Given sec(x)=-5/4 and sinx<0 what is sin(x/2) in proper rationalized form?

Answer 1

secants & sines are only negative in quadrant III

secx = 1/cosx = 1/(-5/3) =-3/5

It's a 3, 4, 5 right triangle, so sinx =-4/5

sin(x/2) = + or - sqr((1-cosx)/2) = sqr((1+3/5)/2) = sqr4/5 = 2sqr(1/5) = 2/sqr5 = about 2/2.2= 0,894

last problem is sin(x/2) = - or + 2/sqr5?

sinx =-4/5, so x = 233.13 degrees

x/2 = 116.57 degrees

sin116.57 = +0.894 or more exactly +2/sqr5 or +2sqr5/5 = +0.4sqr5

x/2 is in quadrant II, where sines are always positive

Answer 2

sec x = -5/4

cos x = -4/5

sin x = -3/5

Now substitute values in the half-angle formula.