  ### Home > A2C > Chapter 8 > Lesson 8.1.6 > Problem8-93

8-93.

You have seen that you can calculate values of the sine function using right triangles formed by a radius of the unit circle. Values of $θ$ that result in $30^\circ − 60^\circ − 90^\circ$ or $45^\circ − 45^\circ − 90^\circ$ triangles are used frequently on exercises and tests because their sines and cosines can be found exactly, without using a calculator. You should learn to recognize these values quickly and easily. The same is true for values of cosθ and sinθ that correspond to the $x$- and $y$-intercepts of the unit circle.

The central angles that correspond to these “special” values of $x$ are $30^\circ, 45^\circ, 60^\circ, 90^\circ, 120^\circ, 135^\circ, 150^\circ, 180^\circ, 210^\circ, 225^\circ, 240^\circ, 270^\circ, 300^\circ, 315^\circ, \text{and}\ 330^\circ$. What these angles have in common is that they are all multiples of $30^{\circ}\ \text{or}\ 45^{\circ}$, and some of them are also multiples of $60^\circ\ \text{or}\ 90^\circ$.

Copy and complete a table like the one below for all special angles between $0^{\circ}\ \text{and}\ 360^{\circ}$.

 Degrees $0$ $30$ $45$ $60$ $90$ $120$ Radians $0$ $\frac { \pi } { 6 }$

Recall that $180^{^{\circ}}$ is $\pi$ radians.  Since $45$ is $\frac{1}{4}$ of $180$,  $45^{^{\circ}}$ is $\frac{1}{4}$ of $\pi$, or $\frac{\pi}{4}$.

All of the 'special' values of $x$ are multiples of either $30^{\circ}\ \text{or}\ 4^{\circ}$.
Knowing just these two values allows you to easily find the rest.

For Example, $225 \div 45 = 5. \text{ So } 225° = 5 \left( \frac{\pi}{4} \right) = \frac{5 \pi}{4} .$

 Degrees $0$ $30$ $45$ $60$ $90$ $120$ $180$ $270$ Radians $0$ $\frac{\pi}{6}$ $\frac{\pi}{4}$ $\pi$ $\frac{3\pi}{2}$