Trigonometry (10th Edition)

Unknown side $a = \sqrt 57$ $\sin B = \frac{8}{11}$ $\cos B = \frac{\sqrt57}{11}$ $\tan B = \frac{8\sqrt 57}{57}$ $\csc B = \frac{11}{8}$ $\sec B = \frac{11\sqrt 57}{57}$ $\cot B = \frac{\sqrt 57}{8}$
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 $\sqrt 57$ $= \frac{11}{\sqrt 57} \times \frac{\sqrt 57}{\sqrt 57} = \frac{11\sqrt 57}{57}$ $\cot B = \frac{1}{\tan B} = \frac{\sqrt 57}{8}$