle instead, the triangle would look like this:

Now the opposite and adjacent sides have swapped. That happens because they are relative to $\theta$:

• The opposite side is the side directly across from angle $\theta$.
• The adjacent side is the side next to angle $\theta$ (but not the hypotenuse).
• The hypotenuse is always the longest side, opposite the right angle, so it doesn’t change.

So, instead of memorizing where the sides appear in a picture, identify opposite and adjacent by looking at the angle you’re working with.

Trigonometric functions in algebra

In algebra, you’ll often use trig functions to relate an angle to side lengths.

• Sometimes you’ll be asked to find a side length when you know an angle and another side.
• Other times you’ll be asked to find the angle $\theta$.

This is the difference between finding a value like $\sin (\theta )$ and finding the angle $\theta$ itself. To solve for $\theta$, you use inverse trig functions.

Solving for a side

Solve for a side of a right triangle using trigonometric functions when you are given one side length and one angle. First, decide which trig function connects:

• a side you know, and
• the side you want to find.

For example:

• If you know the opposite side and want the adjacent side, use tangent (it involves opposite and adjacent).
• If you know the opposite side and want the hypotenuse, use sine (it involves opposite and hypotenuse).

Then set up the equation and solve for the unknown side.

Find the measure of side $x$ in the figure below.

Inverse trig functions

When you solve for $x$ in algebra, you use inverse operations to isolate the variable. For example, if $x$ is multiplied by a number, you divide by that number.

Inverse trig functions work the same way: they let you “undo” a trig function so you can isolate $\theta$. Before using them, make sure your calculator is in degrees.

Inverse trig functions are as follows:

• ${\sin }^{-1}(\theta )$
• ${\cos }^{-1}(\theta )$
• ${\tan }^{-1}(\theta )$

Let’s look at an example.

Find the value of $\theta$ for the following equation:

$$\cos (\theta )=2/3$$

Isolate theta by taking the inverse cosine of both sides:

$$\begin{array}{rl}\theta & ={\cos }^{-1}(2/3)\\ \theta & =48.2^{\circ }\end{array}$$

The measure of angle $\theta$ is $48.2$ degrees.

Key points

SOH CAH TOA. This is an acronym to remind you how to use the trigonometric functions for a right triangle. The first letter of the group indicates the function, while the other letters indicate the sides used in the equation. For functions, S stands for sine, C for cosine, and T for tangent. For sides, O stands for opposite, A stands for adjacent (or next-to), and H stands for hypotenuse.

Using SOH CAH TOA. The first group of three letters stands for Sine Opposite Hypotenuse to remind you that the sine function can be written $\sin (x)=\text{opposite}/\text{adjacent}$. Make sure your calculator is in degrees for all calculations.

Theta. Theta $\theta$ is the symbol used to represent an angle variable instead of $x$. The “opposite” and “adjacent” sides are determined relative to theta. “Opposite” is the side of the triangle that is opposite the angle theta.

Solving for a side. Solve for the value of a side in a right triangle by setting up the trigonometric function that involves a side you are given and the side you want to find. Then, solve for the value of the side.

Inverse trig functions. These are the opposite functions of the normal trig functions. We use them to isolate the variable inside of a trig function.