3.2 The Unit Circle and Circular Functions

The unit circle is a powerful tool for understanding trigonometric functions. It's a circle with a radius of 1 centered at (0,0) that helps us visualize sine, cosine, and tangent. By using the unit circle, we can easily find trig values for any angle.

Key points on the unit circle correspond to common angles like 30°, 45°, and 60°. These coordinates help us calculate trig values quickly. The unit circle also shows how trig functions repeat every 360°, making it easier to work with large angles.

Understanding the Unit Circle

Role of the unit circle

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• Unit circle with radius 1 centered at origin (0, 0) defined by equation $x^2 + y^2 = 1$
• Geometrically represents sine, cosine, and tangent functions on coordinate plane
• X-coordinate corresponds to cosine, y-coordinate to sine of angle
• Ratio y/x gives tangent of angle
• Angles measured counterclockwise from positive x-axis
• Enables calculation of trigonometric values for any angle (0°, 30°, 45°, etc.)

Coordinates on unit circle

• Angle θ determines point (x, y) where x = $\cos θ$ and y = $\sin θ$
• Key points: (1, 0) at 0 rad, $(\sqrt{3}/2, 1/2)$ at π/6 rad, $(\sqrt{2}/2, \sqrt{2}/2)$ at π/4 rad
• Reference angles help find coordinates in other quadrants (II, III, IV)
• Negative angles measured clockwise from positive x-axis

Trigonometric values from unit circle

• Sine: y-coordinate, cosine: x-coordinate on unit circle
• Tangent: ratio of sine to cosine ($\tan θ = \sin θ / \cos θ$)
• Secant, cosecant, cotangent derived from sine and cosine
• Quadrant signs: (+,+) in I, (-,+) in II, (-,-) in III, (+,-) in IV
• Symmetry aids in finding values across quadrants

Periodicity in unit circle

• Trigonometric functions repeat values at regular intervals
• Sine and cosine period: 2π radians or 360°
• Tangent period: π radians or 180°
• Function values recur every full rotation
• Coterminal angles yield identical function values
• Periodicity helps find values for large angles (720°, 1080°)