Introduction to Trigonometry
Cheat sheetClass 10 · Mathematics
⚑ Panic box
Core identity — $\sin^2 A+\cos^2 A=1$
Other identities — $1+\tan^2 A=\sec^2 A$; $1+\cot^2 A=\cosec^2 A$
Triangle sides — label opposite and adjacent from the chosen angle; hypotenuse is always opposite $90^\circ$
Special angles — know the $0^\circ,30^\circ,45^\circ,60^\circ,90^\circ$ table; check undefined entries
1. Ratios & side relationships
| Ratio | Definition in a right triangle | Reciprocal / link |
|---|---|---|
| $\sin A$ | opposite / hypotenuse | $\cosec A=1/\sin A$ |
| $\cos A$ | adjacent / hypotenuse | $\sec A=1/\cos A$ |
| $\tan A$ | opposite / adjacent | $\cot A=1/\tan A=\cos A/\sin A$ |
- Side labels — opposite and adjacent depend on the chosen acute angle; hypotenuse does not.
- Same angle, same ratios — changing the right triangle's size does not change its ratios (similar triangles).
- From one ratio to the rest — treat a given side ratio as lengths in scale $k$; use Pythagoras for the third side, take the positive length, then form each ratio.
2. Special-angle values
| $A$ | $0^\circ$ | $30^\circ$ | $45^\circ$ | $60^\circ$ | $90^\circ$ |
|---|---|---|---|---|---|
| $\sin A$ | $0$ | $\frac12$ | $\frac{1}{\sqrt2}$ | $\frac{\sqrt3}{2}$ | $1$ |
| $\cos A$ | $1$ | $\frac{\sqrt3}{2}$ | $\frac{1}{\sqrt2}$ | $\frac12$ | $0$ |
| $\tan A$ | $0$ | $\frac{1}{\sqrt3}$ | $1$ | $\sqrt3$ | undefined |
| $\cosec A$ | undefined | $2$ | $\sqrt2$ | $\frac{2}{\sqrt3}$ | $1$ |
| $\sec A$ | $1$ | $\frac{2}{\sqrt3}$ | $\sqrt2$ | $2$ | undefined |
| $\cot A$ | undefined | $\sqrt3$ | $1$ | $\frac{1}{\sqrt3}$ | $0$ |
- As $A$ rises from $0^\circ$ to $90^\circ$ — $\sin A$ rises from $0$ to $1$; $\cos A$ falls from $1$ to $0$.
- Range check — $\sin A,\cos A\leq 1$; $\sec A\geq1$ for $0^\circ\leq A<90^\circ$; $\cosec A\geq1$ for $0^\circ<A\leq90^\circ$.
3. Identities & conditions
| Identity | Valid angles | Useful rearrangement |
|---|---|---|
| $\sin^2 A+\cos^2 A=1$ | $0^\circ\leq A\leq90^\circ$ | $\sin^2 A=1-\cos^2 A$; $\cos^2 A=1-\sin^2 A$ |
| $1+\tan^2 A=\sec^2 A$ | $0^\circ\leq A<90^\circ$ | $\sec^2 A-\tan^2 A=1$ |
| $1+\cot^2 A=\cosec^2 A$ | $0^\circ<A\leq90^\circ$ | $\cosec^2 A-\cot^2 A=1$ |
- Identity — an equation true for every angle in the stated domain where its ratios are defined.
- Proving an identity — rewrite ratios using $\sin$ and $\cos$, or convert to the ratios in the identity you need; simplify one side to the other.
- Known ratio → other ratios — use reciprocal links and the identities; for an acute angle, choose positive square roots.
4. Finding a side or angle
| If you need | Then use |
|---|---|
| Opposite and hypotenuse | $\sin A$ |
| Adjacent and hypotenuse | $\cos A$ |
| Opposite and adjacent | $\tan A$ |
| A missing side in a right triangle | Pythagoras' theorem, or a ratio containing the known and unknown sides |
| A special angle from a ratio | Match the ratio in the table; for acute angles, equal sine or cosine values imply equal angles |
5. Traps
Opposite/adjacent are fixed triangle labels→ identify them again whenever the reference angle changes.$\cosec A=\sin^{-1}A$→ $\cosec A=1/\sin A$; inverse sine is different notation.$\sin A$ means $\sin\times A$→ $\sin A$ names the sine of angle $A$.$\tan A$ is always less than $1$→ it equals $1$ at $45^\circ$ and exceeds $1$ above $45^\circ$ up to $90^\circ$.All six ratios exist at $0^\circ$ and $90^\circ$→ division by zero makes $\cosec 0^\circ$, $\cot 0^\circ$, $\tan 90^\circ$, and $\sec 90^\circ$ undefined.$\sin^2 A=(\sin A)^2$ means $\sin(A^2)$→ the square applies to the ratio, not the angle.