# Trig ratios of special triangles

Learn to find the sine, cosine, and tangent of 45-45-90 triangles and also 30-60-90 triangles.
Until now, we have used the calculator to evaluate the sine, cosine, and tangent of an angle. However, it is possible to evaluate the trig functions for certain angles without using a calculator.
This is because there are two special triangles whose side ratios we know! These two triangles are the 45-45-90 triangle and the 30-60-90 triangle.

## The special triangles

30-60-90 triangles
A 30-60-90 triangle is a right triangle with a $30^\circ$ degree angle and a $60^\circ$ degree angle.
Sure! Watch the video below to see Sal derive these ratios.
45-45-90 triangles
A 45-45-90 triangle is a right triangle with two $45^\circ$ degree angles.
Sure! Watch the video below to see Sal derive these ratios.

## The trigonometric ratios of $30^\circ$

We are now ready to evaluate the trig functions of these special angles. Let's start with $30^\circ$.
Study the worked example below to see how this is done.

### What is $\sin(30^\circ)$?

Here's a worked example:
Step 1: Draw the special triangle that includes the angle of interest.
Since the trigonometric ratios of $30^\circ$are defined to be the ratios of the sides of a right triangle with a $30^\circ$ angle, we can start by drawing and labeling the sides of a $30$ - $60$ - $90$ triangle.
Step 2: Label the sides of the triangle according to the ratios of that special triangle.
Step 3: Use the definition of the trigonometric ratios to find the value of the indicated expression.
\begin{aligned} \sin (30^\circ) &= \dfrac{\text{opposite }}{\text{hypotenuse}} \\\\ &= \dfrac{x}{2x} \\\\ &= \dfrac{1\maroonD{\cancel{x}}}{2\maroonD{\cancel{x}}} \\\\ &=\dfrac{1}{2}\end{aligned}
Note that you can think of $x$ as $1 x$ so that it is clear that $\dfrac{x}{2x}=\dfrac{1x}{2x}=\dfrac12$.
Now let's use this method to find $\cos(30^\circ)$ and $\tan(30^\circ)$.

### What is $\cos(30^\circ)$?

Step 1: Draw the special triangle that includes the angle of interest.
Step 2: Label the sides of the triangle according to the ratios of that special triangle.
Step 3: Use the definition of the trigonometric ratios to find the value of the indicated expression.
\begin{aligned} \cos (30^\circ) &= \dfrac{\text{adjacent }}{\text{ hypotenuse} } \\\\ &= \dfrac{x\sqrt{3}}{2x} \\\\ &= \dfrac{\maroonD{\cancel{x}}\sqrt{3}}{2\maroonD{\cancel{x}}} \\\\ &=\dfrac{\sqrt{3}}{2}\end{aligned}

## The trigonometric ratios of $45^\circ$

Let's try this process again with $45^\circ$. Here we can start by drawing and labeling the sides of a 45-45-90 triangle.

### What is $\cos (45^\circ)$?

Remember, the cosine of an angle is the ratio of the adjacent side to the hypotenuse.
Use this definition along with the given triangle to help you find the value.

## The trigonometric ratios of 60$^\circ$

The process of deriving the trigonometric ratios for the special angles $30^\circ$, $45^\circ$, and $60^\circ$ is the same.
While we have not yet explicitly shown how to find the trigonometric ratios of $60^\circ$, we have all of the information we need!

## A summary

We have calculated the trig ratios for $30^\circ$, $45^\circ$, and $60^\circ$. The table below summarizes our results.
$\cos(\theta)$$\sin (\theta)$$\tan( \theta)$
$\theta =30^\circ$$\greenD{\dfrac{\sqrt{3}}{2}}$$\greenD{\dfrac12}$$\greenD{\dfrac{\sqrt{3}}{3}=\dfrac{1}{\sqrt{3}}}$
$\theta = 45^\circ$$\purpleC{\dfrac{\sqrt{2}}{2}=\dfrac{1}{\sqrt{2}}}$$\purpleC{\dfrac{\sqrt{2}}{2}=\dfrac{1}{\sqrt{2}}}$$\purpleC1$
$\theta = 60^\circ$$\greenD{\dfrac12}$$\greenD{\dfrac{\sqrt{3}}{2}}$$\greenD{\sqrt{3}}$
These values tend to occur often in advanced trigonometry problems. Because of this, it is helpful to know them.
Some people choose to memorize these values, but memorization is not necessary. In this article, you derived the values yourself, so hopefully you can re-derive them whenever you need them in the future.