Graphs of trigonometric functions
Also known as: sine and cosine waves, trig graphs
The graphs of sine and cosine are smooth, repeating waves that oscillate between −1 and 1 with period 2π, while tangent produces repeating curves with vertical asymptotes every π units.
The trigonometric functions produce periodic graphs — patterns that repeat forever. The graph of y = sin(x) is a smooth wave passing through the origin, rising to a maximum of 1, falling to a minimum of −1, and completing one full cycle every 2π units. The graph of y = cos(x) is the identical wave shifted left by π/2, so it starts at its maximum value of 1 when x = 0.
Two numbers describe these waves. The amplitude is the height from the midline to a peak — 1 for both parent functions — and the period is the length of one complete cycle, 2π for sine and cosine. In the general form y = a sin(bx), the amplitude is |a| and the period is 2π/|b|, so y = 3 sin(2x) oscillates between −3 and 3 and repeats every π units.
Tangent behaves differently. Because tan(x) = sin(x)/cos(x), the function is undefined wherever cosine equals zero, producing vertical asymptotes at x = π/2, 3π/2, and so on. Between consecutive asymptotes the graph sweeps from −∞ up through 0 to +∞, repeating with period π — half the period of sine and cosine — and it has no amplitude, since it is unbounded.
The ACT math section regularly tests these properties: identifying amplitude and period from an equation or graph, matching a transformed function like y = 2 cos(x) + 1 to its picture, and recalling that sine and cosine stay between −1 and 1 while tangent does not.
Key takeaways
- Sine and cosine graph as waves with amplitude 1 and period 2π; cosine is sine shifted left by π/2.
- In y = a sin(bx), the amplitude is |a| and the period is 2π/|b|.
- Tangent has period π, no amplitude, and vertical asymptotes wherever cosine equals zero.
- The ACT tests reading amplitude, period, and shifts from equations and graphs.