Unit circle

Definition

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is used to define sine, cosine, and tangent functions for all real numbers.

5 Must Know Facts For Your Next Test

1. The coordinates of any point on the unit circle are given by $(\cos(\theta), \sin(\theta))$ where $\theta$ is the angle formed with the positive x-axis.
2. The unit circle allows for easy evaluation of trigonometric functions at common angles such as $0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, and $\frac{\pi}{2}$.
3. Sine and cosine values repeat every $2\pi$ radians (360 degrees) due to the periodic nature of the unit circle.
4. In each quadrant of the unit circle, sine and cosine have specific signs: both positive in Quadrant I, sine positive and cosine negative in Quadrant II, both negative in Quadrant III, and sine negative and cosine positive in Quadrant IV.
5. Trigonometric identities like the Pythagorean identity ($\sin^2(\theta) + \cos^2(\theta) = 1$) can be derived from properties of the unit circle.

Review Questions

• What are the coordinates corresponding to an angle of $45^{\circ}$ (or $\frac{\pi}{4}$ radians) on the unit circle?
• How do you determine which trigonometric functions are positive or negative in each quadrant using the unit circle?
• Explain how to derive the Pythagorean identity using a point on the unit circle.

Related terms

Radian: A measure of angle based on arc length; one radian is equivalent to an angle subtended by an arc equal in length to the radius.

Sine Function: A trigonometric function that gives the y-coordinate of a point on the unit circle corresponding to a given angle.

Cosine Function: A trigonometric function that gives the x-coordinate of a point on the unit circle corresponding to a given angle.