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Question
Find the polar co-ordinates of point whose Cartesian co-ordinates are `(1, sqrt(3))`
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Solution
(x, y) ≡ `(1, sqrt(3))` .......[Given]
Using x = r cos θ and y = r sin θ, where (r, θ) are the required polar co-ordinates, we get
1 = r cos θ, `sqrt(3)` = r sin θ
Now, r = `sqrt(x^2 + y^2)`
= `sqrt(1 + 3)`
= 2
and tan θ = `("r" sin theta)/("r" cos theta)`
= `sqrt(3)/1`
= `sqrt(3)`
= `tan pi/3`
∴ θ = `"n"pi + pi/3`, n ∈ Z .......`[(∵ tan theta = tan alpha "implies"),(theta = "n"pi + alpha"," "n" ∈ "Z")]`
For polar co-ordinates, 0 ≤ θ < 2π
∴ θ = `pi/3` or θ = `pi + pi/3 = (4pi)/3`
But the given point lies in the 1st quadrant.
∴ θ = `pi/3`
∴ The required polar co-ordinates are `(2, pi/3)`.
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