Find the exact value?

Question

cot*theta = 2 (pi < theta < 3*pi/2)

Find the exact value for the following expressions (using double angle formulas):

a) sin 2*theta

b) cos 2*theta

Answer 1

cotθ = 2 so sinθ = - 1/ √5 and cosθ = - 2/ √5 from a 1, 2, √5 right angled triangle, both are negative because the angle is in the third quadrant.

a) sin2θ = 2sinθcosθ

sin2θ = 2 x (- 1/ √5)(- 2/ √5) = 4/5

b) cos2θ = 2cos²θ - 1

cos2θ = 2x(-2/ √5)² - 1 = 3/5

Answer 2

theta = arctan(1/2) + pi

a) sin 2*theta = 2 sin theta * cos theta
= 2 sin (arctan 1/2 + pi) * cos (arctan 1/2 + pi)
= 4/5

b) cos 2*theta = cos^2 theta - sin^2 theta
= 2 cos^2 theta - 1
= 1 - 2*sin^2 theta
= 3/5

Answer 3

Let theta = x
Question a)
1 / tan x = 2
tan x = 1 / 2
sin 2x = 2 sinx.cosx
sin 2x = 2 . (1 / √5).(2 / √5)
sin 2x = 4 / 5

Question b)
cos 2x = cos²x - sin²x
cos 2x = 4/5 - 1 / 5
cos 2x = 3 / 5

Answer 4

The problem is not clear, notation either.
What is cot*theta ?