7.3 Unit Circle

The unit circle is a powerful tool for understanding trigonometric functions. It helps us visualize how sine and cosine values change as angles rotate around the circle. By memorizing key angles and their corresponding values, we can quickly solve trig problems.

Trigonometric functions have specific domains and ranges, which are crucial for graphing and solving equations. Reference angles simplify calculations for angles in different quadrants. Understanding these concepts allows us to tackle more complex problems involving angles and circular motion.

Unit Circle and Trigonometric Functions

Sine and cosine for common angles

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• Common angles in degrees and radians (30° = $\frac{\pi}{6}$, 45° = [object Object],[object Object], 60° = $\frac{\pi}{3}$)
• Sine values for common angles ($\sin 30° = \sin \frac{\pi}{6} = \frac{1}{2}$, $\sin 45° = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, $\sin 60° = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$)
• Cosine values for common angles ($\cos 30° = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$, $\cos 45° = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, $\cos 60° = \cos \frac{\pi}{3} = \frac{1}{2}$)
• Memorize these values to quickly evaluate trigonometric functions for common angles without using a calculator

Domain and range of trigonometric functions

• Domain of sine and cosine functions includes all real numbers in radians $(-\infty, \infty)$ or degrees $(-\infty°, \infty°)$
  • Angle can be any value as it represents rotation around the unit circle
• Range of sine and cosine functions limited to values between -1 and 1, inclusive [object Object],[object Object]
  • Unit circle has a radius of 1, sine and cosine values represent y-coordinate and x-coordinate of a point on the circle
• Understanding domain and range helps determine possible input and output values for trigonometric functions

Reference angles on unit circle

• Reference angle is the acute angle formed between terminal side of given angle and x-axis
• Finding reference angle depends on quadrant of given angle
  • Quadrant I: reference angle same as given angle
  • Quadrant II or III: subtract given angle from 180° or $\pi$ radians
  • Quadrant IV: subtract 360° or [object Object],[object Object] radians from given angle
• Examples of finding reference angle (120° reference angle is 60°, [object Object],[object Object] reference angle is $\frac{\pi}{4}$)
• Reference angles simplify evaluating trigonometric functions for angles in different quadrants

Trigonometric functions in all quadrants

• Evaluate sine and cosine for angles in different quadrants using reference angles and quadrant signs
  1. Determine reference angle
  2. Evaluate sine or cosine of reference angle
  3. Apply sign of function based on quadrant (Quadrant I: both positive, Quadrant II: sine positive, cosine negative, Quadrant III: both negative, Quadrant IV: sine negative, cosine positive)
• Example of evaluating $\sin 120°$
  • Reference angle is 60°
  • $\sin 60° = \frac{\sqrt{3}}{2}$
  • 120° in Quadrant II, sine is positive
  • Therefore, $\sin 120° = \frac{\sqrt{3}}{2}$
• Mastering trigonometric functions in all quadrants essential for solving complex problems involving angles and triangles

Unit Circle Properties and Motion

• The unit circle is centered at the origin (0,0) of the coordinate plane
• The radius of the unit circle is always 1 unit
• Circular motion around the unit circle represents the periodic nature of trigonometric functions
• The concept of periodicity in trigonometric functions is related to the complete rotation around the unit circle
• Angular velocity describes the rate of change of the angle as a point moves around the unit circle