### Home > INT3 > Chapter 9 > Lesson 9.1.6 > Problem9-79

9-79.

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 frequently used because their sine and cosine values can be expressed exactly, without using a calculator. You should learn to recognize these values when you see them. The same is true for values of $\text{cos}\left(θ\right)$ and $\text{sin}\left(θ\right)$ that correspond to the $x$- and $y$-intercepts of the unit circle.

The central angles that correspond to these “special” values of $θ$ are $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ, 120^\circ, 135^\circ, 150^\circ, 180^\circ, 210^\circ, 225^\circ, 240^\circ, 270^\circ, 300^\circ, 315^\circ, \text{and }330^\circ$. What these angles have in common is that they are all multiples of $30^\circ$ or $45^\circ$.

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

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

$\text{Recall that}\:180^\circ\text{is } \pi\text{ radians. Since }45 \text{ is one-fourth of }180,45^\circ\text{ is one-fourth of }\pi,\text{or }\frac{\pi}{4}.$

All of the 'special' values of $x$ are multiples of either $30^\circ$ or $45^\circ$.
Knowing just these two values allows you to easily find the rest.

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

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