# A problem in Inverse Circular Functions in Trigonometry

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1. May 22, 2017

### Wrichik Basu

1. The problem statement, all variables and given/known data:

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

2. Relevant equations:

All relevant formulae on inverse circular functions may be used.

3. The attempt at a solution:

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

2. May 22, 2017

### ehild

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 $\sqrt{1-\sin^2(θ)}$

3. May 22, 2017

### BvU

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.

4. May 22, 2017

### 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.

5. May 22, 2017

### Wrichik Basu

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.

6. May 22, 2017

### Wrichik Basu

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

7. May 22, 2017

### ehild

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)

Last edited: May 22, 2017
8. May 22, 2017

### Ray Vickson

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.

9. May 22, 2017

### Wrichik Basu

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.

10. May 25, 2017

### 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).