Question

Given a and b are first quadrant angles, sin a=5/13 and cos b=3/5 evaluate tan(a-b)1) 33/562) -33/563) -21/56

Answer 1

The angles given are

$\sin a=\frac{5}{13},\cos b=\frac{3}{5}$

Determine the value of cos a

$\cos a=\sqrt[]{1-\frac{25}{169}}=\frac{12}{13}$

Determine the value of sin b

$\sin b=\sqrt[]{1-\frac{9}{25}}=\frac{4}{5}$

The value of tan a is

$\tan \text{ a=}\frac{\frac{5}{13}}{\frac{12}{13}}=\frac{5}{13}\times\frac{13}{12}=\frac{5}{12}$

The value of tan b is

$\tan b=\frac{\frac{4}{5}}{\frac{3}{5}}=\frac{4}{3}$

The value of tan(a-b) is

$\tan (a-b)=\frac{\tan a-tanb}{1+\tan a\tan b}$$\tan (a-b)=\frac{\frac{5}{12}-\frac{4}{3}}{1+\frac{5}{12}\times\frac{4}{3}}=\frac{\frac{5-16}{12}}{1+\frac{5}{9}}=\frac{\frac{-11}{12}}{\frac{14}{9}}=-\frac{11}{12}\times\frac{9}{14}=-\frac{11}{4}\times\frac{3}{14}=-\frac{33}{56}$$\tan (a-b)=-\frac{33}{56}$

Hence the correct option is 2.