Question 189534
sin theta= 12/13, 3pi/2<theta<2pi find sin2theta,cos2theta,tan2theta?

I will use t for theta.  

So, let t = theta for short.

sin(t)= 12/13

We make a right triangle knowing that sine equals opposite side divided by hypotenuse.

We need to find one of the legs of this right triangle.

Let x = the missing leg.

x^2 + (12)^2 = (13)^2

x^2 + 144 = 169

x^2 = 169 - 144

x^2 = 25

x = 5

The missing leg is 5.

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We now need to know what sin2(t), cos2(t) and tan2(t) stand for.

cosine = adjacent side of right triangle divided by hypotenuse.

tangent = opposite side of right triangle divided by the adjacent side of the right triangle.

sin2(t) = 2sin(t) times cos(t)

I will do this one only.  I then will give you the information you need to solve the other two trig expressions.

sin2(t) = 2sin(t) times cos(t)

Before I can go on, I need to know what cos(t) stands for.  To do so, I go back to the right triangle knowing that cosine = adjacent side/hypotenuse.

sin(t) was given to be 12/13.

If cosine = adjacent side/hypotenuse, then cos(t) = 5/13.

I now have everything to solve for sin2(t).

sin2(t) = 2(12/13) times 5/13

sin2(t) = 120/169

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You will need to find the rest.

Here is the data you need:

tangent = 12/5

cos2(t) = cos^2 (t) - sin^2 (t)

tan2(t) = [2tan(t)]/[1 - tan^2 (t)]

I gave you everything you need.

All you have to do and plug and chug.

Can you take it from here?