Question 6.3.7: Converting Equations from Polar to Rectangular Form Convert ...
Converting Equations from Polar to Rectangular Form
Convert each polar equation to a rectangular equation in x and y:
a. r = 5 b. θ = \frac{π}{4} c. r = 3 csc θ d. r = -6 cos θ.
In each case, let’s express the rectangular equation in a form that enables us to recognize its graph.
a. We use r² = x² + y² to convert the polar equation r = 5 to a rectangular equation.
r = 5 This is the given polar equation.
r² = 25 Square both sides.
x² + y² = 25 Use r² = x² + y² on the left side.
The rectangular equation for r = 5 is x² + y² = 25. The graph is a circle with center at (0, 0) and radius 5.
b. We use \tan θ =\frac{y}{x} to convert the polar equation θ =\frac{\pi}{4} to a rectangular equation in x and y.
θ =\frac{\pi}{4} This is the given polar equation.
\tan θ =\tan\frac{\pi}{4} Take the tangent of both sides.
tan θ = 1 \tan\frac{\pi}{4}=1
\frac{y}{x}=1 Use \tan θ=\frac{y}{x} on the left side.
y = x Multiply both sides by x.
The rectangular equation for θ =\frac{\pi}{4} is y = x. The graph is a line that bisects quadrants I and III. Figure 6.28 shows the line drawn in a polar coordinate system.
c. We use r sin θ = y to convert the polar equation r = 3 csc θ to a rectangular equation. To do this, we express the cosecant in terms of the sine.
r = 3 csc θ This is the given polar equation.
r=\frac{3}{\sin θ} \csc θ=\frac{1}{\sin θ}
r sin θ = 3 Multiply both sides by sin θ.
y = 3 Use r sin θ = y on the left side.
The rectangular equation for r = 3 csc θ is y = 3. The graph is a horizontal line three units above the x-axis. Figure 6.29 shows the line drawn in a polar coordinate system.
d. To convert r = -6 cos θ to rectangular coordinates, we multiply both sides by r. Then we use r² = x² + y² on the left side and r cos θ = x on the right side.
r = -6 cos θ This is the given polar equation.
r² = -6r cos θ Multiply both sides by r.
x² + y² = -6x Convert to rectangular coordinates: r² = x² + y² and r cos θ = x.
x² + 6x + y² = 0 Add 6x to both sides.
x² + 6x + 9 + y² = 9 Complete the square on x: \frac{1}{2}·6=3 and 3² =9.
(x + 3)² + y² = 9 Factor.
The rectangular equation for r = -6 cos θ is (x + 3)² + y² = 9. This last equation is the standard form of the equation of a circle, (x – h)² + (y – k)² = r², with radius r and center at (h, k). Thus, the graph of (x + 3)² + y² = 9 is a circle with center at (-3, 0) and radius 3.
