Answer
$a=\sqrt{21}$
$\sin B=\dfrac{2}{5}$ $;$ $\cos B=\dfrac{\sqrt{21}}{5}$ $;$ $\tan B=\dfrac{2\sqrt{21}}{21}$
$\csc B=\dfrac{5}{2}$ $;$ $\sec B=\dfrac{5\sqrt{21}}{21}$ $;$ $\cot B=\dfrac{\sqrt{21}}{2}$
Work Step by Step
$b=2$ $,$ $c=5$
Since $\angle C$ is the right angle, $c$ is the hypotenuse of this triangle and $b$ is a cathetus.
Use the Pythagorean Theorem to obtain the unknown cathetus $a$:
$a=\sqrt{c^{2}-b^{2}}$
$a=\sqrt{5^{2}-2^{2}}$
$a=\sqrt{25-4}$
$a=\sqrt{21}$
Proceed to find the six trigonometric functions for angle $B$:
$\sin B=\dfrac{opposite}{hypotenuse}=\dfrac{2}{5}$
$\cos B=\dfrac{adjacent}{hypotenuse}=\dfrac{\sqrt{21}}{5}$
$\tan B=\dfrac{opposite}{adjacent}=\dfrac{2}{\sqrt{21}}=\dfrac{2\sqrt{21}}{21}$
$\csc B=\dfrac{hypotenuse}{opposite}=\dfrac{5}{2}$
$\sec B=\dfrac{hypotenuse}{adjacent}=\dfrac{5}{\sqrt{21}}=\dfrac{5\sqrt{21}}{21}$
$\cot B=\dfrac{adjacent}{opposite}=\dfrac{\sqrt{21}}{2}$
