Answer
$a=\sqrt {57}$
$\sin B=\frac{8}{11}$
$\cos B=\frac{\sqrt {57}}{11}$
$\tan B=\frac{8\sqrt {57}}{57}$
$\csc B=\frac{11}{8}$
$\sec B=\frac{11\sqrt {57}}{57}$
$\cot B=\frac{\sqrt {57}}{8}$
Work Step by Step
$a=\sqrt {c^{2}-b^{2}}=\sqrt {11^{2}-8^{2}}=\sqrt {57}$
$\sin B=\frac{\text{side opposite}}{\text{hypotenuse}}=\frac{b}{c}=\frac{8}{11}$
$\cos B=\frac{\text{side adjacent}}{\text{hypotenuse}}=\frac{a}{c}=\frac{\sqrt {57}}{11}$
$\tan B=\frac{\text{side opposite}}{\text{side adjacent}}=\frac{b}{a}=\frac{8}{\sqrt {57}}=\frac{8\sqrt {57}}{57}$
The cosecant, secant, and cotangent ratios are reciprocals of the sine, cosine, and tangent values, respectively, so we have
$\csc B=\frac{11}{8}$
$\sec B=\frac{11}{\sqrt {57}}=\frac{11\sqrt {57}}{57}$
$\cot B=\frac{\sqrt {57}}{8}$
