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 3030^\circ degree angle and a 6060^\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 4545^\circ degree angles.
Sure! Watch the video below to see Sal derive these ratios.

The trigonometric ratios of 3030^\circ

We are now ready to evaluate the trig functions of these special angles. Let's start with 3030^\circ.
Study the worked example below to see how this is done.

What is sin(30)\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 3030^\circare defined to be the ratios of the sides of a right triangle with a 3030^\circ angle, we can start by drawing and labeling the sides of a 3030 - 6060 - 9090 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.
sin(30)=opposite hypotenuse=x2x=1x2x=12\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 xx as 1x1 x so that it is clear that x2x=1x2x=12\dfrac{x}{2x}=\dfrac{1x}{2x}=\dfrac12.
Now let's use this method to find cos(30)\cos(30^\circ) and tan(30)\tan(30^\circ).

What is cos(30)\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.
cos(30)=adjacent  hypotenuse=x32x=x32x=32\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}

What is tan(30)\tan(30^\circ)?

The trigonometric ratios of 4545^\circ

Let's try this process again with 4545^\circ. Here we can start by drawing and labeling the sides of a 45-45-90 triangle.

What is cos(45)\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.

What is sin(45)\sin(45^\circ)?

What is tan(45)\tan (45^\circ)?

The trigonometric ratios of 60^\circ

The process of deriving the trigonometric ratios for the special angles 3030^\circ, 4545^\circ, and 6060^\circ is the same.
While we have not yet explicitly shown how to find the trigonometric ratios of 6060^\circ, we have all of the information we need!

What is cos(60)\cos(60^\circ)?

What is sin(60)\sin(60^\circ)?

What is tan(60)\tan(60^\circ)?

A summary

We have calculated the trig ratios for 3030^\circ, 4545^\circ, and 6060^\circ. The table below summarizes our results.
cos(θ)\cos(\theta)sin(θ)\sin (\theta)tan(θ)\tan( \theta)
θ=30 \theta =30^\circ32\greenD{\dfrac{\sqrt{3}}{2}}12\greenD{\dfrac12}33=13\greenD{\dfrac{\sqrt{3}}{3}=\dfrac{1}{\sqrt{3}}}
θ=45\theta = 45^\circ22=12\purpleC{\dfrac{\sqrt{2}}{2}=\dfrac{1}{\sqrt{2}}}22=12\purpleC{\dfrac{\sqrt{2}}{2}=\dfrac{1}{\sqrt{2}}}1\purpleC1
θ=60\theta = 60^\circ12\greenD{\dfrac12}32\greenD{\dfrac{\sqrt{3}}{2}}3\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.
Loading