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Multiple Angles - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Multiple angles in terms of arctan and arccos is considered one the most difficult concept.

  • 20 Questions around this concept.

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QuestionEASY

Considering only the principal values of inverse trigonometric functions, the number of positive real values of x satisfying $\tan ^{-1}(x)+\tan ^{-1}(2 x)=\frac{\pi}{4}$ is :

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Concepts Covered - 4

Multiple angles in terms of arcsin

Multiple angles in terms of arcsin

 

\\\mathrm{1.\;\;\:2\;sin^{-1}x=\left\{\begin{matrix} \sin ^{-1}(2 x \sqrt{1-x^{2}}),& &\frac{1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}}\\ \\ \pi-\sin ^{-1}(2 x \sqrt{1-x^{2}}),&& x>\frac{1}{\sqrt{2}}\\ \\ -\pi-\sin ^{-1}(2 x \sqrt{1-x^{2}}),&&x<-\frac{1}{\sqrt{2}} \end{matrix}\right.}\\\\\\\\\mathrm{2.\;\;\;3\:\sin^{-1}x=\left\{\begin{matrix} \sin ^{-1}\left(3 x-4 x^{3}\right),& & -\frac{1}{2} \leq x \leq \frac{1}{2}\\ \\ \pi-\sin ^{-1}\left(3 x-4 x^{3}\right), & & x>\frac{1}{2}\\ \\ -\pi-\sin ^{-1}\left(3 x-4 x^{3}\right) & & x :-\frac{1}{2} \end{matrix}\right.}

Multiple angles in terms of arccos

Multiple angles in terms of  arccos

 

\\\mathrm{1.\;\;\;\2\;cos^{-1}x=\left\{\begin{matrix} \cos ^{-1}\left(2 x^{2}-1\right), & &\text { if } 0 \leq x \leq 1\\ \\ 2\pi-\cos ^{-1}\left(2 x^{2}-1\right),& &\text { if }-1 \leq x \leq 0 \end{matrix}\right.}\\\\\\\mathrm{2.\;\;\;3\;\cos^{-1}x=\left\{\begin{matrix} \cos ^{-1}\left(4 x^{3}-3 x\right), & & \text { if } \frac{1}{2} \leq x \leq 1\\\\ 2\pi-\cos ^{-1}\left(4 x^{3}-3 x\right),& &\text { if }-\frac{1}{2} \leq x \leq \frac{1}{2} \\\\ 2\pi+\cos ^{-1}\left(4 x^{3}-3 x\right),& & \text { if }-1 \leq x \leq-\frac{1}{2} \end{matrix}\right.}

Multiple angles in terms of arctan and arcsin

Multiple angles in terms of  arctan and arcsin 

 

 

\\\mathrm{\;\;\;2\;\tan^{-1}x=\left\{\begin{matrix} \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right), & & \text { if }-1 \leq x \leq 1\\ \\ \pi-\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right),& &\text { if }\;\;x>1 \\\\ -\pi-\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right),& & \text { if }\;\;x<-1 \end{matrix}\right.}


 



Multiple angles in terms of arctan and arccos

Multiple angles in terms of  arctan and arccos

 

\mathrm{\;\;\;2\tan^{-1}x=\left\{\begin{matrix} \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right),& &\text { if }\;\; 0 \leq x<\infty \\ \\ -\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right),& & \text { if }-\infty<x \leq 0 \end{matrix}\right.}



 

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Multiple angles in terms of arcsin
Multiple angles in terms of arccos
Multiple angles in terms of arctan and arcsin

Study other Related Concepts

Multiple Angles Current Topic

Measuring Angles
Trigonometric Ratios
Trigonometric Identities
Sign of Trigonometric Functions
Graphs of General Trigonometric Functions
Trigonometric Ratios of Allied Angles
Trigonometric Ratios of Compound Angles
Sum-to-Product and Product-to-Sum Formulas
Product To Sum Formulas
Double Angle Formulas

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Reference Books

Multiple angles in terms of arcsin

Mathematics for Joint Entrance Examination JEE (Advanced) : Trigonometry

Page No. : 7.26

Line : 17

Multiple angles in terms of arccos

Mathematics for Joint Entrance Examination JEE (Advanced) : Trigonometry

Page No. : 7.26

Line : 42

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