The polar equation for the line containing the origin that makes an angle 30∘ with the positive x− axis.
Solution:
The polar equation of the line containing the origin that makes an angle 30∘ with the positive x− axis is θ=π6
Given information:
The line passes through the origin and makes an angle 30∘ with the positive x− axis.
The general equation of the line containing the origin that makes an angle θ with the positive x− axisis y=mx, where m=tanθ is the slope
The equation of the line containing the origin that makes an angle 30∘ with the positive x− axis is y=mx=tan(30∘)x=x3 .
To convert the rectangular equation into polar equation, substitute polar coordinates x=rcosθ,y=rsinθ
⇒rsinθ=rcosθ3
⇒sinθ=cosθ3
⇒sinθcosθ=13
⇒tanθ=13
Apply tan−1 both sides
⇒θ=tan−1(13)
⇒θ=π6
Therefore, the polar equation of the line containing the origin that makes an angle 30∘ with the positive x− axis is θ=π6 .
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