Question

The point (r,theta) in polar coordinates is (7,5) in rectangular coordinates. What is the point (2r, theta + pi/2) in rectangular coordinates? So I tried this problem and got r = sqrt(74). But while finding the value of theta, I got it equal to cot(5/7) which ends up being some decimal number but I need it to be expressed as a square root or fraction in order to continue because I have to have my answer in exact form. Any help?

Answer 1

r = sqrt (74)
theta = tan^-1 (5/7)
leave it that way, do not go to degrees

2 r = 2 sqrt(74)

draw this axis system to figure out tan theta+pi/2

we are in Quadrant 2 theta degrees from the y axis
so now tan of new theta = 7/-5
and we have the point
( 2 sqrt 74 , tan^-1 (-7/5)

now from the drawing you can see that
x = -5 (2) = -10
and
y = 7 (2) = 14
so we are at
(-10,14)

Answer 2

For (r,θ),
x = r cosθ = 7
y = r sinθ = 5

For the new point, we have

x = 2r cos(θ+π/2) = -2r sinθ
y = 2r sin(θ+π/2) = 2r cosθ

so, rectangular coordinates for (2r,θ+π/2) are (-10,14)