Find the exact polar coordinates of the points of intersection of these polar equations: r=8sin(theta) and r=4csc(theta)

Question

When I plot it on a graph I can see the points of intersection at (-4,4) and (4,4), But I can not seem to figure out how to solve this by hand.

Answer 1

r = 8sinx

r = 4cscx = 4/sinx

8sinx = 4/sinx

2sinx = 1/sinx

2sin^2(x) = 1

sin^2(x) = 1/2

sinx = + or - 1/sqr2

x = pi/4 or 3pi/4

r = 8sinx = 8/sqr2 - 8/sqr2 = 4sqr2 or -4sqr2

(4,4), (4,-4)

the 1st and 2nd quadrants have r=4sqr2 >0

the seemingly 3rd and 4th quadrants have r=-4sqr2 <0 which makes it 1st & 2nd quadrants

polar coordinates are (4sqr2, pi/4), (4sqr2, 3pi/4), (-4sqr2, 5pi/4), (-4sqr2, 7pi/4) but the last two are the same as the first two

so just (4sq2, pi/4) and (4sqr2, 3pi/4) are the polar coordinates of the intersection points