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

9-79.
1. 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 30º-60º-90º or 45º-45º-90º 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 cos(θ) and sin(θ) that correspond to the x- and y-intercepts of the unit circle. Homework Help ✎

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

Copy and complete a table like the one below for all special angles between 0º and 360º.

3.  Degrees 0 30 45 60 90 120 Radians 0

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

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

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