# Roots of implicit equations

1. Jan 30, 2009

### tom92373

I was working on double integrals when I came across the equation: x^(3/2)=sin(x).
There was no noticeable way to isolate the equation for x without having a function of x equal to x. I am wondering how to isolate equations involving logs, sines, etc when it is given in an implicit form.
Using a computer, I was able to get an approximation of 0 and 8.02... How do I get the EXACT value of x?

2. Jan 30, 2009

### qntty

I don't think that can be solved analytically, numerical approximations are the best you can get.

3. Jan 30, 2009

### CRGreathouse

There's no reason to expect a nice solution to that equation.

To 500 decimal places:
0.80280373173788931551183532460400044122266891061652741081013964565691641862577997739822547061430396268572323604994666281323668533410644604205801464291930503518478667486487218236513935782397374909479614327907963131119225878971201268489647029085385407187785694454923172056331593018083775727247023723969536341968998158469732909155080566871504200160137298683450160853972584968512566509877215100019308073835565249990882682850748486897243599882872536008937760137965323934876164878700580114920356083682742718

4. Jan 31, 2009

### DuncanM

How did you get an answer to 500 decimal places?!

5. Feb 1, 2009