Graph r=1+3sin(x)

Problem

r=1+3*sin(θ)

Solution

1. Identify the type of polar equation. The equation is in the form r=a+b*sin(θ) where a=1 and b=3 Since |a|<|b| the graph is a limaçon with an inner loop.

2. Determine the symmetry of the graph. Because the equation involves sin(θ) the graph is symmetric with respect to the vertical axis θ=π/2 (the y-axis).

3. Find the intercepts and key points by evaluating r at quadrantal angles.

θ=0⇒r=1+3*sin(0)=1

θ=π/2⇒r=1+3*sin(π/2)=4

θ=π⇒r=1+3*sin(π)=1

θ=(3*π)/2⇒r=1+3*sin((3*π)/2)=−2

4. Locate the inner loop by finding where r=0

1+3*sin(θ)=0

sin(θ)=−1/3

θ≈3.48,5.94* radians

5. Sketch the curve starting from θ=0 The graph moves from (1,0) up to (4,π/2) back down to (1,π) passes through the origin to form an inner loop reaching a distance of 2 units at θ=(3*π)/2 and returns to (1,2*π)

Final Answer

r=1+3*sin(θ)

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