Chapter 14 – Trigonometry (JEE)
1. Introduction to Trigonometry
Trigonometry is the study of relationships between angles and sides of a triangle.
In JEE, trigonometry is not just a chapter—it is a **tool used everywhere**:
calculus, coordinate geometry, vectors, complex numbers, and physics.
2. Trigonometric Ratios
In a right-angled triangle:
| Ratio | Definition |
|---|---|
| $\sin\theta$ | $\dfrac{\text{Perpendicular}}{\text{Hypotenuse}}$ |
| $\cos\theta$ | $\dfrac{\text{Base}}{\text{Hypotenuse}}$ |
| $\tan\theta$ | $\dfrac{\text{Perpendicular}}{\text{Base}}$ |
| $\csc\theta$ | $\dfrac{1}{\sin\theta}$ |
| $\sec\theta$ | $\dfrac{1}{\cos\theta}$ |
| $\cot\theta$ | $\dfrac{1}{\tan\theta}$ |
3. Fundamental Trigonometric Identities
$\sin^2\theta + \cos^2\theta = 1$
$1 + \tan^2\theta = \sec^2\theta$
$1 + \cot^2\theta = \csc^2\theta$
These three identities are the **base of all trigonometric simplifications**.
4. Reciprocal and Quotient Identities
$\sin\theta = \dfrac{1}{\csc\theta}, \quad \cos\theta = \dfrac{1}{\sec\theta}$
$\tan\theta = \dfrac{\sin\theta}{\cos\theta}, \quad \cot\theta = \dfrac{\cos\theta}{\sin\theta}$
5. Trigonometric Ratios of Standard Angles
| $\theta$ | $0^\circ$ | $30^\circ$ | $45^\circ$ | $60^\circ$ | $90^\circ$ |
|---|---|---|---|---|---|
| $\sin\theta$ | 0 | $\frac12$ | $\frac{1}{\sqrt2}$ | $\frac{\sqrt3}{2}$ | 1 |
| $\cos\theta$ | 1 | $\frac{\sqrt3}{2}$ | $\frac{1}{\sqrt2}$ | $\frac12$ | 0 |
| $\tan\theta$ | 0 | $\frac{1}{\sqrt3}$ | 1 | $\sqrt3$ | Not defined |
6. Trigonometric Functions of Negative Angles
$\sin(-\theta) = -\sin\theta$
$\cos(-\theta) = \cos\theta$
$\tan(-\theta) = -\tan\theta$
$\sin\theta$ and $\tan\theta$ are **odd functions**,
$\cos\theta$ is an **even function**.
7. Allied Angles
$\sin(90^\circ-\theta)=\cos\theta$
$\sin(90^\circ+\theta)=\cos\theta$
$\sin(180^\circ-\theta)=\sin\theta$
$\sin(180^\circ+\theta)=-\sin\theta$
All allied angle formulas must be **memorised perfectly** for JEE.
8. Trigonometric Identities (Important for JEE)
$\frac{1}{\sin\theta} - \frac{1}{\csc\theta} = 0$
$(1+\tan\theta)(1-\tan\theta) = 1 - \tan^2\theta$
9. Trigonometric Equations
A trigonometric equation contains trigonometric functions of unknown angles.
$\sin x = \sin \alpha \Rightarrow x = n\pi + (-1)^n \alpha$
$\cos x = \cos \alpha \Rightarrow x = 2n\pi \pm \alpha$
10. General Solutions
Always write **general solutions** in JEE unless a specific interval is given.
11. Heights and Distances
Problems involving angles of elevation and depression are solved using trigonometry.
12. Trigonometric Inequalities
Trigonometric inequalities are solved using:
- Graphs
- Standard interval method
13. Periodicity of Trigonometric Functions
Period of $\sin x, \cos x$ = $2\pi$
Period of $\tan x, \cot x$ = $\pi$
14. Inverse Trigonometric Functions (Overview)
Inverse trigonometric functions undo trigonometric functions with restricted domains.
15. Common JEE Traps
- Forgetting sign of allied angles
- Ignoring domain in inverse functions
- Not writing general solution
- Mixing degrees and radians
16. Final Revision Checklist
You have mastered trigonometry if you can:
- Simplify any trigonometric expression
- Use identities correctly
- Solve trigonometric equations
- Handle allied angles confidently
- Apply trigonometry in word problems