3.3 Circular Functions and Their Graphs

Sine and cosine as circular functions

On the unit circle, sin(θ) and cos(θ) give the y- and x-coordinates of a point traveling around the circle. As θ increases, these values cycle repeatedly, producing periodic wave graphs. Both functions have period 2π, amplitude 1, and range [-1, 1] since the unit circle has radius 1.

Transformations: amplitude, period, midline

For y = a·sin(b(x−h)) + k: |a| is the amplitude (vertical stretch), 2π/|b| is the period (horizontal stretch/compression), k shifts the midline vertically, and h shifts the graph horizontally (phase shift). Increasing |b| compresses the graph, making cycles happen faster.

Key features & tangent's behavior

Sine and cosine graphs have maximums, minimums, and zeros spaced evenly through each cycle, useful for modeling periodic phenomena like tides or sound waves. Tangent (sin/cos) has period π, no amplitude, and vertical asymptotes wherever cos(θ) = 0, since division by zero is undefined there.

Key terms

Amplitude of y = a·sin(bx)

|a| — half the distance between the maximum and minimum values.

Period formula for sine/cosine

2π/|b|, where b is the horizontal stretch/compression factor.

Midline of y = a·sin(x) + k

y = k, the horizontal line halfway between max and min.

Tangent function key features

Period π, no amplitude, vertical asymptotes where cos(x) = 0.

Relationship between sine and cosine graphs

cos(x) = sin(x + π/2); cosine is sine shifted left by π/2.

Try an AP-style question

What is the period of y = sin x?

1. π
2. π/2
3. 2π
4. 1