Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation cos-(x) – 2sin-1(x) = cos-1(2x) is equal to:

Ask & Answer Question

Maths Class 12th Inverse Trigonometric Functions

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation cos^-(x) – 2sin^-¹(x) = cos^-¹(2x) is equal to:

Option 1 -

Option 2 -

Option 3 -

$\frac{1}{2}$

Option 4 -

$- \frac{1}{2}$

2 Views | Posted 3 months ago

Asked by Shiksha User

1 Answer

Answered by

Vishal Baghel | Contributor-Level 10

3 months ago

Correct Option - 1

Detailed Solution:

cos^-1 x = 2 sin^-1 x = cos^-1 2x

$c o s^{- 1} x - 2 (\frac{π}{2} - c o s^{- 1} x) = c o s^{- 1} 2 x$

$c o s (3 c o s^{- 1} x) = c o s (π + c o s^{- 1} 2 x)$

$4 x^{3} - 3 x = - 2 x$

$4 x^{3} = x \Rightarrow x = 0 \pm \frac{1}{2}$

All satisfy the original equation

Sum = $- \frac{1}{2} + 0 + \frac{1}{2} = 0$

cos-1 x = 2 sin-1 x = cos-1 2x <math> <mrow> <mi>c</mi> <mi>o</mi> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>−</mo> <mi>c</mi> <mi>o</mi> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>c</mi> <mi>o</mi> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mn>2</mn> <mi>x</mi> </mrow> </math>        <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1815296578"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo> (</mo> <mrow> <mn>3</mn> <mi>c</mi> <mi>o</mi> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo> (</mo> <mrow> <mi>π</mi> <mo>+</mo> <mi>c</mi> <mi>o</mi> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math>   <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1815296579"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mn>4</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mrow> </math>       <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1815296580"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mn>4</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>⇒</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>±</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>     <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1028" DrawAspect="Content" ObjectID="_1815296581"> </o:OLEObject></xml><! [endif]-> All satisfy the original equationSum = <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1029" DrawAspect="Content" ObjectID="_1815296582"> </o:OLEObject></xml><! [endif]-> <math> <mrow> <mo>−</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>+</mo> <mn>0</mn> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </mrow> </math>

0 Upvotes 0 Downvotes

Get authentic answers from experts, students and alumni that you won't find anywhere else

On Shiksha, get access to

65k Colleges
1.2k Exams
682k Reviews
1800k Answers

Community GuidelinesUser Points System

QuestionsDiscussionsTags

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation cos^-(x) – 2sin^-¹(x) = cos^-¹(2x) is equal to:

1 Answer

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question