1. General Solutions of Trigonometric Equations
- $\sin \theta = \sin \alpha \implies \theta = n\pi + (-1)^n \alpha$, where $n \in Z$.
- $\cos \theta = \cos \alpha \implies \theta = 2n\pi \pm \alpha$, where $n \in Z$.
- $\tan \theta = \tan \alpha \implies \theta = n\pi + \alpha$, where $n \in Z$.
- $\sin^2 \theta = \sin^2 \alpha \implies \theta = n\pi \pm \alpha$, where $n \in Z$.
- $\cos^2 \theta = \cos^2 \alpha \implies \theta = n\pi \pm \alpha$, where $n \in Z$.
- $\tan^2 \theta = \tan^2 \alpha \implies \theta = n\pi \pm \alpha$, where $n \in Z$.
2. Solutions of Triangle
- Relation between Cartesian and Polar Co-ordinates:
$$ x = r \cos \theta, \quad y = r \sin \theta $$
- The Sine Rule:
In $\Delta ABC$:
$$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R $$
Where $R$ is the circumradius of $\Delta ABC$.
- The Cosine Rule:
In $\Delta ABC$:
$$ \begin{aligned} a^2 &= b^2 + c^2 - 2bc \cos A \implies \cos A = \frac{b^2 + c^2 - a^2}{2bc} \\ b^2 &= c^2 + a^2 - 2ca \cos B \implies \cos B = \frac{c^2 + a^2 - b^2}{2ca} \\ c^2 &= a^2 + b^2 - 2ab \cos C \implies \cos C = \frac{a^2 + b^2 - c^2}{2ab} \end{aligned} $$
- The Projection Rule:
In $\Delta ABC$:
$$ \begin{aligned} a &= b \cos C + c \cos B \\ b &= c \cos A + a \cos C \\ c &= a \cos B + b \cos A \end{aligned} $$
- Half Angle Formulae:
In $\Delta ABC$, if $a + b + c = 2s$:
$$ \begin{aligned} \sin\left(\frac{A}{2}\right) &= \sqrt{\frac{(s-b)(s-c)}{bc}} \\ \cos\left(\frac{A}{2}\right) &= \sqrt{\frac{s(s-a)}{bc}} \\ \tan\left(\frac{A}{2}\right) &= \sqrt{\frac{(s-b)(s-c)}{s(s-a)}} \end{aligned} $$
(Similar formulae apply for angles $B$ and $C$).
- Heron's Formula:
$$ A(\Delta ABC) = \sqrt{s(s-a)(s-b)(s-c)} $$
- Napier's Analogy:
$$ \begin{aligned} \tan\left(\frac{B-C}{2}\right) &= \frac{(b-c)}{(b+c)} \cot\left(\frac{A}{2}\right) \\ \tan\left(\frac{C-A}{2}\right) &= \frac{(c-a)}{(c+a)} \cot\left(\frac{B}{2}\right) \\ \tan\left(\frac{A-B}{2}\right) &= \frac{(a-b)}{(a+b)} \cot\left(\frac{C}{2}\right) \end{aligned} $$
3. Inverse Trigonometric Functions
Domain and Range (Principal Values)
| Function | Domain | Range (Principal Value) |
|---|---|---|
| $\sin^{-1} x$ | $[-1, 1]$ | $[-\frac{\pi}{2}, \frac{\pi}{2}]$ |
| $\cos^{-1} x$ | $[-1, 1]$ | $[0, \pi]$ |
| $\tan^{-1} x$ | $\mathbb{R}$ | $(-\frac{\pi}{2}, \frac{\pi}{2})$ |
| $\text{cosec}^{-1} x$ | $\mathbb{R} - (-1, 1)$ | $[-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\}$ |
| $\sec^{-1} x$ | $\mathbb{R} - (-1, 1)$ | $[0, \pi] - \{\frac{\pi}{2}\}$ |
| $\cot^{-1} x$ | $\mathbb{R}$ | $(0, \pi)$ |
Properties of Inverse Trigonometric Functions
- $\sin^{-1} x = \text{cosec}^{-1}\left(\frac{1}{x}\right)$ for $-1 \leq x \leq 1$ and $x \neq 0$.
- $\cos^{-1} x = \sec^{-1}\left(\frac{1}{x}\right)$ for $-1 \leq x \leq 1$ and $x \neq 0$.
- $\tan^{-1} x = \cot^{-1}\left(\frac{1}{x}\right)$ for $x > 0$.
- $\sin^{-1}(-x) = -\sin^{-1}(x)$ for $-1 \leq x \leq 1$.
- $\cos^{-1}(-x) = \pi - \cos^{-1}(x)$ for $-1 \leq x \leq 1$.
- $\tan^{-1}(-x) = -\tan^{-1}(x)$ for all $x \in \mathbb{R}$.
- $\text{cosec}^{-1}(-x) = -\text{cosec}^{-1}(x)$ for $|x| \geq 1$.
- $\sec^{-1}(-x) = \pi - \sec^{-1}(x)$ for $|x| \geq 1$.
- $\cot^{-1}(-x) = \pi - \cot^{-1}(x)$ for all $x \in \mathbb{R}$.
- $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$ for $-1 \leq x \leq 1$.
- $\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}$ for all $x \in \mathbb{R}$.
- $\text{cosec}^{-1} x + \sec^{-1} x = \frac{\pi}{2}$ for $x \geq 1$.
- $\tan^{-1} x + \tan^{-1} y = \tan^{-1}\left(\frac{x+y}{1-xy}\right)$ if $x > 0, y > 0$ and $xy < 1$.
- $\tan^{-1} x + \tan^{-1} y = \pi + \tan^{-1}\left(\frac{x+y}{1-xy}\right)$ if $x > 0, y > 0$ and $xy > 1$.
- $\tan^{-1} x + \tan^{-1} y = \frac{\pi}{2}$ if $x > 0, y > 0$ and $xy = 1$.
- $\tan^{-1} x - \tan^{-1} y = \tan^{-1}\left(\frac{x-y}{1+xy}\right)$ if $x > 0, y > 0$.