Question

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

Answer 1

Step-by-step 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=√(1-cos^2\theta)`

Where you have;

`\theta=arccos(x)`
`sin(arccos(x))=√(1-x^2)`

ANSWER:

Option D:

`√(1-x^2)`