Trig - Converting polar equation to rectangular

Question

r = 2sin(Theta + pi/4) to rectangular

Answer 1

Using the sine angle addition identity sin(x + y) = sin(x)cos(y) + cos(x)sin(y), we can say that r = 2sin(θ+ π/4) is the same as r = 2[sin(θ)cos(π/4) + cos(θ)sin(π/4)] and since cos(π/4) and sin(π/4) both equal √2/2, the equation becomes r = 2sin(θ)(√2/2) + 2cos(θ)(√2/2) or r = √2sin(θ) + √2cos(θ)

Now, using the "conversion to rectangular coordinates" equations:

y = rsin(θ)

x = rcos(θ)

by substitution, we get:

r = √2y/r + √2x/r

Multiplying through by r, we get:

r² = √2y + √2x

By using the "conversion to rectangular coordinates" equation x² + y² = r², this becomes:

x² + y² = √2y + √2x

Moving all to the left side of the equation, we get:

x² - √2x + y² - √2y = 0

Completing the square, we get:

(x² - √2x + (√2/2)²) + y² - √2y + (√2/2)²) = (√2/2)² + (√2/2)² or

(x - √2/2)² + (y - √2/2)² = 1