Convert the following Cartesian equations to polar equations

Convert the following Cartesian equations to polar equations, x^2 - y^2 = 1 y = x^2 Convert the following polar equations to Cartesian equations, r(theta) = b/sin(theta) - m cos(theta) r(theta) = a sin(2 theta)/2(cos^3 theta + sin^3 theta)

Solution

Solution:3(a)

x^2 - y^2 = 1

x = r cos & y = r sin

(r cos )^2 - (r sin )^2 = 1
r^2 cos^2 - r^2 sin^2 = 1
r^2 (cos^2 - sin^2 ) = 1
r^2 cos 2 = 1
r^2 = sec 2

Solution :3(b)

y = x^2

y=rsin and x=rcos
then:
rsin = (rcos)^2
rsin = r^2cos^2
sin = rcos^2

r = sin / cos^2

r = tan sec

Solution:4(a)

r() = b/(sin - m cos)

r (sin - m cos) = b

r sin - m r cos = b

we know that

x = r cos & y = r sin

y - mx = b

y = mx + b