Answer
$\sin\theta=\dfrac{\sqrt{5}}{5}$
$\cos\theta=\dfrac{2\sqrt 5}{5}$
$\tan\theta=\dfrac{1}{2}$
$\cot\theta=2$
$\sec\theta=\dfrac{\sqrt 5}{2}$
$\csc\theta=\sqrt 5$
Work Step by Step
Let's note:
$h$=the hypotenuse
$o$=the opposite side of angle $\theta$
$a$=the adjacent side of angle $\theta$
We are given:
$a=2$
$h=\sqrt 5$
Determine the opposite side, using the Pythagorean Theorem:
$h^2=o^2+a^2$
$(\sqrt 5)^2=o^2+2^2$
$o^2=1$
$o=\pm1$
Since $o$ is never negative, then $o=\sqrt5$.
Determine the $6$ trigonometric functions of angle $\theta$:
$\sin\theta=\dfrac{o}{h}=\dfrac{1}{\sqrt 5}=\dfrac{\sqrt{5}}{5}$
$\cos\theta=\dfrac{a}{h}=\dfrac{2}{\sqrt 5}=\dfrac{2\sqrt 5}{5}$
$\tan\theta=\dfrac{o}{a}=\dfrac{1}{2}$
$\cot\theta=\dfrac{a}{o}=\dfrac{2}{1}=2$
$\sec\theta=\dfrac{h}{a}=\dfrac{\sqrt 5}{2}$
$\csc\theta=\dfrac{h}{o}=\dfrac{\sqrt 5}{1}=\sqrt 5$
