Question

How do you simplify the expression `cos(arcsin(x/5))`?

Answer 1

`cos(arcsin(\frac{x}{5})) = \frac{sqrt{25-x^2}}{5}`.

Explanation:

Firstly, note that for the range of `arcsin(\frac{x}{5})`, the cosine function is purely positive.

Thus, I can say,

`cos(arcsin(\frac{x}{5})) = sqrt{1-sin^2(arcsin(\frac{x}{5}))}`, and also for the domain of x, `sin(arcsin(\frac{x}{5})) = \frac{x}{5}` since the arcsine function outputs values ranging from `-\frac{pi}{2}` to `\frac{pi}{2}`.

This can be verified graphically, the red one being `cos(arcsin(\frac{x}{5}))`, green `cos(\frac{x}{5})`, and blue `arcsin(\frac{x}{5})`:

Therefore,

`cos(arcsin(\frac{x}{5})) = sqrt{1-\frac{x^2}{25})`

`= \frac{sqrt{25-x^2}}{5}`.