Turn an angle into a wave.

One radian subtends an arc equal in length to the radius. A full turn is 2π radians. On the unit circle, the point at angle θ has coordinates ( $\cos\theta$ , $\sin\theta$ ), so signs depend on the quadrant.

In y = a sin(b(x − h)) + k, amplitude is |a| and period is 2π/|b|. The midline is y = k. When solving an equation, include every solution in the stated interval. Arc length uses s = rθ only when θ is in radians.

INTERACTIVE MODEL

Try it. Watch it change.

Angle θ / radians1

θ = 0.3183π = 57.2958° · cos θ = 0.5403 · sin θ = 0.8415

Use the angle slider or rotate the point. The circle has radius 1; horizontal and vertical projections give cosine and sine.

Explore: Move the angle slider around the circle, then switch to Sine graph to explore amplitude, period and shifts.

Degrees and radians

180^\circ=\pi\text{ radians}

Arc length

s=r\theta

Period of a sine or cosine function

T=\frac{2\pi}{|b|}

WORKED EXAMPLE

Solve $\sin\theta$ = $\frac{1}{2}$ for 0 ≤ θ ≤ 2π.

The reference angle is π/6.
Sine is positive in quadrants I and II.
θ = π/6 or $\frac{5\pi}6$ .

Assumed knowledge

Right-triangle ratios and graph transformations.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Convert radians and degrees.
Use unit-circle values and transformed sine and cosine graphs.

QCAA Mathematical Methods 2025 v1.3 · p. 18 ↗

READY TO TRY IT?

Turn an angle into a wave.

Solve sin⁡θ\sin\thetasinθ = 12\frac{1}{2}21​ for 0 ≤ θ ≤ 2π.

Put the idea to work.

Solve $\sin\theta$ = $\frac{1}{2}$ for 0 ≤ θ ≤ 2π.