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Question
Find the polar coordinates of the point whose Cartesian coordinates are `(1, - sqrt(3))`.
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Solution
Here x = 1 and y = `- sqrt(3)`
∴ The point lies in the fourth quadrant.
Let the polar coordinates be (r, θ).
Then r2 = x2 + y2 = `(1)^2 + (- sqrt(3))^2` = 1 + 3 = 4
∴ r = 2 ...[ ∵ r > 0]
`cos θ = x/r = (1)/(2)`
and `sin θ = y/r = - sqrt(3)/(2)`
∴ tan θ = `- sqrt(3)`
Since, the point lies in the fourth quadrant and 0 ≤ θ < 2π
tan θ = `- sqrt(3) = - tan π/3`
= `tan(2π - π/3)` ...[ ∵ tan(2π – θ) = –tan θ]
= `tan (5π)/3`
∴ θ = `(5π)/3`
∴ The polar coordinates of the given point are `(2, (5π)/3)`.
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