The six trigonometric functions are defined for any angle using the unit circle (radius $1$, centred at the origin). If the terminal side of angle $\theta$ meets the circle at the point $P(x, y)$, then:
The remaining three are reciprocals: $\csc\theta = \dfrac{1}{\sin\theta}$, $\sec\theta = \dfrac{1}{\cos\theta}$, $\cot\theta = \dfrac{1}{\tan\theta} = \dfrac{\cos\theta}{\sin\theta}$. Because $x$ and $y$ never exceed $1$ in magnitude on the unit circle, $-1 \le \sin\theta \le 1$ and $-1 \le \cos\theta \le 1$ always.
The sign of each function depends only on the signs of $x$ and $y$ in that quadrant — remembered as "All Silver Tea Cups" (All positive in Q1, Sin in Q2, Tan in Q3, Cos in Q4):
| Quadrant | Positive functions | $\sin$ | $\cos$ | $\tan$ |
|---|---|---|---|---|
| I ($0$ to $90^\circ$) | all | $+$ | $+$ | $+$ |
| II ($90$ to $180^\circ$) | $\sin,\csc$ | $+$ | $-$ | $-$ |
| III ($180$ to $270^\circ$) | $\tan,\cot$ | $-$ | $-$ | $+$ |
| IV ($270$ to $360^\circ$) | $\cos,\sec$ | $-$ | $+$ | $-$ |
From $x^2 + y^2 = 1$ on the unit circle come the three Pythagorean identities:
The sum and difference formulae let you break a compound angle apart:
Setting $B = A$ gives the double-angle formulae, with three equivalent forms for $\cos 2A$:
The last two forms of $\cos 2A$ rearrange into the half-angle identities $\sin^2 A = \dfrac{1 - \cos 2A}{2}$ and $\cos^2 A = \dfrac{1 + \cos 2A}{2}$.
Deeper Insight — one definition generates the entire formula sheet: Students often try to memorise dozens of identities as separate facts, but they all descend from a single source — the unit-circle point $(\cos\theta, \sin\theta)$. The Pythagorean identities are literally just $x^2 + y^2 = 1$ rewritten, and dividing that one equation by $\cos^2\theta$ or $\sin^2\theta$ produces the other two for free. The double-angle formulae are not new either: they are the sum formulae with $B$ replaced by $A$, and the three faces of $\cos 2A$ come from substituting $\sin^2 = 1 - \cos^2$ into one another. Even the signs across quadrants are not arbitrary rules to cram — they simply read off whether $x$ and $y$ are positive or negative where the terminal side lands. If you internalise the unit circle and the two sum formulae, you can reconstruct the rest in seconds under exam pressure, which is far safer than recalling a memorised list and hoping you got a sign right.