Trig functions from a given cosine

Text

Given that $\cos x=\dfrac{3}{5}$ and $x$ lies in the first quadrant, compute $\sin x$, $\tan x$, $\sec x$, $\csc x$ and $\cot x$.

Solution

From the fundamental identity, $\sin^{2}x=1-\cos^{2}x=1-\tfrac{9}{25}=\tfrac{16}{25}$. In the first quadrant sine is positive, so $\sin x=\tfrac45$. Consequently: $\tan x=\frac{\sin x}{\cos x}=\frac{4/5}{3/5}=\frac43,\quad \sec x=\frac{1}{\cos x}=\frac53,\quad \csc x=\frac{1}{\sin x}=\frac54,\quad \cot x=\frac{1}{\tan x}=\frac34.$ $\boxed{\,\sin x=\tfrac45,\ \tan x=\tfrac43,\ \sec x=\tfrac53,\ \csc x=\tfrac54,\ \cot x=\tfrac34\,}$