Quick Revision Notes: Class 11 Mathematics

Chapter: Trigonometric Functions (त्रिकोणमितीय फलन)

An angle is the amount of rotation of a revolving ray about a fixed point.

Degree Measure: A full rotation is divided into $360^\circ$.
- $1^\circ = 60'$ (minutes)
- $1' = 60''$ (seconds)
Radian Measure: The angle subtended at the centre by an arc of length equal to the radius ($r$) of the circle is $1$ radian ($1^c$).
Relation between Degree and Radian:
- $\pi \text{ radians} = 180^\circ$
- $1 \text{ radian} = \frac{180^\circ}{\pi} \approx 57^\circ 16'$
- $1^\circ = \frac{\pi}{180} \text{ radians} \approx 0.01746 \text{ radians}$
Arc Length and Radius:
- $l = r\theta$ (where $\theta$ is in radians)

For a right-angled triangle with angle $\theta$:

$\sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{P}{H}$
$\cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{B}{H}$
$\tan \theta = \frac{\text{Perpendicular}}{\text{Base}} = \frac{P}{B} = \frac{\sin \theta}{\cos \theta}$
$\csc \theta = \frac{1}{\sin \theta} = \frac{H}{P}$
$\sec \theta = \frac{1}{\cos \theta} = \frac{H}{B}$
$\cot \theta = \frac{1}{\tan \theta} = \frac{B}{P} = \frac{\cos \theta}{\sin \theta}$

First Quadrant (I - All +): All trigonometric functions are positive. ($0$ to $\pi/2$)
Second Quadrant (II - Sin/Cosec +): $\sin \theta$ and $\csc \theta$ are positive, others are negative. ($\pi/2$ to $\pi$)
Third Quadrant (III - Tan/Cot +): $\tan \theta$ and $\cot \theta$ are positive, others are negative. ($\pi$ to $3\pi/2$)
Fourth Quadrant (IV - Cos/Sec +): $\cos \theta$ and $\sec \theta$ are positive, others are negative. ($3\pi/2$ to $2\pi$)

For angles like $(-\theta)$, $(\frac{\pi}{2} \pm \theta)$, $(\pi \pm \theta)$, $(2\pi \pm \theta)$, etc.:

$\sin(-\theta) = -\sin \theta$, $\cos(-\theta) = \cos \theta$, $\tan(-\theta) = -\tan \theta$
$\sin(\pi - \theta) = \sin \theta$, $\cos(\pi - \theta) = -\cos \theta$
$\sin(\pi + \theta) = -\sin \theta$, $\cos(\pi + \theta) = -\cos \theta$
$\sin(2\pi - \theta) = -\sin \theta$, $\cos(2\pi - \theta) = \cos \theta$
$\sin(\frac{\pi}{2} - \theta) = \cos \theta$, $\cos(\frac{\pi}{2} - \theta) = \sin \theta$
$\sin(\frac{\pi}{2} + \theta) = \cos \theta$, $\cos(\frac{\pi}{2} + \theta) = -\sin \theta$

$\sin(x + y) = \sin x \cos y + \cos x \sin y$
$\sin(x - y) = \sin x \cos y - \cos x \sin y$
$\cos(x + y) = \cos x \cos y - \sin x \sin y$
$\cos(x - y) = \cos x \cos y + \sin x \sin y$
$\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}$
$\tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}$
$\cot(x + y) = \frac{\cot x \cot y - 1}{\cot y + \cot x}$
$\cot(x - y) = \frac{\cot x \cot y + 1}{\cot y - \cot x}$
$\tan(x + y + z) = \frac{\tan x + \tan y + \tan z - \tan x \tan y \tan z}{1 - \tan x \tan y - \tan y \tan z - \tan z \tan x}$

Product into Sum/Difference:

$2 \sin x \cos y = \sin(x + y) + \sin(x - y)$
$2 \cos x \sin y = \sin(x + y) - \sin(x - y)$
$2 \cos x \cos y = \cos(x + y) + \cos(x - y)$
$-2 \sin x \sin y = \cos(x + y) - \cos(x - y)$ (or $2 \sin x \sin y = \cos(x - y) - \cos(x + y)$)

Sum/Difference into Product:

$\sin x + \sin y = 2 \sin\left(\frac{x + y}{2}\right) \cos\left(\frac{x - y}{2}\right)$
$\sin x - \sin y = 2 \cos\left(\frac{x + y}{2}\right) \sin\left(\frac{x - y}{2}\right)$
$\cos x + \cos y = 2 \cos\left(\frac{x + y}{2}\right) \cos\left(\frac{x - y}{2}\right)$
$\cos x - \cos y = -2 \sin\left(\frac{x + y}{2}\right) \sin\left(\frac{x - y}{2}\right)$

For $2x$:
- $\sin 2x = 2 \sin x \cos x = \frac{2 \tan x}{1 + \tan^2 x}$
- $\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x = \frac{1 - \tan^2 x}{1 + \tan^2 x}$
- $\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}$
For $3x$:
- $\sin 3x = 3 \sin x - 4 \sin^3 x$
- $\cos 3x = 4 \cos^3 x - 3 \cos x$
- $\tan 3x = \frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x}$
Half-Angle Relations:
- $1 + \cos 2x = 2 \cos^2 x$
- $1 - \cos 2x = 2 \sin^2 x$

Equations involving trigonometric functions are called trigonometric equations.

General Solutions:
1. If $\sin \theta = 0$, then $\theta = n\pi$, where $n \in \mathbb{Z}$ (Integer).
2. If $\cos \theta = 0$, then $\theta = (2n + 1)\frac{\pi}{2}$, where $n \in \mathbb{Z}$.
3. If $\tan \theta = 0$, then $\theta = n\pi$, where $n \in \mathbb{Z}$.
4. If $\sin \theta = \sin \alpha$, then $\theta = n\pi + (-1)^n \alpha$, where $n \in \mathbb{Z}$.
5. If $\cos \theta = \cos \alpha$, then $\theta = 2n\pi \pm \alpha$, where $n \in \mathbb{Z}$.
6. If $\tan \theta = \tan \alpha$, then $\theta = n\pi + \alpha$, where $n \in \mathbb{Z}$.