Question

If sin(x)=} " align="absmiddle" class="latex-formula"> what is cos(x) and tan(x)? Explain your steps in complete sentences.

Answer 1

The solution is not unique, but given that $\sin x=\dfrac12$, by the Pythagorean identity we have two solutions for $\cos x$:

$\sin^2x+\cos^2x=1\implies\cos x=\pm\sqrt{1-\sin^2x}=\pm\dfrac{\sqrt3}2$

Then from this, we have two corresponding solutions for $\tan x$. By definition,

$\tan x=\dfrac{\sin x}{\cos x}$

and so we can have, in the case of $\cos x>0$.

$\tan x=\dfrac{\frac12}{\frac{\sqrt3}2}=\dfrac1{\sqrt3}$

or $\tan x=-\dfrac1{\sqrt3}$ in the case that $\cos x$.