Given tan theta= 1/2, pi< theta < 3/2 pi, find sin 2 theta.

Question

I have no idea how to do this problem. Please show me the steps and the answer in simplest form. 

Answer 1

Solve for theta using inverse tan of 1/2. You can do this on your calculator be selecting tan^-1(1/2).

That will give you a value 26.565 degrees. They specify that they want the value to be in between pi and 3/2 pi so you can just add 180 degrees to your answer and get 206.565 degrees. Now plug in Sin(2*206.565) = .8.

Answer 2

An alternative method without the use of calculator is to use the definition of the trigonometric ratios.

Since tan θ = 1/2 = opposite/adjacent,

opposite side =1

adjacent side = 2

hypotenuse = √5

sin θ  = 1/√5

cos θ = 2/√5

The identity sin2θ = 2 sinθ. cosθ = 2 . 1/√5 . 2/√5 = 4/5