Unit Circle Notes

Unit Circle Basics

  • The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in the Cartesian coordinate system.

Angles

  • Common angles in degrees and radians:

    • 30° = π6\frac{\pi}{6} radians
    • 45° = π4\frac{\pi}{4} radians
    • 60° = π3\frac{\pi}{3} radians
    • 90° = π2{\pi}{2} radians
    • 180° = π\pi radians
    • 270° = {\frac{3\pi}{2} radians
    • 360° = 2π2\pi radians

  • Reference angles are used to find the trigonometric values of angles in different quadrants.

Coordinates on the Unit Circle

  • Coordinates are in the form (cosine, sine) or (x, y).

  • Key coordinates:

    • (0°, 1) or (0,1) : (1,0)(1, 0)
    • (90°, {\frac{\pi}{2}) or (\frac{π}{2}, ):):(0, 1)
    • (180°, \pi)or() or (\pi):):(-1, 0)
    • (270°, {\frac{3\pi}{2})or() or (\frac{3\pi}{2}):):(0, -1)

All Denominators are Two!

  • The statement "all denominators are two" likely refers to the common form of coordinates on the unit circle where many coordinates involve a denominator of 2, especially when dealing with square roots.

    • Example: \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)

CAST

  • CAST rule indicates which trigonometric functions are positive in each quadrant.
    • C (Quadrant IV): Cosine is positive.
    • A (Quadrant I): All trigonometric functions are positive.
    • S (Quadrant II): Sine is positive.
    • T (Quadrant III): Tangent is positive.

Finger Method

  • The "finger method" is a technique to remember the sine and cosine values for common angles (30°, 45°, 60°).

  • Always (Cosine, Sine)

Common Angles and Coordinates

  • 30° (\frac{\pi}{6}):

    • Positive values, coordinates: \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)

  • 45° (\frac{\pi}{4}):
    *Coordinates: \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)

  • 60° (\frac{\pi}{3}):

    • Coordinates: \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)

Reference points in radians.

  • \frac{\pi}{6},,\frac{5\pi}{6},,{\frac{7\pi}{6},,{\frac{11\pi}{6}

  • {\frac{\pi}{4},,{\frac{3\pi}{4},,{\frac{5\pi}{4},,{\frac{7\pi}{4}

  • {\frac{\pi}{3},,{\frac{2\pi}{3},,{\frac{4\pi}{3},,{\frac{5\pi}{3}$$