Question

Using the standard domain for the sine function, which of the following expressions corresponds to the composite function sin(accosx)?

Answer 1

EXPLANATION:

We are given the composite function;

$sin(arccosx)$

We are required to determine the expression that corresponds to this.

To do that, we shall recall the Pythagorean identity, and that is;

$sin^2\theta+cos^2\theta=1$$\begin{gathered} If\text{ }\theta=arccos(x) \\ Then\text{ }\theta\text{ }\in[0,\pi] \end{gathered}$

And;

$sin\theta\ge0$

Therefore;

$sin(arccos(x))=sin\theta$

Going back to our Pythagorean identity;

$\begin{gathered} sin^2\theta+cos^2\theta=1 \\ sin^2\theta=1-cos^2\theta \end{gathered}$

Take the square root of both sides;

$sin\theta=\sqrt{1-cos^2\theta}$

Where you have;

$\theta=arccos(x)$$sin(arccos(x))=\sqrt{1-x^2}$

ANSWER:

Option D:

$\sqrt{1-x^2}$