Given that sin theta 45 and sec theta 0 identify which qua

Given that sin theta = -4/5 and sec theta > 0, identify which quadrant theta lies in and find: (a) cot theta (b) cos (-theta) (c) cos(theta + 7 pi/6) (d) tan(2 theta) (e) sin(theta/2)

Solution

sinx<0 and secx>0 only in quadrant IV

sin^2x + cos^2x = 1

16/25 + cos^2x = 1

cos^2x = 3/5 as cosine is positive in IV quadrant

a) Cotx = cosx/sinx = (3/5)/(-4/5) = -3/4

b) cos(-x) = cosx = 3/5

c) Tanx = 1/cotx = -4/3

tan2x = (2tanx)/(1-tan^2x)

=(-2*4/3)/1-(16/9)
= (-8/3)/(-7/3)

= 8/7