📐 Ratios & Standard Angles

The sine values simply count up across the five key angles as $0,\ \tfrac12,\ \tfrac{1}{\sqrt2},\ \tfrac{\sqrt3}{2},\ 1$:

• sin: $\sin0^\circ=0,\ \ \sin30^\circ=\tfrac12,\ \ \sin45^\circ=\tfrac{1}{\sqrt2},\ \ \sin60^\circ=\tfrac{\sqrt3}{2},\ \ \sin90^\circ=1$

• cos: the very same list read backwards — $\cos0^\circ=1,\ \ \cos30^\circ=\tfrac{\sqrt3}{2},\ \ \cos45^\circ=\tfrac{1}{\sqrt2},\ \ \cos60^\circ=\tfrac12,\ \ \cos90^\circ=0$

• tan: just sine divided by cosine — $\tan0^\circ=0,\ \ \tan30^\circ=\tfrac{1}{\sqrt3},\ \ \tan45^\circ=1,\ \ \tan60^\circ=\sqrt3,\ \ \tan90^\circ=\infty$ (undefined).

The other three ratios are simply flips of these. The reciprocal relations are $\sec\theta=\dfrac{1}{\cos\theta}$, $\csc\theta=\dfrac{1}{\sin\theta}$ and $\cot\theta=\dfrac{1}{\tan\theta}=\dfrac{\cos\theta}{\sin\theta}$. So once the table is in your head, every ratio is one division away.

Two angles that add to $90^\circ$ are complementary, and a $90^\circ-\theta$ angle swaps a ratio for its co-function: $\sin(90^\circ-\theta)=\cos\theta$, $\cos(90^\circ-\theta)=\sin\theta$, $\tan(90^\circ-\theta)=\cot\theta$ and $\sec(90^\circ-\theta)=\csc\theta$. This is why a product like $\tan\theta\cdot\cot(90^\circ-\theta)$ quietly becomes $\tan\theta\cdot\tan\theta$, and why pairs of angles that sum to $90^\circ$ cancel each other out.

One habit to drill: always square the value before multiplying. For instance $\tan^2 60^\circ=(\sqrt3)^2=3$, never $\sqrt3$ — forgetting this is the most common slip in the whole topic.