Radian

You already know that we are measuring angles in degrees, but in mathematics there is another type of measurement for angles and that is called radian.

One whole circle, $360°$ is equal to $2\pi rad$, and the half of the circle, $180°$ is equal to $\mathrm{\pi } \mathrm{rad}$. We can express any angles given in degrees using radians. For example $90°$ is equal to $\frac{\mathrm{\pi }}{2} rad$. 

We can convert between these two types of measurements. $\mathrm{\pi } \mathrm{rad}= 180°$ is the conversion number, similarly to converting hours to minutes with the conversion number $1$ hour$=60$ minutes.

If we want to change radian to degree, first we have to divide by $\pi$ and multiply by $180$.

 $degree=\frac{180\times radian}{\mathrm{\pi }}$

If we want to change degree to radian, first we have to divide by $180$ and multiply with $\pi$.

$radian=\frac{\mathrm{\pi }\times degree}{180}$

You don’t need to memorize these rules above, because if you know that $180°$ is equal to $\mathrm{\pi } \mathrm{rad}$ you will be able to deduce them.

Unit circle

Let’s see a circle with center $O$ and radius $1$. This is the unit circle. With the unit circle we can measure any angles.

There's a hand going around anti-clockwise. The measurement starts from the positive side of the horizontal axis, so that is $0$ degrees. When it is on the positive side of the vertical axis, it becomes $90$ degrees, or $\frac{\mathrm{\pi }}{2}$ radians.

If an angle is greater than $360$ degrees / $2\mathrm{\pi }$ radians, then it just means that the hand went around and it starts from $0$ again. If we take $1$ and a half circle we measured $360°+180°=540°$ or $2\mathrm{\pi } \mathrm{rad}+\mathrm{\pi } \mathrm{rad}= 3\mathrm{\pi } \mathrm{rad}$.

Using the unit circle you can find the sine and cosine of an angle. The projection of the end point of the hand belonging to an angel $x$ on the vertical axis is $\sin \left(x\right)$ and the projection of the endpoint of the hand belonging to the same angle $x$ on the horizontal axis is $\cos \left(x\right)$.

In other words $\cos \left(x\right)$ is the $x$ coordinate of the end point of the hand and $\sin \left(x\right)$ is the $y$ coordinate of the end point of the hand.

Exact values

It can be difficult to solve a trigonometric equation if you are not allowed to use your calculator. This is why we recommend you to memorize these exact values of trigonometric functions.

To be able to memorise the values better, we suggest that you memorise the options for sine and cosine. These are $0, \frac{1}{2}, \frac{1}{\sqrt{2}}, \frac{\sqrt{3}}{2}$  and $1$.

They are always in pairs, for example

when $\sin (30°)=\frac{1}{2}$  then $\cos (30°)=\frac{\sqrt{3}}{2}$  

when $\cos (60°)=\frac{1}{2}$  then $\sin (60°)=\frac{\sqrt{3}}{2}$.

For $45°$ they both give the same value, that is $\frac{1}{\sqrt{2}}$. The other two options, $0$ and $1$ are easy to read from the unit circle.

The values for $\tan \left(x\right)$ can be calculated by using the values of sine and cosine and considering $\tan \left(x\right)$ as $\frac{\sin \left(x\right)}{\cos \left(x\right)}$.