Question

If `cos(theta) = (x-1)/(x+1)`, what is sin and tan of theta?

Answer 1

`sintheta=(2sqrtx)/(x+1), tantheta=(2sqrtx)/(x-1)`

Explanation:

Here is a sketch to clarify.

Given `costheta=(x-1)/(x+1)=a/h`

We can identify the adjacent side (x-1) and the hypotenuse (x+1)

To find the opposite side (o) use `color(blue)"Pythagoras' theorem"`

For this right triangle then.

`(x+1)^2=(x-1)^2+o^2`

expand using FOIL and collect like terms

`rArrx^2+2x+1=x^2-2x+1+o^2`

`rArro^2=cancel(x^2)-cancel(x^2)+2x+2xcancel(+1)cancel(-1)=4x`

now `o^2=4xrArro=sqrt(4x)=2sqrtx`

`rArrsintheta=(2sqrtx)/(x+1)" and " tantheta=(2sqrtx)/(x-1)`