Question

What is the value of `sin(arcsec(x/3))`?

Answer 1

It can be simplified to `|sqrt(x^2 -9)/x|`

Explanation:

If we let:

`theta = arcsec(x/3)`

Then

`sectheta = x/3`

Since `sectheta` by definition is defined by `"hypotenuse"/"adjacent"`, we know the hypotenuse measures `x` and the side adjacent measures `3`. This means that the side opposite `theta` measures `sqrt(x^2 - 9)`.

Furthermore, `sintheta = sqrt(x^2 - 9)/x`.

Therefore, the expression in question can be simplified to

`sin(arcsec(x/3)) = sin(theta) = sqrt(x^ 2 - 9)/x`.

But if we observe the graph of `f(x) = sin(arcsec(x/3))`, we notice that `y> 0`. So the correct answer is `sin(arcsec(x/3)) = |sqrt(x^2 -9)/x|`.

Hopefully this helps!