Express the equation in rectangular form

a) r = 2sin (2Θ)

b) r = 4cos (3Θ)

a. sqrt (x^2+y^2) * (x^2+y^2) = 4xy

b. (x^2+y^2) ^2 = 4x{ (x^2 - 3 y^2}

Step-by-step explanation:

a) r = 2sin (2Θ)

r * = 2 sin 2 theta

We know sin 2theta = 2 sin theta cos theta

r = 2 * 2 sin theta cos theta

sin theta = y/r and cos theta = x/r

r = 4 y/r * x/r

r = 4xy / r^2

Multiply each side by r^2

r^3 = 4xy

r * r^2 = 4xy

We know r = sqrt (x^2+y^2) and r^2 = (x^2+y^2)

sqrt (x^2+y^2) * (x^2+y^2) = 4xy

b) r = 4cos (3Θ)

we know that cos 3 theta = cos^3 (theta) - 3 sin^2 (theta) cos (theta)

r = 4 * cos^3 (theta) - 3 sin^2 (theta) cos (theta)

Factor out a cos theta

r = 4 * cos (theta) { cos^2 (theta) - 3 sin^2 (theta) }

We know that sin theta = y/r and cos theta = x/r

r = 4 (x/r) { (x/r) ^2 - 3 (y/r) ^2}

r = 4 * (x/r) { (x^2/r^2 - 3 y^2/r^2}

Multiply by r

r^2 = 4x { (x^2/r^2 - 3 y^2/r^2}

Multiply by r^2

r^2 * r^2 = 4x { (x^2 - 3 y^2}

r^4 = 4x{ (x^2 - 3 y^2}

We know r^2 = (x^2+y^2)

(x^2+y^2) ^2 = 4x{ (x^2 - 3 y^2}