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

May 22, 2016

$\sin \theta = \frac{2 \sqrt{x}}{x + 1} , \tan \theta = \frac{2 \sqrt{x}}{x - 1}$

#### Explanation:

Here is a sketch to clarify.

Given $\cos \theta = \frac{x - 1}{x + 1} = \frac{a}{h}$

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

To find the opposite side (o) use $\textcolor{b l u e}{\text{Pythagoras' theorem}}$

For this right triangle then.

${\left(x + 1\right)}^{2} = {\left(x - 1\right)}^{2} + {o}^{2}$

expand using FOIL and collect like terms

$\Rightarrow {x}^{2} + 2 x + 1 = {x}^{2} - 2 x + 1 + {o}^{2}$

$\Rightarrow {o}^{2} = \cancel{{x}^{2}} - \cancel{{x}^{2}} + 2 x + 2 x \cancel{+ 1} \cancel{- 1} = 4 x$

now ${o}^{2} = 4 x \Rightarrow o = \sqrt{4 x} = 2 \sqrt{x}$

$\Rightarrow \sin \theta = \frac{2 \sqrt{x}}{x + 1} \text{ and } \tan \theta = \frac{2 \sqrt{x}}{x - 1}$