Evaluate the following problems on polar coordinates and complex numbers. (a) Co

Question

Evaluate the following problems on polar coordinates and complex numbers. (a) Convert polar equation r^2 - 2r sin theta = 0 to its equivalent Cartesian equation. Then identify and describe the graph. (b) Convert Cartesian equation x^2 + y^2 - 4x = 0 into its equivalent polar equation. (c) Plot the graph and list all possible coordinates of the point (-2, - pi/3) (d) Let z_1 = 1 + i and z_2 = squareroot 3 - i: determine the following. z_1 + z_2 = k, find the Cartesian equation of k. Express k in polar form.

Explanation / Answer

z1 = 1+ i
z2 = sqrt(3) - i
we are to find z1+z2 = k
k = z1+z2= = 1+i + sqrt(3) - i = 1+sqrt (3) +0*i = 1+sqrt(3)
Its cartesian equation is (1+sqrt(3),0)
'k' in cartesian form: since y component =0, theta = sin -1 (y/x) = sin -1 0 = 0
R = sqrt (x2 + y2)
= sart(x2) = x
Thus, k in polar form is (r, theta) = (1+sqrt(3) , 0 )  

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