Question

Express the following Cartesian coordinates in polar coordinates in at least two different ways. $$(1, \sqrt{3})$$

Short answer

Answer: The Cartesian coordinates (1, √3) can be expressed in polar coordinates as (2, π/3) and (2, 7π/3).

Step-by-step solution

Finding the radius

Find the radius r by using the Pythagorean theorem. \(r = \sqrt{x^2 + y^2}=\sqrt{1^2 + (\sqrt{3})^2}=\sqrt{1 + 3} = \sqrt{4} = 2\)

Finding the angle (First way)

To find the angle θ, use the tangent function and inverse tangent function. \(\tan{θ} = \frac{y}{x}\), so \(θ = \tan^{-1}(\frac{y}{x})=\tan^{-1}(\frac{\sqrt{3}}{1})= \tan^{-1}(\sqrt{3})=60^{\circ}\) Now convert the angle to radians: \(60^{\circ} = \frac{\pi}{3}\) radians. So, the first set of polar coordinates is \((2, \frac{\pi}{3})\).

Finding the angle (Second way)

Taking into account the periodicity of the tangent function, we can add an integer multiple of \(2\pi\) radians to the angle. So, let's add \(2\pi\) radians: \(\frac{\pi}{3} + 2\pi = \frac{\pi}{3} + \frac{6\pi}{3} = \frac{7\pi}{3}\). So, the second set of polar coordinates is \((2, \frac{7\pi}{3})\).