Answer
$sin~\theta=\frac{opp}{hyp}=\frac{4}{\sqrt {41}}=\frac{4\sqrt {41}}{41}$
$cos~\theta=\frac{adj}{hyp}=\frac{5}{\sqrt {41}}=\frac{5\sqrt {41}}{41}$
$cot~\theta=\frac{adj}{opp}=\frac{5}{4}$
$csc~\theta=\frac{hyp}{opp}=\frac{\sqrt {41}}{4}$
$sec~\theta=\frac{hyp}{adj}=\frac{\sqrt {41}}{5}$
Work Step by Step
$tan~\theta=\frac{opp}{adj}$
$\frac{4}{5}=\frac{opp}{adj}$
Use the pythagorean theorem to find the opposite side of $\theta$
$hyp^2=4^2+5^2$
$hyp^2=16+25=41$
$hyp=\sqrt {41}$
$sin~\theta=\frac{opp}{hyp}=\frac{4}{\sqrt {41}}=\frac{4\sqrt {41}}{41}$
$cos~\theta=\frac{adj}{hyp}=\frac{5}{\sqrt {41}}=\frac{5\sqrt {41}}{41}$
$cot~\theta=\frac{adj}{opp}=\frac{5}{4}$
$csc~\theta=\frac{hyp}{opp}=\frac{\sqrt {41}}{4}$
$sec~\theta=\frac{hyp}{adj}=\frac{\sqrt {41}}{5}$
