The unit circle is a powerful tool for understanding trigonometric functions. It helps visualize how sine and cosine values change as angles rotate around the circle. By memorizing key angles and their coordinates, you can quickly find trig values for common angles.

Reference angles are a handy shortcut for finding trig values of any angle. By relating angles to their acute counterparts and applying quadrant rules, you can determine sine and cosine values for any angle on the unit circle.

Unit Circle and Trigonometric Functions

Sine and cosine for common angles

• Unit circle has radius 1 centered at origin (0, 0)
  • Angles measured counterclockwise from positive x-axis
  • x-coordinate of point on unit circle represents cosine of angle
  • y-coordinate of point on unit circle represents sine of angle
• Common angles in degrees and radian equivalents:
  • 30° = $\frac{\pi}{6}$ radians
  • 45° = $\frac{\pi}{4}$ radians
  • 60° = $\frac{\pi}{3}$ radians
• Sine and cosine values for common angles:
  • 30° or $\frac{\pi}{6}$: $\sin(\frac{\pi}{6}) = \frac{1}{2}$, $\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$
    • At 30°, point on unit circle is ($\frac{\sqrt{3}}{2}$, $\frac{1}{2}$)
  • 45° or $\frac{\pi}{4}$: $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$
    • At 45°, point on unit circle is ($\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$)
  • 60° or $\frac{\pi}{3}$: $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$, $\cos(\frac{\pi}{3}) = \frac{1}{2}$
    • At 60°, point on unit circle is ($\frac{1}{2}$, $\frac{\sqrt{3}}{2}$)

Domain and range of trigonometric functions

• Domain of sine and cosine functions is all real numbers
  • In radians, domain is $$(-\infty, \infty)$ $
  • In degrees, domain is also $(-\infty, \infty)$
  • Any angle can be plotted on unit circle
• Range of sine and cosine functions limited to interval [-1, 1]
  • x and y coordinates of point on unit circle cannot exceed 1 or be less than -1
  • For sine function: $-1 \leq \sin(\theta) \leq 1$
    • Minimum value of sine is -1 (270° or $\frac{3\pi}{2}$)
    • Maximum value of sine is 1 (90° or $\frac{\pi}{2}$)
  • For cosine function: $-1 \leq \cos(\theta) \leq 1$
    • Minimum value of cosine is -1 (180° or $\pi$)
    • Maximum value of cosine is 1 (0° or 0, 360° or $2\pi$)
• Sine and cosine are examples of periodic functions, repeating their values every 2π radians or 360°

Applying Reference Angles

Reference angles in unit circle

• Reference angles are acute angles (less than 90°)
  • Formed by terminal side of given angle and x-axis
  • Find reference angle by measuring smallest angle between terminal side and x-axis
• Quadrants and effect on sine and cosine values:
  1. Quadrant I (0° to 90°): both sine and cosine positive
     • All angles have same sine and cosine as reference angle
  2. Quadrant II (90° to 180°): sine positive, cosine negative
     • Sine same as reference angle, cosine opposite of reference angle
  3. Quadrant III (180° to 270°): both sine and cosine negative
     • Sine and cosine both opposite of reference angle
  4. Quadrant IV (270° to 360°): sine negative, cosine positive
     • Sine opposite of reference angle, cosine same as reference angle
• Steps to determine trigonometric function values using reference angles:
  1. Identify quadrant in which angle lies
  2. Find reference angle by measuring smallest angle between terminal side and x-axis
  3. Determine sine and cosine values for reference angle
  4. Apply sign rules based on quadrant to sine and cosine values found in step 3
     • Example: For 210°, reference angle is 30°. In Quadrant III, so sine and cosine are negatives of 30° values. $\sin(210°) = -\frac{1}{2}$, $\cos(210°) = -\frac{\sqrt{3}}{2}$

Advanced Trigonometric Concepts

• Trigonometric identities: Fundamental relationships between trigonometric functions that are always true
• Phase shift: Horizontal translation of a trigonometric function's graph
• Angular velocity: Rate of change of an angle with respect to time in rotating systems
• Parametric equations: Represent curves using functions of an independent parameter, often used with trigonometric functions to describe circular or periodic motion