Question

Write the composed trigonometric function sin(arctan x) in terms of x

Answer 1

Let
∅=arc tan x

we know that

∅ is an angle whose tangent function is

tan ∅=x/1

Considering the sides of the right triangle. 

We have 

opposite side =x,

adjacent side =1

and 

hypotenuse =√(x²+1)

Therefore 

the sine of this angle =opposite side/hypotenuse

sin ∅=x/√(x²+1)-----> the sign of sine ∅ will depend on the quadrant in which it is located

value of sin ∅ is positive--------->  ∅ belong to the first or second quadrant

value of sin ∅ is negative--------->  ∅ belong to the third or fourth quadrant

the answer is

x/√(x²+1)

Answer 2

The composed trigonometric function sin(arctan x) in terms of x is  $y=\frac{x}{\sqrt{x^2+1} }$.

Given function is:

$y=sin(tan^{-1} x)$......(1)

What is an inverse trigonometric function?

It is the inverse function of trigonometric functions in a restricted domain.

Let us say,

$tan^{-1} x =p$.....(2)

So, equation 1 becomes,

y =sinp.....(3)

From equation 2,

$x =tanp$

$tanp =\frac{x}{1}$

x is the opposite side

1 is the adjacent side

So hypotenuse will be $\sqrt{x^{2} +1}$

So, $sinp =\frac{x}{\sqrt{x^2+1} }$

Therefore, the composed trigonometric function sin(arctan x) in terms of x is  $y=\frac{x}{\sqrt{x^2+1} }$.

To get more about inverse trigonometric functions visit:

https://brainly.com/question/1143565