Trigonometric Functions

Graphing Sine & Cosine

Graphing sine and cosine functions and identifying amplitude and period.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

From a Circle to a Wave

Recall the unit circle from Algebra II: as an angle θ sweeps all the way around, the point (cos θ, sin θ) traces the circle once and returns to exactly where it started. Before reading on, think about what plotting sin θ against θ itself — not against cos θ — should look like as θ keeps increasing past a full revolution.

Since a full revolution returns sin θ to the exact same values it already passed through, the graph of y = sin(x) repeats forever — a genuinely periodic wave, not a one-time curve. The same is true for y = cos(x).

Definition — Amplitude, Period, and Midline

For y = a · sin(b(x − h)) + k (or cosine): the amplitude is |a| (how far the wave rises and falls from its center), the period is 2π/|b| (the horizontal length of one complete cycle), the midline is y = k (the wave's vertical center), and h is the phase shift.

Amplitude scales the unit circle's radius of 1 by a factor of |a|, stretching the wave's height without changing how often it repeats. The period formula comes from the fact that a full revolution is always 2π radians (from Algebra II's radian-measure reasoning) — multiplying the input by b compresses (or stretches) how quickly x sweeps through that same 2π-radian revolution, finishing a full cycle in 2π/b instead.

Worked Example — Graphing a Scaled Sine Function

Graph y = 3sin(x) and identify its amplitude and period. Amplitude: |3| = 3 (the wave rises to 3 and falls to −3). Period: 2π/1 = 2π (unchanged from the parent function, since b = 1).

Worked Example — Graphing a Compressed Sine Function

Graph y = 2sin(3x) and identify its amplitude and period. Amplitude: |2| = 2. Period: 2π/3 (the wave completes a full cycle three times faster than the parent function).

Worked Example — Graphing Cosine with a Phase Shift

Graph y = cos(x − π/2) and describe its transformation. Amplitude 1, period 2π (both unchanged), shifted right by π/2 (a phase shift), matching the standard cosine wave delayed by a quarter cycle.

Graph Visualizer

Trace modePoint mode (click graph to plot)

Show x^2

Color

Expression

Domain & range

x min

x max

y min

y max

Precision (decimal places)2

Evaluate a point

Evaluate at x =

x^2 = 0

Tip

Sketch one full cycle first, using the period to mark where it starts and ends, then repeat that same shape in both directions — a periodic function's entire infinite graph is just one cycle copied over and over.

Common Mistakes

Using b directly as the period, instead of 2π/b.
b controls how compressed or stretched the wave is, but the actual period is 2π divided by b — a larger b means a shorter period, not a longer one.
Confusing amplitude with the maximum y-value reached, when a vertical shift (k) is also present.
Amplitude is the distance from the midline to the peak, not the peak's actual height — with a vertical shift, the true maximum is k plus the amplitude, not the amplitude alone.

Key Takeaways

Sine and cosine graphs repeat forever, since the unit circle itself repeats every full revolution.
Amplitude scales the wave's height; period (2π/|b|) controls how quickly it repeats.
A phase shift moves the wave horizontally, the same way any function's horizontal shift works.

Summary

Sine and cosine graph as smooth, repeating waves directly derived from the unit circle. The next lesson graphs tangent, whose shape looks completely different from sine and cosine's smooth curve.

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