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

#### DavidB

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?

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