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Pure · Trigonometry

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Graphs of sine, cosine and tangent

The shapes of y = sin x, cos x and tan x from the unit circle — period, amplitude, symmetry and asymptotes — and why reading them off the graph is how you find ALL the solutions to a trig equation.

Pure · Trigonometry

Graphs of sine, cosine and tangent

The shapes of y = sin x, cos x and tan x from the unit circle — period, amplitude, symmetry and asymptotes — and why reading them off the graph is how you find ALL the solutions to a trig equation.

Why it works

Sine and cosine are the shadow of a point going round in a circle. Picture a point on a circle of radius 11, starting on the right and turning anticlockwise through an angle xx. Its height above the centre is sin⁡x\sin x; its horizontal position is cos⁡x\cos x. As the point keeps going round, those two coordinates rise and fall between −1-1 and +1+1, repeating every full turn — every 360°360°. That is the whole shape of both graphs.
  • y=sin⁡xy = \sin x starts at 00, climbs to 11 at 90°90°, back to 00 at 180°180°, down
to −1-1 at 270°270°, and home to 00 at 360°360°.
  • y=cos⁡xy = \cos x starts at 11 (the point begins on the right), drops to 00 at
90°90°, to −1-1 at 180°180°, 00 at 270°270°, and back to 11 at 360°360°.

They are the same wave, shifted: cos⁡x=sin⁡(x+90°)\cos x = \sin(x + 90°). Both have period 360°360° (they repeat) and amplitude 11 (they reach ±1\pm 1).50100150200250300350-1-0.50.51x°y(Sine in indigo, cosine in pink.) Two symmetries save time: sine has rotational symmetry about the origin — it is odd, sin⁡(−x)=−sin⁡x\sin(-x) = -\sin x — while cosine is a mirror image in the yy-axis — it is even, cos⁡(−x)=cos⁡x\cos(-x) = \cos x.

Tangent is a different animal, because tan⁡x=sin⁡xcos⁡x\tan x = \dfrac{\sin x}{\cos x}. It is zero wherever sin⁡x=0\sin x = 0 (at 0°,180°,360°0°, 180°, 360°), but it blows up wherever cos⁡x=0\cos x = 0 (at 90°90° and 270°270°) — dividing by zero gives vertical asymptotes. Between them it sweeps from −∞-\infty to +∞+\infty, repeating every 180°180° (half the period of the others), with no amplitude — no ceiling.

Why this matters: an equation like sin⁡x=0.5\sin x = 0.5 doesn't have one answer, it has one per cycle, and your calculator only ever gives the first. The graph — or the symmetry it encodes — is how you find all the solutions in the range. It's the single most common place marks are dropped.