Find the exact value of the following expressions Sin tan1 3

Find the exact value of the following expressions. Sin [tan^-1 (-3/8)] tan [cos^-1 (-2/5)]

Solution

sin[tan^-1(-3/8)]

let tan^-1(-3/8)=t

therefore tan t=-3/8

here opposite=-3 and adjacent= 8

therefore hypotenuse= sqrt(9+ 64)=sqrt 73

therefore sin t= -3/sqrt 73= -3sqrt73/73

hence sin[tan^-1(-3/8)]=-3sqrt73/73

tan[cos^-1(-2/5))]

Let cos^-1(-2/5)=t

therefore cos t= -2/5

adjacent=-2 and hypotenuse=5

Therefore opposite= sqrt(25-16)= 3

hence tan t= 3/-2

tan[cos^-1(-2/5)]= 3/-2