Graph r=3cos(x)

Problem

r=3*cos(θ)

Solution

1. Identify the type of polar equation. The equation r=a*cos(θ) represents a circle in the polar coordinate system.

2. Determine the diameter and center. For r=3*cos(θ) the diameter of the circle is a=3 Since it is a cosine function, the circle is symmetric with respect to the polar axis (the x-axis) and passes through the origin.

3. Calculate key points by substituting values for θ

θ=0⇒r=3*cos(0)=3

θ=π/4⇒r=3*cos(π/4)≈2.12

θ=π/2⇒r=3*cos(π/2)=0

θ=(3*π)/4⇒r=3*cos((3*π)/4)≈−2.12

θ=π⇒r=3*cos(π)=−3

4. Convert to rectangular coordinates to verify the shape. Use the identities x=r*cos(θ) and r2=x2+y2

r2=3*r*cos(θ)

x2+y2=3*x

x2−3*x+y2=0

5. Complete the square for the x terms to find the standard form of the circle.

(x−3/2)2+y2=(3/2)2

6. Sketch the graph. The graph is a circle centered at the rectangular coordinates (3/2,0) with a radius of 3/2

Final Answer

r=3*cos(θ)* is a circle centered at *(r,θ)=(1.5,0)* with diameter *3

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