Find sin tan 1 a2 A aSquareroot a2 4 B a C 1Squareroot2 4

Find sin (tan 1 a/2) A. a/Squareroot a^2 + 4 B. a C. 1/Squareroot^2 + 4 D. 2/Squareroot a^2 + 4

Solution

sin[tan^-1(a/2)]

let arctan(a/2) = y
by definition
tany = (a/2)
bring tany in terms of siny
siny/cosy = (a/2)
siny = (a/2) cosy
squaring
sin^2(y) = (a/2)^2*cos^2(y)
sin^2(y) = (a/2)^2*(1 - sin^2(y))
sin^2(y) = (a/2)^2 - (a/2)^2sin^2(y)
sin^2(y) + (a/2)^2sin^2(y) = (a/2)^2
sin^2(y)[1 + (a/2)^2] = (a/2)^2
sin^2(y) = (a/2)^2 /(1+(a/2)^2)
siny = (a/2)/sqrt(1+(a/2)^2)]
so sin(arctan(a/2)) = siny = a/sqrt(4+a^2)]