Crash Course — Unit 3: Trigonometric and Polar Functions

• Periodic Functions
  €” Functions that repeat their y-values at regular intervals, which lets us model cyclical patterns like a person's height on a moving Ferris wheel.
• The Unit Circle
  €” The ultimate cheat sheet; a circle with radius 1 where any point's coordinates (x, y) directly give you cos(θ) and sin(θ).
• Sinusoidal Graphs
  €” The wave-like graphs of sine and cosine, created by "unwrapping" the unit circle's coordinates as the angle θ increases.
• Sinusoidal Transformations (A, B, C, D)
  €” The four parameters that control a wave's height (Amplitude), width (Period), and position (Phase & Vertical Shifts).
• Modeling with Sinusoids
  €” Using a sinusoidal function to create an equation that predicts real-world data, like the temperature in Chicago throughout the year.
• Tangent & Reciprocal Functions
  €” tan(θ), csc(θ), sec(θ), and cot(θ) are all defined by sine and cosine, creating graphs with unique features like vertical asymptotes.

• Inverse Trig Functions
  €” These functions work backward, finding the angle when you know the trig ratio (e.g., arcsin(0.5) = ?). They require a restricted range to be true functions.
• Trigonometric Identities
  €” Fundamental equations, like the Pythagorean Identity sin²(θ) + cos²(θ) = 1, that help you simplify expressions and solve equations.
• Solving Trig Equations
  €” Finding all possible angle solutions for an equation, which often involves using identities, factoring, and remembering the function's periodic nature.
• Polar Coordinates (r, θ)
  €” A new way to locate points using a distance from the origin (r) and an angle (θ), which is perfect for describing rotational or circular motion.
• Graphing Polar Functions
  €” Creating intricate graphs like cardioids and roses by plotting how a radius r changes in response to a sweeping angle θ.

Key Formulas / Terms

• Sinusoidal Function Form
  f(x) = A sin(B(x - C)) + D

  • Amplitude: |A| (distance from midline to max/min)
  • Period: 2π / |B| (length of one full cycle)
  • Phase Shift: C (horizontal shift)
  • Vertical Shift / Midline: y = D
• Unit Circle Definitions
  For a point (x, y) at angle θ on the unit circle:

  • cos(θ) = x
  • sin(θ) = y
  • tan(θ) = y/x
• Reciprocal & Quotient Identities
  • csc(θ) = 1/sin(θ)
  • sec(θ) = 1/cos(θ)
  • cot(θ) = 1/tan(θ) = cos(θ)/sin(θ)
• Pythagorean Identity
  sin²(θ) + cos²(θ) = 1
• Coordinate Conversion
  • Polar to Rectangular: x = r cos(θ) and y = r sin(θ)
  • Rectangular to Polar: r² = x² + y² and tan(θ) = y/x

Exam Traps

• Trap
  Confusing the period formula. Students often incorrectly use B as the period. · Counter: The period is 2π / |B|. Think of B as the frequency—the number of cycles that fit into a 2π interval. You always have to divide.
• Trap
  Messing up the phase shift when B isn't 1. For cos(2x + π), the shift is NOT -π. · Counter: Always factor out the B value first. cos(2x + π) becomes cos(2(x + π/2)). The phase shift is -π/2, or π/2 to the left.
• Trap
  Giving only one solution to a trig equation like cos(x) = 1/2. · Counter: Remember the repeating wave. First, find all solutions within one cycle, [0, 2π). In this case, x = π/3 and x = 5π/3. Then, add + 2πk to each to represent all possible solutions, unless the problem restricts the domain.
• Trap
  Forgetting the restricted ranges for inverse trig functions. · Counter: Burn this into your memory: arcsin(x) and arctan(x) answers must be in [-π/2, π/2] (Quadrants I & IV). arccos(x) answers must be in [0, π] (Quadrants I & II). Your calculator knows this, and you need to as well.
• Trap
  Mixing up rectangular and polar plotting. Plotting (r, θ) of (2, π/4) at the (x, y) point (2, 0.785). · Counter: Treat them as instructions. First, rotate to the angle θ (like pointing a compass). Then, move r units out from the origin along that angle line. For (2, π/4), you rotate 45 degrees, then walk 2 units out.