In-Depth Notes on the Unit Circle

The Unit Circle

Definition of the Unit Circle

  • A unit circle is a circle with a radius of 1 unit.
  • Each point P(x, y) on the circle has the property that the radius r equals 1.

Trigonometric Functions Defined on the Unit Circle

  • The trigonometric functions sine, cosine, and tangent for an angle θ are defined as:
    • sin⁡(θ)=yr\sin(θ) = \frac{y}{r}
    • cos⁡(θ)=xr\cos(θ) = \frac{x}{r}
    • tan⁡(θ)=yx\tan(θ) = \frac{y}{x}
  • Since the radius is 1 ( =1"), these simplify to:
    • sin⁡(θ)=y\sin(θ) = y
    • cos⁡(θ)=x\cos(θ) = x
    • tan⁡(θ)=yx\tan(θ) = \frac{y}{x}

Special Angles on the Unit Circle

  • Key angles to memorize in degrees and their equivalent in radians include:
    • 0° (0 radians)
    • 30° (π/6 radians)
    • 45° (π/4 radians)
    • 60° (π/3 radians)
    • 90° (π/2 radians)
    • 180° (π radians)
    • 270° (3π/2 radians)
    • 360° (2π radians)
Quadrantal Angles
  • Angles of 0°, 90°, 180°, and 270° lie on the axes and are termed quadrantal angles.
Trigonometric Values for Special Angles
  • Here is a table of trigonometric values for some special angles:
    • | Angle | Degrees | Radians | sin⁡(θ)\sin(θ) | cos⁡(θ)\cos(θ) | tan⁡(θ)\tan(θ) |
      |---------|---------|-----------|--------------|---------------|----------------|
      | 0° | 0 | 0 | 0 | 1 | 0 |
      | 30° | π/6 | √3/2 | 1/2 | √3/3 |
      | 45° | π/4 | √2/2 | √2/2 | 1 |
      | 60° | π/3 | √1/2 | √3/2 | √3 |
      | 90° | π/2 | 1 | 0 | undefined |
      | 180° | π | 0 | -1 | 0 |
      | 270° | 3π/2 | -1 | 0 | undefined |
      | 360° | 2π | 0 | 1 | 0 |

Labeling the Unit Circle

  • When labeling the unit circle, each angle corresponds to specific ordered pairs based on the sine and cosine values:
    • For example:
    • (0°, 1) corresponds to (0, 1)
    • (90°, 0) corresponds to (1, 0)
    • (180°, -1) corresponds to (-1, 0)
  • Remember to adjust the coordinates based on which quadrant the angle lies in.

Summary of Signs in the Quadrants

  • The signs of trigonometric functions for each quadrant are:
    • 1st Quadrant: \sin(θ) > 0, \cos(θ) > 0, \tan(θ) > 0
    • 2nd Quadrant: \sin(θ) > 0, \cos(θ) < 0, \tan(θ) < 0
    • 3rd Quadrant: sin⁡(θ)<0,cos⁡(θ)<0,tan⁡(θ)>0\sin(θ) < 0, \cos(θ) < 0, \tan(θ) > 0
    • 4th Quadrant: \sin(θ) < 0, \cos(θ) > 0, \tan(θ) < 0