Class 12 Maths Inverse Trigonometric Functions Important Questions with Answers

 

Inverse Trigonometric Functions Worksheets with Answers

Inverse Trigonometry is one of the most important topics in class 12 maths NCERT syllabus. Inverse Trigonometric functions Notes, Inverse Trigonometric functions formula list, Inverse Trigonometric functions Properties ,Inverse Trigonometric functions substitution tricks and Inverse Trigonometric functions topic-wise worksheets of most important questions are prepared by our experts for the students to practice the questions and prepare themselves for the final exams. These worksheets are prepared strictly according to NCERT syllabus and examination pattern so they can help students to improve their subject knowledge prepare for exams.

Class 12 students should practice these questions and answers given here for Inverse Trigonometric functions which will help you to improve your knowledge of all topics of class 12 maths Inverse Trigonometric functions.

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INVERSE TRIGONOMETRIC FUNCTION NOTES   

INVERSE OF A FUNCTION:

$\begin{array}{l}\text{Let }\text{f}:X\to Y\text{be a one -one onto(Bijective) function,} \\ \text{For each }y\in Y,\text{}\text{there}\text{exists a unique elements }x\in X\text{such that } \\ f\left( x \right)=y \\ \text{so we define a new function denoted by }{f}^{-1},\text{ called the inverse of }f. \\ f^{-1}:Y\to X:f^{-1}\left( y \right)=x\iff f\left( x \right)=y \\ domain f^{-1}=range \left( f \right)andRange{f}^{-1}= domain \left( f \right) \end{array}$

PRINCIPAL VALUE BRANCHES:

Trigonometric Function are, in general not one –one onto (bijective). Therefore, their inverse do not exist. But if we restricted their domains, we can make them one – one and onto. For example Sine function has an inverse on $\left[ -\frac{\pi }{2},\frac{\pi }{2}\right]$, the inverse function is called arc sine function read as , sine inverse . 

NOTE $si{n}^{-1}x={\left( sinx \right)}^{-1}$

Following substitutions are used to write inverse trigonometric functions in simplest form 
$$\begin{array}{ccc}S.No.& Expression& Substitution\\ 1& \sqrt{a^2-x^2}& x=a\sin \theta orx=a\cos \theta \\ 2& \sqrt{a^2+x^2}& x=a\tan \theta orx=a\cot \theta \\ 3& \sqrt{x^2-a^2}& x=a\sec \theta orx=a\cos ec\theta \\ 4& \sqrt{a+x}or\sqrt{a-x}& x=a\cos \theta orx=a\cos 2\theta \\ 5& \sqrt{1+x^2}\pm \sqrt{1-x^2},\sqrt{\frac{1+x^2}{1-x^2}},\sqrt{\frac{1-x^2}{1+x^2}}& x^2={\cos }^2\theta \\ 6& \sqrt{a^2+x^2}\pm \sqrt{a^2-x^2},\sqrt{\frac{a^2+x^2}{a^2-x^2}},\sqrt{\frac{a^2-x^2}{a^2+x^2}}& x^2=a^2\cos 2\theta \\ 7& \sqrt{1+x}\pm \sqrt{1-x},\sqrt{\frac{1-x}{1+x}},\sqrt{\frac{1+x}{1-x}}& x=\cos 2\theta \\ 8& \sqrt{a+x}\pm \sqrt{a-x},\sqrt{\frac{a+x}{a-x}},\sqrt{\frac{a-x}{a+x}}& x=a\cos 2\theta \end{array}$$

$$\begin{array}{l}\end{array}$$PROPERTIES ​OF ​INVERSE​ TRIGONOMETRIC​ FUNCTION

$$\begin{array}{l}\text{Property 1}\\ si{n}^{-1}\left(\sin x\right)=x,x\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right]\\ {\cos }^{-1}\left(\cos x\right)=x,x\in \left[0,\frac{\pi }{2}\right]\\ {\tan }^{-1}\left(\tan x\right)=x,x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\\ \cos e{c}^{-1}\left(\cos ecx\right)=x,x\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right]-\left\{0\right\}\\ {\sec }^{-1}\left(\sec x\right)=x,x\in \left[0,\pi \right]-\left\{\frac{\pi }{2}\right\}\\ {\cot }^{-1}\left(\cot x\right)=x,x\in \left(0,\pi \right)\\ \\ Property2\\ \sin \left({\sin }^{-1}x\right)=x,x\in \left[-1,1\right]\\ \cos \left({\cos }^{-1}x\right)=x,x\in \left[-1,1\right]\\ \tan \left({\tan }^{-1}x\right)=x,x\in R\\ \cos ec\left(\cos e{c}^{-1}x\right)=x,x\in R-\left[-1,1\right]\\ \sec \left({\sec }^{-1}x\right)=x,x\in R-\left[-1,1\right]\\ \cot \left({\cot }^{-1}x\right)=x,x\in R\\ \\ Property3\\ {\sin }^{-1}\left(\frac{1}{x}\right)=\cos e{c}^{-1}x,\left(x\ge 1orx\le -1\right)\\ {\cos }^{-1}\left(\frac{1}{x}\right)={\sec }^{-1}x,\left(x\ge 1orx\le -1\right)\\ {\tan }^{-1}\left(\frac{1}{x}\right)={\cot }^{-1}x,\left(x>0\right)\\ \\ \\ \end{array}$$ $$\begin{array}{l}Property4\\ {\sin }^{-1}\left(-x\right)=-{\sin }^{-1}x,x\in \left[-1,1\right]\\ {\tan }^{-1}\left(-x\right)=-{\tan }^{-1}x,x\in R\\ \cos e{c}^{-1}\left(-x\right)=-\cos e{c}^{-1}x,\left|x\right|\ge 1\\ \\ Property5\\ {\cos }^{-1}\left(-x\right)=\pi -{\cos }^{-1}x,x\in \left[-1,1\right]\\ {\sec }^{-1}\left(-x\right)=\pi -{\sec }^{-1}x,\left|x\right|\ge 1\\ {\cot }^{-1}\left(-x\right)=\pi -{\cot }^{-1}x,x\in R\\ \\ Property6\\ {\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2},x\in \left[-1,1\right]\\ {\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2},x\in R\\ {\sec }^{-1}x+\cos e{c}^{-1}x=\frac{\pi }{2},\left|x\right|\ge 1\\ \\ Property7\\ {\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\left(\frac{x+y}{1-xy}\right),xy<1\\ {\tan }^{-1}x-{\tan }^{-1}y={\tan }^{-1}\left(\frac{x-y}{1+xy}\right),xy>-1\\ 2{\tan }^{-1}x={\tan }^{-1}\left(\frac{2x}{1-x^2}\right),\left|x\right|<1\end{array}$$ $$ $$\begin{array}{l}Property8\\ 2{\tan }^{-1}x={\sin }^{-1}\left(\frac{2x}{1+x^2}\right),\left|x\right|<1\\ 2{\tan }^{-1}x={\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right),\left|x\right|\ge 0\\ 2{\tan }^{-1}x={\tan }^{-1}\left(\frac{2x}{1-x^2}\right),\left|x\right|<1\\ \\ Property9\\ 2{\sin }^{-1}x={\sin }^{-1}\left[2x\sqrt{1-x^2}\right],-\frac{1}{\sqrt{2}}\le x\le \frac{1}{\sqrt{2}}\\ \\ Property10\\ 2{\cos }^{-1}x={\cos }^{-1}\left(2x^2-1\right)\\ \\ Property11\\ {\sin }^{-1}x+{\sin }^{-1}y={\sin }^{-1}\left[x\sqrt{1-y^2}+y\sqrt{1-x^2}\right]\\ {\sin }^{-1}x-{\sin }^{-1}y={\sin }^{-1}\left[x\sqrt{1-y^2}-y\sqrt{1-x^2}\right]\\ {\cos }^{-1}x+{\cos }^{-1}y={\cos }^{-1}\left[xy-\sqrt{1-x^2}\sqrt{1-y^2}\right]\\ {\cos }^{-1}x-{\cos }^{-1}y={\cos }^{-1}\left[xy+\sqrt{1-x^2}\sqrt{1-y^2}\right]\end{array}$$ $$$$

If the branch of an inverse trigonometric function is not specifically mentioned, then we consider the principal branch only.

Topic wise worksheets:

• Inverse Trigonometric Functions worksheet 1 with Answers
  
• Inverse Trigonometric Functions worksheet 2 with Answers