Sine, Cosine & Tangent Graphs
The unit circle turns sin, cos and tan into functions for every angle — periodic waves whose whole shape, symmetry and behaviour come straight from a point moving round a circle of radius 1.
From triangle to unit circle
So far, angles have lived inside triangles, capped at $90°$. But $\sin$, $\cos$ and $\tan$ are really functions defined for every angle — including obtuse, negative, and angles beyond $360°$. To see how, we move from the triangle to the unit circle: a circle of radius $1$ centred at the origin. For a point on the circle at angle $\theta$ (measured anticlockwise from the positive $x$-axis), its coordinates are exactly $(\cos\theta,\ \sin\theta)$.
That single definition unlocks everything. As $\theta$ sweeps around the circle, the $x$-coordinate traces out the cosine graph and the $y$-coordinate traces the sine graph — smooth waves that repeat every $360°$. The values simply keep going round, which is why the graphs are periodic.
Everything about the sine and cosine graphs falls out of the circle. They oscillate between −1 and 1 because that's the range of a coordinate on a unit-radius circle. They repeat every 360° because going once round returns you to the start. $\sin$ starts at $0$ (the point $(1,0)$ has height $0$) while $\cos$ starts at $1$ (that same point has $x$-coordinate $1$) — the two graphs are the same wave shifted by $90°$. You never have to memorise the graphs; you can read them straight off a point moving round the circle.
The graphs and their key features
$y = \sin\theta$: starts at $0$, peaks at $1$ (at $90°$), back to $0$ (at $180°$), down to $-1$ (at $270°$), repeats every $360°$.
$y = \cos\theta$: starts at $1$, down to $0$ (at $90°$), $-1$ (at $180°$), repeats every $360°$ — identical to sine but shifted left by $90°$.
$y = \tan\theta$: quite different from the other two — it repeats every $180°$ (not $360°$) and tends to $\pm\infty$ as $\theta$ approaches $90°, 270°, \ldots$ (vertical asymptotes), because $\tan\theta = \tfrac{\sin\theta}{\cos\theta}$ is undefined wherever $\cos\theta = 0$.
Sine and cosine: range $[-1,1]$, period $360°$, no asymptotes, defined for every real angle. Tangent: range is all real numbers, period $180°$, vertical asymptotes at $90°, 270°, \ldots$ wherever $\cos\theta = 0$. Knowing these three facts about each graph answers most "state the period / range" exam questions without a calculator.
Worked example 1
No calculator needed here — questions like this test whether you can picture the wave, not whether you can evaluate it. Sketch a quick wave in the margin before answering; it turns a memory question into a reading-off question.
Worked example 2
Rather than a calculator, use how the sine wave is built from the circle: it's symmetric about $90°$, and negative in the third and fourth quadrants.
Every angle's sine relates back to an acute reference angle (here $30°$), with the sign decided by which part of the graph — which quadrant — you're in. The next lesson turns this quadrant-by-quadrant sign pattern into a single memorable diagram. Learn to picture the wave and you rarely need to compute more than one value.
Worked example 3
It's tempting to apply the $360°$ period from sine and cosine out of habit — but tan repeats twice as often, every $180°$. Forgetting this is one of the most common slips when angles beyond the first quadrant appear.
Check yourself
Given that $\cos 25° = 0.906$ (3 s.f.), use the symmetry of the cosine graph to write down the value of $\cos 335°$, and explain why $\cos(-25°) = \cos 25°$.
(Hint: 335° = 360° − 25°; cosine is symmetric about θ = 0° and θ = 360°.)