Question

Find the points of horizontal and vertical tangency to the polar curve. r=a sin(theta). 0 <= theta < pi. The answers have to be ordered from smallest to largest. I got 1. (a,0) 2. (a,pi/2) for the horizontal tans and 1. (a/2, 1/2) 2. (-a/2, 1/2) for the vertical tans. Are these answers correct? Please explain. I found dy = asin(2theta) and dx = acos(2theta)

I suspect that the vertical tans should be (a/sqrt2, pi/4) and (a/sqrt2, 3pi/4)

Answer 1

also interval is o to pi

Answer 2

Your answers are almost all good, but don't know if your ordered pair refers to (x,y), or (r,theta). There seems to be a mix of the two.
The rule of thumb is: whenever in doubt, plot the graph of the function. Graphs don't lie!

If in doubt, work out the following:
Let t=\(\theta\), then
r(t)=a sin(t) [given]
x(t)=r(t)cos(t)=a sin(t)cos(t),
y(t)=r(t)sin(t)=a sin(t)sin(t)
equate x'(t) and y'(t) to zero to get the horizontal tangents.
The answers should be obvious from the above graph.