Trigonometric Ratios and Graphs

In a right-angled triangle, $\sin \theta =\frac{\text{opp}}{\text{hyp}}$, $\cos \theta =\frac{\text{adj}}{\text{hyp}}$, $\tan \theta =\frac{\text{opp}}{\text{adj}}$ (SOHCAHTOA). But this only works for angles between $0^{\circ }$ and $90^{\circ }$.

The $\text{unit circle}$ — radius 1, centred at the origin — removes that limit. For any angle $\theta$ measured from the positive $x$-axis, the point on the circle has coordinates $(\cos \theta ,\sin \theta )$. Now $\sin$ and $\cos$ are defined for every angle, including obtuse, reflex and negative ones, and their signs follow the quadrant the point lands in.

Plot those coordinates against the angle and you get the trig graphs. $y=\sin x$ and $y=\cos x$ are smooth waves oscillating between $-1$ and $1$, repeating every $360^{\circ }$ — they have $\text{amplitude}$ 1 and $\text{period}$ $360^{\circ }$. They are identical apart from a shift: $\cos$ is $\sin$ moved $90^{\circ }$ left.

$y=\tan x$ is different: it repeats every $180^{\circ }$, has no maximum or minimum, and shoots off to infinity at $90^{\circ },270^{\circ },\dots$ where $\cos x=0$ (giving vertical $\text{asymptotes}$).

The graphs’ symmetry links angles together. For instance $\sin (180^{\circ }-x)=\sin x$ and $\cos (-x)=\cos x$ — relationships you can read straight off the curves. These are what let you find every solution once you have one.

You are also expected to know the $\text{exact values}$ at the standard angles $0^{\circ },30^{\circ },45^{\circ },60^{\circ },90^{\circ }$ — the surd values such as $\sin 60^{\circ }=\frac{\sqrt{3}}{2}$ — which come from the $30$–$60$–$90$ and $45$–$45$–$90$ triangles.