# For the function `y=sqrt(144-x^(2))/x` `x=12 sin theta` , `theta in (- pi/2, pi/2)`   by using sustitute indicated find an equivalent function of theta and rewrite the resulting function in terms...

For the function

`y=sqrt(144-x^(2))/x`

`x=12 sin theta` , `theta in (- pi/2, pi/2)`

by using sustitute indicated find an equivalent function of theta

and rewrite the resulting function in terms of x using an inverse trig function

lemjay | High School Teacher | (Level 2) Senior Educator

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`y=sqrt(144-x^2)/x`  (Let this be EQ1.)

`x=12sintheta`    (Let this be EQ2.)

To get the function theta(x), plug-in EQ2 to EQ1.

`y=sqrt(144-x^2)/x`

`y=sqrt(144-(12sintheta)^2)/(12sin theta)`

`y=sqrt(144-144sin^2theta)/(12sin theta)`

Factor out the GCF of the radicand which is 144.

`y=sqrt(144(1-sin^2theta))/(12 sin theta)`

Use the Pythagorean identity to simplify the numerator. So, `1-sin^2theta=cos^2theta` .

`y=sqrt(144cos^2theta)/(12sin theta)`

`y=(12costheta)/(12sintheta)`

`y=(cos theta)/(sin theta)`

Since `cos theta/sintheta=cot theta` , this simplifies to:

`y=cot theta`

Then, take the inverse of cotangent to isolate theta.

`cot^(-1)y=theta`

To express this in terms of x, plug-in EQ1.

`cot^(-1)(sqrt(144-x^2)/x)=theta`

Hence, the function of `theta` in terms of x is

`theta(x)=cot^(-1)(sqrt(144-x^2)/x)` .