Trig graphs

Smooth waves between −1 and 1: y = sin x and y = cos x are smooth waves with range −1 ≤ y ≤ 1 and period 360° (2π) — they repeat every full turn. sin starts at 0 (rising); cos starts at 1.

Same wave, shifted: cos x is just sin x shifted left by 90° — same shape, different starting point.

Repeats every 180°, with asymptotes: y = tan x is different: it has period 180° (π), no maximum or minimum (its range is all real numbers), and vertical asymptotes where cos x = 0 (at 90°, 270°, …).

Why the asymptotes?: tan x = sin x / cos x, so wherever cos x = 0 the function blows up — those are the vertical asymptotes.

No amplitude: tan has no amplitude (it isn't bounded). Amplitude only applies to sin and cos.

Height and repeat-length: Amplitude = half the distance from max to min (how tall the wave is). Period = how far along the x-axis before it repeats. Read both straight off a graph.

Amplitude from max & min: Amplitude = (max − min)/2; the midline sits at (max + min)/2.

Maxes, mins, and crossings: On a basic sine wave: maximum at 90°, zero at 0°/180°/360°, minimum at 270°. Cosine: max at 0°/360°, zeros at 90°/270°, min at 180°. Knowing these anchors lets you sketch quickly.

Use symmetry: Sine and cosine are symmetric — once you know one max and the period, the rest follow by spacing.