Question

If tan x=3/4 and π 14 years ago

Answer 1

1 + tan^2 x = sec^2 x
1 + (3/4)^2 = 1/cos^2 x
1 + 9/16 = 1/cos^2x
cos^2 x = 16/25 Hence cos x = 4/5 or -4/5. Since x is between pi and 3pi / 2, cos x is negative, therefore cos x = -4/5.

Also, tan x = sin x / cos x, which means that sin x = tan x cos x = (3/4)(-4/5) = -3/5.

Thus

a) sin 2x = 2 sin x cos x = 2 (-3/5) (-4/5) = 24/25.

b) cos 2x = 1 - 2 sin^2 x = 1 - 2 (-3/5)^2 = 1 - 2 (9/25) = 1 - 18/25 = 7/25.

c) tan 2x = sin 2x / cos 2x = (24/25) / (7/25) = 24/7

e) cos x = 1 - 2sin^2 (x/2)
2sin^2 (x/2) = 1 - cos x = 1 - (-4/5) = 9/5
sin^2 (x/2) = 9/10
sin (x/2) = 3/sqrt(10) or -3/sqrt(10).

Since x is between pi and 3pi/2, x/2 is between pi/2 and 3pi/4, and hence sin(x/2) is positive; therefire sin (x/2) = 3/sqrt(10).

d) cos x = 2cos^2(x/2) - 1
2cos^2(x/2) = 1 + cos x = 1 + (-4/5) = 1/5
cos^2(x/2) = 1/10
cos(x/2) = 1/sqrt(10) or -1/sqrt(10)

x/2 is between pi/2 and 3pi/4; hence cos(x/2) is negative. Hence cos(x/2) = -1/sqrt(10).

f) tan(x/2) = sin(x/2) / cos (x/2) = (3/sqrt(10)) / (-1/sqrt(10)) = -3.