Answer
Unknown side $a = \sqrt 57 $
$\sin B = \frac{8}{11}$
$\cos B = \frac{\sqrt57}{11}$
$\tan B = \frac{8\sqrt 57}{57}$
$\csc B = \frac{11}{8}$
$\sec B = \frac{11\sqrt 57}{57}$
$\cot B = \frac{\sqrt 57}{8}$
Work Step by Step
Given $b=8$ and $c=11$
By Pythagorean theorem, $a^{2} +b^{2} = c^{2}$
$a = \sqrt ( c^{2} - b^{2})$
$ = \sqrt ( 11^{2} - 8^{2})$
$ = \sqrt ( 121 - 64) = \sqrt57$
Trigonometric functions for angle B.
$\sin B = \frac{Side Opposite To B}{hypotenuse}= \frac{b}{c} = \frac{8}{11}$
$\cos B = \frac{Side adjacentTo B}{hypotenuse}= \frac{a}{c} = \frac{\sqrt57}{11}$
$\tan B = \frac{Side Opposite To B}{Side adjacentTo B} = \frac{b}{a} = \frac{8}{\sqrt 57}$
Rationalize the denominator, Multiply and divide by $\sqrt 57$
$\tan B = \frac{8}{\sqrt 57} \times \frac{\sqrt 57}{\sqrt 57}$
$\tan B = \frac{8\sqrt 57}{57}$
$\csc B = \frac{1}{\sin B} =\frac{11}{8}$
$\sec B = \frac{1}{\cos B} =\frac{11}{\sqrt 57}$
Rationalize the denominator, Multiply and divide by $\sqrt 57$
$= \frac{11}{\sqrt 57} \times \frac{\sqrt 57}{\sqrt 57} = \frac{11\sqrt 57}{57}$
$\cot B = \frac{1}{\tan B} = \frac{\sqrt 57}{8}$
