A problem in Inverse Circular Functions in Trigonometry

[Wrichik Basu]

Homework Statement

:[/B]

Solve for $x$: $$ \sin ^{-1} {x} +\sin ^{-1} {(1-x)} =\cos ^{-1} {x} $$
Answer given: $0$ or $\frac {1}{2}$.

Homework Equations

:[/B]

All relevant formulae on inverse circular functions may be used.

The Attempt at a Solution

:[/B]

Please see the pic below.

I can't solve with the huge expression under root. How do I simplify that?

 

[ehild]

Homework Statement

:[/B]

Solve for $x$: $$ \sin ^{-1} {x} +\sin ^{-1} {(1-x)} =\cos ^{-1} {x} $$
Answer given: $0$ or $\frac {1}{2}$.

Homework Equations

:[/B]

All relevant formulae on inverse circular functions may be used.

The Attempt at a Solution

:[/B]

Please see the pic below.

View attachment 204003

I can't solve with the huge expression under root. How do I simplify that?

It is impossible to read your picture. If you want help from others, take the effort to type in your work.
Anyway, take the cosine of both sides, and replace cos(θ) with $\sqrt{1-\sin^2(θ)}$

 

[BvU]

I can't solve with the huge expression under root. How do I simplify that?

First you want it typed so others can read it.
Then you want it correct.
Under square root in last line you should have $$\left [ 4x^2(1-x^2)\right] * \left[ -x(x-2)\right] $$and then you want to solve.

 

[ehild]

From the fourth line, you get cos^-1(2x²-1) = π - cos^-1(1-x)
Take the cosine of both sides. Use the addition formula for cosine: cos(a-b)= cos(a)cos(b)+sin(a)sin(b). You get a simple quadratic equation for x, without complicated square roots.

 

[Wrichik Basu]

First you want it typed so others can read it.
Then you want it correct.
Under square root in last line you should have $$\left [ 4x^2(1-x^2)\right] * \left[ -x(x-2)\right] $$and then you want to solve.

I never said I wanted it to be typed. I did not have the copy at hand to change the image. I can't always sit with the copy, waiting to see if others can read from the image. I can read from the image very well. Perhaps you can't understand my handwriting. That's a different issue. I'll change the pic once I reach home.

 

[Wrichik Basu]

From the fourth line, you get cos^-1(2x²-1) = π - cos^-1(1-x)
Take the cosine of both sides. Use the addition formula for cosine: cos(a-b)= cos(a)cos(b)+sin(a)sin(b). You get a simple quadratic equation for x, without complicated square roots.

I'll try that, but wait a bit. I'll try once I reach home.

 

[ehild]

I never said I wanted it to be typed. I did not have the copy at hand to change the image. I can't always sit with the copy, waiting to see if others can read from the image. I can read from the image very well. Perhaps you can't understand my handwriting. That's a different issue. I'll change the pic once I reach home.

It is the policy here, that we type in the problems and solutions. It is very impolite to force the helpers to read blurry pictures of gray writing on gray background. (Quite many helpers are old people. I am 75)
Read https://www.physicsforums.com/threads/guidelines-for-students-and-helpers.686783/ "Executive Summary"

 

[Ray Vickson]

I never said I wanted it to be typed. I did not have the copy at hand to change the image. I can't always sit with the copy, waiting to see if others can read from the image. I can read from the image very well. Perhaps you can't understand my handwriting. That's a different issue. I'll change the pic once I reach home.

Many helpers say to themselves "If this person cannot be bothered to type out the work (diagrams excepted) I cannot be bothered to help". Like it or not, that is reality.

 

[Wrichik Basu]

It is the policy here, that we type in the problems and solutions. It is very impolite to force the helpers to read blurry pictures of gray writing on gray background. (Quite many helpers are old people. I am 75)
Read https://www.physicsforums.com/threads/guidelines-for-students-and-helpers.686783/ "Executive Summary"

Many helpers say to themselves "If this person cannot be bothered to type out the work (diagrams excepted) I cannot be bothered to help". Like it or not, that is reality.

Give me some time, I'll type it out, and pardon me for my behaviour. I was actually very angry with one of my teachers who scolded me even though I did nothing wrong, but only pointed out his error. I had lost my mind. Sorry, please give me some time.

 

[haruspex]

Well, I'm only 68, so I'll have a go at reading the image.
You were doing fine until the last line. You've done something weird in expanding the terms inside the square root. You should e.g. end up with being able to remove a factor x (x=0 being a solution).