Translation and then a rotation of a triangle

I have a triangle and the vertices were A(3,−1), B(−2,1) and C(2,3). I made a translation on a triangle (in another part of the paper) and now its new

vertices are A(5,−2), B(0,0) and C(4,2).

Next, I must rotate the triangle so that the line BA lies along the positive x-axis.

Here is the questions:

(3) Let rθ be the rotation that completes the required transformation, where θ lies in the interval (−π,π]. Find the exact values of tan θ,cos θand sin θ, and hence write down a formal definition of rθ using two-line notation. (There is no need to work out the value of the angle θ.)

- This is where I have the problems. I worked out as follows, but that is a lot of typing numbers so the answers I found are:

cos(θ)= 5*sqrt(29)/29

sin(θ)= 2*sqrt(29)/29

tan(θ)=2/5

-- Is this correct?

If so, I then tried to make the formal definition and I arrived with (but I can not show the right arrow shape!):

r(θ): R*2-->R*2

(x,y) |--> (x cos θ - y sin θ, x sin θ + y cos θ)

So I replace all of this with the values at the above paragraph (a lot of typing of the surds, so I don't be putting it all here the working out). We do not show matrices as we are not covering them yet in class, so I must always write with surds.

I finish with the (x,y) values of

(25*sqrt(29)/29 + 45*sqrt(29)/29, 10*sqrt(29)/29 - 10*sqrt(29)/29)

This makes coordinates ( (sqrt(29), 0).

-- But this does not look correct to me. Particular on the drawing I made. Please advise if I am doing this correct.

(4) Find the coordinates of the images of A' and C' under the rotation rθ. Give your answers as exact values. This means surds.

- But I just don't know how to do this question. I wrote a lot of workings and find different answers each time and they don't look right on the drawing that I made of the triangle in the x axis.

-- Can you please help me as I am spending so many hours on this today and yesterday and I go not very far!

(5) Write down a formal definition of the composite transformation: that is the result of the translation followed by the rotation above.

- I can not make this definition without getting the above question 2 well done.

Please help someone who can!

Brigitte