Hi, I need to find the domain and rnge of the following:

Question

1. X= 1/2 (cos theta + theta sin theta)
Y=1/2 (sin theta - theta cos theta)

total guess here, but here goes
Domain cos theta = +/- 1 and domain sin theta = +/- 1 so domain theta sin theta = 0- +/-infinity
so domain of X= 1/2 (cos theta + theta sin theta) = 1/2 - +/- infinity
Then range with similar logic of Y=1/2 (sin theta - theta cos theta) = 0 - +/- infinity

Last, domain if x = cot theta

Thank yu

Range using similar logic

Answer 1

As usual, the domain is the set of values for x where the function is defined.

x = 1/2 (cosθ + θsinθ)
domain is all real numbers.
range is all real numbers, since θsinθ oscillates between θ and -θ.

y = 1/2 (sinθ - θcosθ)
same domain and range as for x

This graph is a pair of spirals, starting at (1,0) and expanding outwards, clockwise for t<0, counterclockwise for t>0