Any Way to Reduce SQRT(1 - sin(theta))?

Sep 2006
104
15
Between my ears
I am making some wild guesses at approximating functions for a curve on which I am working, and am examining functions for which sin(\(\displaystyle \theta\)) may take various powers.

For example, say

sin(\(\displaystyle \theta\)) = \(\displaystyle (1 - x^{2})^{(1/4)}\)

The corresponding value of cosine would be

cos(\(\displaystyle \theta\)) = \(\displaystyle \sqrt {(1 - ((1 - x^{2})^{(1/4)})^{2})}\) = \(\displaystyle \sqrt {(1 - (1 - x^{2})^{(1/2)})}\) = \(\displaystyle \sqrt {(1 - \sqrt{(1 - x^{2})})}\)

Or, another curve I am looking at:

sin(\(\displaystyle \theta\)) = \(\displaystyle (1 - x^{2})^{(3/4)}\)

How to reduce the corresponding expression for cosine?

Or, another one, simply that :

cos(\(\displaystyle \theta\)) = \(\displaystyle \sqrt{(1 - sin(\theta))}\)

Is there any way to simplify these expressions to eliminate the radical?

I was looking at expressing the sine functions in terms of r*exp(x\(\displaystyle \theta\)), but didn't get far with it. Perhaps I am overlooking something simple. Any advice?
 
Jan 2009
463
138
Above you're using Pythagorean Formula, so getting the square root is inescapable.

You could use Euler's Formula and isolate cos
1575903347706.png

Alternatively, you could express cos as a phase shift of sin. (first one in this list)
1575903481789.png
 
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