So I've been asked to find $\sin(\theta)$ and $\cos(\theta)$ when $\tan(\theta)=\cfrac{12}{5}$; my question is if $\tan (\theta)=\cfrac{\sin (\theta) }{\cos (\theta)}$ does this mean that because $\tan (\theta)=\cfrac{12}{5}$ then $\sin (\theta) =12$ and $\cos(\theta)=5$?
It doesn't seem to be the case in this example, because $\sin (\theta)\ne 12 $ and $\cos (\theta)\ne 12 $.
Can someone tell me where the error is in my thinking?
Best Answer
Just because $\frac{a}{b}=\frac{c}{d},$ it does not necessarily follow that $a=c$ or that $b=d.$
e.g. $\frac{2}{3}=\frac{4}{6}$, but $2 \neq 4$ and $3 \neq 6$.
In your case, since $\tan(\theta)=\frac{12}{5},$ we have that $$\frac{\sin(\theta)}{\cos(\theta)}=\frac{12}{5} \iff \boxed{5 \sin(\theta)=12\cos(\theta)},$$ which is not the same thing as saying that $\color{red}{\sin(\theta)=12 \ \rm{and} \ \cos(\theta)=5 \ (\rm{wrong})}.$