📝 Chapter 5: Trigonometric Functions

State amplitude, period, phase shift, vertical shift for y = -3sin(2x - π/2) + 5

Sketch two full cycles of y = 4cos(πx/3) - 1. Label max, min, and zeros.

Write the equation of the reciprocal functions: a) csc(x) in terms of sin(x) b) State domain restrictions for sec(x)

Solve on [0, 2π): a) 2sin(x) + √3 = 0 b) cos²(x) = 3/4

Solve: tan²(x) - 3 = 0 on [0, 2π). State all solutions.

Solve: 2sin²(x) - sin(x) - 1 = 0 on [0, 2π). Explain your method.

Find all solutions to sin(2x) = cos(x) on [0, 2π). Show algebraic steps.

Two sinusoidal functions intersect at the origin. One has period π and the other has period 2π/3. When is the next point where they intersect?

Determine the equation of a sinusoidal function that has: max at (π/4, 7), min at (3π/4, 1), and is a sine function.

Compare and contrast the graphs of y = sin(x) and y = csc(x). Include: domain, range, period, asymptotes, and how one is derived from the other.

Explain how to determine the equation of a sinusoidal function from its graph. Use a specific example with a labeled diagram.

A student says "sin(x) = 0.5 has two solutions." In what context is this true? In what context could it have infinitely many solutions? Explain clearly.

Write a guide explaining how the parameters a, k, d, c in y = a·sin(k(x-d))+c each affect the graph. Include a transformation table.

The depth of water in a harbour is modeled by d(t) = 4sin(πt/6) + 8, where d is in metres and t is hours after midnight. a) What is the maximum depth and when does it first occur? b) A boat needs 5m clearance. During what hours can it safely enter?

The temperature in a city varies sinusoidally over a year. The max temperature is 28°C in July (month 7) and the min is -8°C in January (month 1). a) Write a cosine model T(m). b) During which months is temperature above 20°C?

A pendulum swings according to θ(t) = 0.2cos(4πt), where θ is the angle in radians and t is in seconds. a) What is the period of one swing? b) How many complete swings in 10 seconds?

Two musicians play notes modeled by y₁ = sin(440·2πt) and y₂ = sin(442·2πt). The "beat" they hear has frequency |440-442|/2 = 1 Hz. Explain what happens when you add these two waves (use the sum-to-product identity if possible).