Question

find the exact value of tan x/2, givin that sin x=8/17 and 90degrees 15 years ago

Answer 1

For this problem you need to use your trig half-angle formulas
sin(x/2) = sqrt((1-cosx)/2)
cos(x/2) = sqrt((1+cosx)/2)
tanx = sinx/cosx
so tan(x/2) = sin(x/2)/cos(x/2)
->tan(x/2) = sqrt((1-cosx)/2) / sqrt((1+cosx)/2)
But now the function is defined in terms of cosx and we are given value of sinx so we use a trig identity
sin^2 + cos^2 = 1 -> cosx = sqrt(1-sinx^2)
-> cosx = sqrt(1-(8/17)^2) = sqrt((17^2-64)/17^2) = 15/17
sub that into the above equation for cosx
with a little simplifying
->tan(x/2) = sqrt(1/17) / sqrt(16/17)
= (1/sqrt(17))*(sqrt(17)/sqrt(16)) = 1/4

Answer 2

actually tan(x/2) = -1/4 due to restrictions on x
remember 1/sqrt(16) = +-(1/4)