Answer
$a=\sqrt {95}$
$\sin B=\frac{7}{12}$
$\cos B=\frac{\sqrt {95}}{12}$
$\tan B=\frac{7\sqrt {95}}{95}$
$\csc B=\frac{12}{7}$
$\sec B=\frac{12\sqrt {95}}{95}$
$\cot B=\frac{\sqrt {95}}{7}$
Work Step by Step
$a=\sqrt {c^{2}-b^{2}}=\sqrt {12^{2}-7^{2}}=\sqrt {95}$
$\sin B=\frac{\text{side opposite}}{\text{hypotenuse}}=\frac{b}{c}=\frac{7}{12}$
$\cos B=\frac{\text{side adjacent}}{\text{hypotenuse}}=\frac{a}{c}=\frac{\sqrt {95}}{12}$
$\tan B=\frac{\text{side opposite}}{\text{side adjacent}}=\frac{b}{a}=\frac{7}{\sqrt {95}}=\frac{7\sqrt {95}}{95}$
The cosecant, secant, and cotangent ratios are reciprocals of the sine, cosine, and tangent values, respectively, so we have
$\csc B=\frac{12}{7}$
$\sec B=\frac{12}{\sqrt {95}}=\frac{12\sqrt {95}}{95}$
$\cot B=\frac{\sqrt {95}}{7}$
