The Power-Amplification Formula for Sin and Cos: How to Handle Negative Inputs?

[Jhenrique]

Homework Statement

If exist a formula for reducion of power for sin and cos:

So, is possible to define a formula of "amplification of power" for sin and cos?

$$\sqrt{\sin(x)} = ?$$$$\sqrt{\cos(x)} = ?$$

Homework Equations

The Attempt at a Solution

None attempt well succeful or relevant.

 

[Ray Vickson]

Homework Statement

If exist a formula for reducion of power for sin and cos:

So, is possible to define a formula of "amplification of power" for sin and cos?

$$\sqrt{\sin(x)} = ?$$$$\sqrt{\cos(x)} = ?$$

Homework Equations

The Attempt at a Solution

None attempt well succeful or relevant.

Since $\sqrt{\sin(x)}$ and $\sqrt{\cos(x)}$ do not exist (as real quantities) whenever the sin or cos are < 0, any kind of formula would need to be genuinely weird. If there were formulas at all they would need to have branch cuts in the complex plane.

 

[Jhenrique]

I didn't understand your answer.

 

[haruspex]

I didn't understand your answer.

Ray is saying, suppose f(x) = √(sin(x)), where f(x) is a real function. How will it give the right answer when sin(x) is negative?