Unit Circle Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Using the unit circle, find the exact values of sin⁡\sin, cos⁡\cos, and tan⁡\tan for θ=π2\theta=\dfrac{\pi}{2} and θ=π\theta=\pi.

Solution

  1. 1
    θ=π2\theta=\frac{\pi}{2}: point (0,1)(0,1) on unit circle. cos⁡π2=0\cos\frac{\pi}{2}=0, sin⁡π2=1\sin\frac{\pi}{2}=1, tan⁡π2=1/0\tan\frac{\pi}{2}=1/0 (undefined).
  2. 2
    θ=π\theta=\pi: point (−1,0)(-1,0). cos⁡π=−1\cos\pi=-1, sin⁡π=0\sin\pi=0, tan⁡π=0/(−1)=0\tan\pi=0/(-1)=0.

Answer

θ=π2\theta=\frac{\pi}{2}: cos⁡=0,sin⁡=1,tan⁡=\cos=0, \sin=1, \tan= undefined; θ=π\theta=\pi: cos⁡=−1,sin⁡=0,tan⁡=0\cos=-1, \sin=0, \tan=0
The axes of the unit circle give special values. At π/2\pi/2 (top of circle), x=0x=0 makes tan⁡\tan undefined. At π\pi (left of circle), y=0y=0 makes tan⁡=0\tan=0. These are the values at quadrantal angles.

About Unit Circle

The circle of radius 1 centered at the origin in the coordinate plane, used to define trigonometric functions for all angles.

Learn more about Unit Circle →

More Unit Circle Examples

Example 1 easy

Verify that the point [formula] lies on the unit circle and identify the angle [formula].

Example 2 medium

Find [formula], [formula], and [formula] for [formula] using the unit circle. Identify which quadran

Example 4 hard

Use the unit circle to prove the Pythagorean identity [formula] and derive [formula].

← All Unit Circle Examples Unit Circle Concept