Pre-Calculus - Week 10

The Unit Circle and Trigonometric Functions

The unit circle extends trigonometry beyond right triangles to all real numbers. Every angle — including negatives and those larger than 360° — has a sine and cosine given by the coordinates of a point on the unit circle. This definition is what calculus uses when it differentiates sin(x) and cos(x), and it opens the door to modeling waves, oscillations, and periodic motion.

Learning Objectives

- Convert between degrees and radians. - Define sine and cosine using the unit circle coordinates. - Evaluate trig functions at all standard unit circle angles. - State the period, amplitude, domain, and range of y = sin(x) and y = cos(x). - Graph y = A*sin(Bx + C) + D and y = A*cos(Bx + C) + D with all transformations. - Graph y = tan(x) and identify its period and asymptotes. - Define even and odd trig functions and apply symmetry.

1. Radians

Radian measure: the angle subtended by an arc equal in length to the radius. Conversion: Degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π. Key conversions: 0° = 0, 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2 120° = 2π/3, 135° = 3π/4, 150° = 5π/6, 180° = π 210° = 7π/6, 225° = 5π/4, 270° = 3π/2, 360° = 2π Example: Convert 120° to radians. 120 * π/180 = 2π/3. Convert 5π/4 to degrees: 5π/4 * 180/π = 225°.

2. The Unit Circle Definition

The unit circle is the circle x^2 + y^2 = 1 (center at origin, radius 1). For an angle θ in standard position (vertex at origin, initial side on positive x-axis), the terminal side intersects the unit circle at point (x, y). Definitions: cos(θ) = x-coordinate of the terminal point. sin(θ) = y-coordinate of the terminal point. tan(θ) = y/x = sin(θ)/cos(θ). This works for ANY angle — not just angles in right triangles.

3. Standard Unit Circle Values

Memorize these (x, y) = (cos θ, sin θ) pairs: θ=0: (1,0) θ=π/6: (sqrt(3)/2, 1/2) θ=π/4: (sqrt(2)/2, sqrt(2)/2) θ=π/3: (1/2, sqrt(3)/2) θ=π/2: (0,1) θ=2π/3: (-1/2, sqrt(3)/2) θ=3π/4: (-sqrt(2)/2, sqrt(2)/2) θ=5π/6: (-sqrt(3)/2, 1/2) θ=π: (-1,0) θ=3π/2: (0,-1) θ=2π: (1,0) The pattern: x-values (cosines) go from 1 down to 0 to -1 to 0 and back. The y-values (sines) trace the reverse pattern shifted by π/2.

4. Properties of Sine and Cosine

Period: both sin and cos repeat every 2π. sin(θ + 2π) = sin(θ) and cos(θ + 2π) = cos(θ). Symmetry: sin is ODD: sin(-θ) = -sin(θ). Symmetric about origin. cos is EVEN: cos(-θ) = cos(θ). Symmetric about y-axis. Domain: all real numbers. Range: [-1, 1]. Maximum of sin and cos: 1. Minimum: -1. Zeros of sin(θ): θ = nπ (n any integer). Zeros of cos(θ): θ = π/2 + nπ.

5. Graphing y = sin(x)

Key points for one period [0, 2π]: (0, 0), (π/2, 1), (π, 0), (3π/2, -1), (2π, 0). The sine wave starts at 0, rises to 1, returns to 0, drops to -1, returns to 0. Amplitude = 1 (max distance from midline). Period = 2π. Midline: y = 0. The shape repeats to the left and right forever.

6. Graphing y = A*sin(Bx + C) + D

A: amplitude = |A|. If A < 0, the graph is reflected over the midline. B: period = 2π/|B|. C: phase shift = -C/B (positive shifts left, negative shifts right). D: vertical shift (midline moves to y = D). Example: y = 3*sin(2x - π) + 1. Amplitude = 3. Period = 2π/2 = π. Phase shift = π/2 (right). Midline: y = 1. Key points: start at x = π/2 (phase shift), each quarter period is π/4.

7. Graphing y = cos(x) and Transformations

Key points for y = cos(x) on [0, 2π]: (0, 1), (π/2, 0), (π, -1), (3π/2, 0), (2π, 1). Cosine starts at its maximum, unlike sine which starts at 0. Example: Graph y = -2*cos(x/2). Amplitude = 2 (reflected). Period = 2π/(1/2) = 4π. Key points: (0, -2), (π, 0), (2π, 2), (3π, 0), (4π, -2). Relationship: cos(x) = sin(x + π/2). Cosine is just sine shifted left by π/2.

8. Graphing y = tan(x)

tan(x) = sin(x)/cos(x). Undefined when cos(x) = 0. Vertical asymptotes at x = π/2 + nπ. Period = π (half that of sin and cos). Domain: all reals except x = π/2 + nπ. Range: all real numbers. Key points on [-π/2, π/2] (one period): (-π/3, -sqrt(3)), (-π/4, -1), (0, 0), (π/4, 1), (π/3, sqrt(3)). The graph rises from -infinity near the left asymptote to +infinity near the right asymptote, passing through the origin.

9. Negative Angles and Coterminal Angles

Coterminal angles share the same terminal side. Add or subtract 360° (or 2π) to find them. Example: 390° and 30° are coterminal (390 - 360 = 30). Example: -45° and 315° are coterminal (-45 + 360 = 315). For evaluating trig: sin(-30°) = -sin(30°) = -1/2 (sine is odd). cos(-60°) = cos(60°) = 1/2 (cosine is even). sin(390°) = sin(30°) = 1/2 (coterminal with 30°).

10. Arc Length and Sector Area

For a circle of radius r and central angle θ (in radians): Arc length: s = rθ. Sector area: A = (1/2)r^2*θ. Example: A circle of radius 5 has a central angle of π/3. Arc length: s = 5*(π/3) = 5π/3 ≈ 5.24. Sector area: A = (1/2)(25)(π/3) = 25π/6 ≈ 13.09. Note: θ must be in radians for these formulas. If given degrees, convert first.

11. Common Mistakes

- Using degree formulas with radian arguments or vice versa. - Misreading the period: period = 2π/B, not 2π*B. - Forgetting the phase shift is -C/B (not just C). - Thinking the unit circle only covers 0° to 360°: it works for all real numbers. - Confusing the amplitude (always positive) with the leading coefficient A (which can be negative).