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Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.Write the composed trigonometric function sin (arctan x) in terms of x. Explain your steps and/orshow your work. Remember to rationalize the denominator if necessary.

Answer :

Given:

[tex]\sin(\tan^{-1}x)[/tex]

To write:

The trigonometric function in terms of x.

Explanation:

Let us take,

[tex]y=\sin(\tan^{-1}x).............(1)[/tex]

Let us assume that,

[tex]tan^{-1}x=p..............(2)[/tex]

So, the function becomes

[tex]y=\sin p............(3)[/tex]

From the equation (2),

[tex]\begin{gathered} x=\tan p \\ i.e)\tan p=\frac{x}{1}=\frac{Oppsite}{Adjacent} \end{gathered}[/tex]

Using Pythagoras theorem,

[tex]\begin{gathered} Hyp^2=Opp^2+Adj^2 \\ Hyp^2=x^2+1^2 \\ Hyp=\sqrt{x^2+1} \end{gathered}[/tex]

So, equation 3 becomes,

[tex]\begin{gathered} y=\sin p \\ =\frac{Opp}{Hyp} \\ y=\frac{x}{\sqrt{x^2+1}} \end{gathered}[/tex]

Therefore, the composed trigonometric function in terms of x is,

[tex]\sin(\tan^{-1}x)=\frac{x}{\sqrt{x^2+1}}[/tex]

Final answer:

The composed trigonometric function in terms of x is,

[tex]\sin(\tan^{-1}x)=\frac{x}{\sqrt{x^2+1}}[/tex]

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