inverse-trigonometry

Domain-Range Table

Here, family 1,2 are only used in order to group functions with similar domain and range together for better understanding.

Family 1			Family 2
Function	Domain	Range	Function	Domain	Range
${\sin }^{-1}(x)$	$[-1,1]$	$\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$	${\cos }^{-1}(x)$	$[-1,1]$	$[0,\pi ]$
${\csc }^{-1}(x)$	$\mathbb{R}-(-1,1)$	$\left[-\frac{\pi }{2},\frac{\pi }{2}\right]-\{0\}$	${\sec }^{-1}(x)$	$\mathbb{R}-(-1,1)$	$[0,\pi ]-\{\frac{\pi }{2}\}$
${\tan }^{-1}(x)$	$\mathbb{R}$	$\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$	${\cot }^{-1}(x)$	$\mathbb{R}$	$(0,\pi )$

Family 1			Family 2
Function	Domain	Range	Function	Domain	Range
$\sin (x)$	$\mathbb{R}$	$[-1,1]$	$\cos (x)$	$\mathbb{R}$	$[-1,1]$
$\csc (x)$	$\mathbb{R}-\{k\pi \}$	$\mathbb{R}-(-1,1)$	$\sec (x)$	$\mathbb{R}-\left\{\frac{\pi }{2}+k\pi \right\}$	$\mathbb{R}-(-1,1)$
$\tan (x)$	$\mathbb{R}-\left\{\frac{\pi }{2}+k\pi \right\}$	$\mathbb{R}$	$\cot (x)$	$\mathbb{R}-\{k\pi \}$	$\mathbb{R}$

Inverse Trigonometric function of -x

Family 1	Family 2
${\sin }^{-1}(-x)=-{\sin }^{-1}(x)$	${\cos }^{-1}(-x)=\pi -{\cos }^{-1}(x)$
${\csc }^{-1}(-x)=-{\csc }^{-1}(x)$	${\sec }^{-1}(-x)=\pi -{\sec }^{-1}(x)$
${\tan }^{-1}(-x)=-{\tan }^{-1}(x)$	${\cot }^{-1}(-x)=\pi -{\cot }^{-1}(x)$

ITF Conversion Formulas

1. ${\sin }^{-1}(\frac{1}{x})={\csc }^{-1}(x)x\ge 1\text{OR}x\le -1$
2. ${\cos }^{-1}(\frac{1}{x})={\sec }^{-1}(x)x\ge 1\text{OR}x\le -1$
3. ${\sec }^{-1}(\frac{1}{x})={\cos }^{-1}(x)x\ge 1\text{OR}x\le -1$
4. ${\cot }^{-1}(\frac{1}{x})={\tan }^{-1}(x)x>0$
5. ${\tan }^{-1}(\frac{1}{x})={\cot }^{-1}(x)x>0$
6. ${\tan }^{-1}(\frac{1}{x})=-\pi +{\cot }^{-1}(x)x<0$

Single Angle, Multiple ITC Formulas

1. ${\sin }^{-1}(x)+{\cos }^{-1}(x)=\frac{\pi }{2}x\in [-1,1]$
2. ${\tan }^{-1}(x)+{\cot }^{-1}(x)=\frac{\pi }{2}x\in \mathbb{R}$
3. ${\csc }^{-1}(x)+{\sec }^{-1}(x)=\frac{\pi }{2}|x|\ge 1$

Multiple Angle Formulas

1. ${\tan }^{-1}(x)+{\tan }^{-1}(y)$
   • ${\tan }^{-1}(x)+{\tan }^{-1}(y)={\tan }^{-1}(\frac{x+y}{1-xy})xy<1$
   • ${\tan }^{-1}(x)+{\tan }^{-1}(y)=\frac{\pi }{2}xy=1$
   • ${\tan }^{-1}(x)+{\tan }^{-1}(y)=\pi +{\tan }^{-1}(\frac{x+y}{1-xy})xy>1$
2. ${\tan }^{-1}(x)-{\tan }^{-1}(y)={\tan }^{-1}(\frac{x-y}{1+xy})xy>-1$
3. ${\sin }^{-1}(x)+{\sin }^{-1}(y)$
   • ${\sin }^{-1}(x)+{\sin }^{-1}(y)={\sin }^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2})x^2+y^2\le 1$
   • ${\sin }^{-1}(x)+{\sin }^{-1}(y)=\pi -{\sin }^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2})x^2+y^2\ge 1$
   • ${\sin }^{-1}(x)-{\sin }^{-1}(y)={\sin }^{-1}(x\sqrt{1-y^2}-y\sqrt{1-x^2})x\ge 0,y\ge 0$
4. ${\cos }^{-1}(x)+{\cos }^{-1}(y)={\cos }^{-1}(xy-\sqrt{1-x^2}\cdot \sqrt{1-y^2})x\ge 0,y\ge 0$
5. ${\cos }^{-1}(x)-{\cos }^{-1}(y)$
   • ${\cos }^{-1}(x)-{\cos }^{-1}(y)={\cos }^{-1}(xy+\sqrt{1-x^2}\cdot \sqrt{1-y^2})x\le y$
   • ${\cos }^{-1}(x)-{\cos }^{-1}(y)=-{\cos }^{-1}(xy+\sqrt{1-x^2}\cdot \sqrt{1-y^2})x\ge y$

ITF Multiple Formulas

1. $2{\tan }^{-1}(x)={\tan }^{-1}\left(\frac{2x}{1-x^2}\right)\text{for}|x|<1$
2. $2{\tan }^{-1}(x)=\pi +{\tan }^{-1}\left(\frac{2x}{1-x^2}\right)\text{for}x>1$
3. $2{\tan }^{-1}(x)=-\pi +{\tan }^{-1}\left(\frac{2x}{1-x^2}\right)\text{for}x<-1$
4. $2{\tan }^{-1}(x)={\sin }^{-1}\left(\frac{2x}{1+x^2}\right)\text{for}|x|\le 1$
5. $2{\tan }^{-1}(x)=\pi +{\sin }^{-1}\left(\frac{2x}{1+x^2}\right)\text{for}x>1$
6. $2{\tan }^{-1}(x)=-\pi +{\sin }^{-1}\left(\frac{2x}{1+x^2}\right)\text{for}x<-1$
7. $2{\tan }^{-1}(x)={\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)\text{for}x\ge 0$
8. $2{\tan }^{-1}(x)=-{\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)\text{for}x\le 0$
9. $3{\tan }^{-1}(x)$
   • $3{\tan }^{-1}(x)={\tan }^{-1}\left(\frac{3x-x^3}{1-3x^2}\right)\text{for}\frac{-1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}}$
   • $3{\tan }^{-1}(x)=\pi +{\tan }^{-1}\left(\frac{3x-x^3}{1-3x^2}\right)\text{for}x>\frac{1}{\sqrt{3}}$
   • $3{\tan }^{-1}(x)=-\pi +{\tan }^{-1}\left(\frac{3x-x^3}{1-3x^2}\right)\text{for}x<\frac{-1}{\sqrt{3}}$