Question

Express the following Cartesian coordinates in polar coordinates in at least two different ways. $$(-1,0)$$

Short answer

Answer: The two polar coordinates representations for the given Cartesian coordinates (-1,0) are (1,π) and (1,3π).

Step-by-step solution

Calculate the radial distance

We will find the radial distance from the origin to the point using the formula: $$r = \sqrt{x^2 + y^2}$$ $$r = \sqrt{(-1)^2 + (0)^2}$$ $$r = \sqrt{1}$$ $$r = 1$$

Calculate the angle (first representation)

We'll calculate the angle \(\theta\) using the formula: $$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$ Since we have \(x=-1\) and \(y=0\), the angle is: $$\theta = \tan^{-1}\left(\frac{0}{-1}\right)$$ $$\theta = \tan^{-1}(0)$$ $$\theta = 0$$ Now we add $$\pi$$ to the angle to get the representation in the second quadrant: $$\theta = 0 + \pi$$ $$\theta = \pi$$ The first representation of the polar coordinates is: $$(r,\theta) = (1,\pi)$$

Calculate the angle (second representation)

Now we'll find another representation of the polar coordinates by adding $$2\pi$$ to the angle: $$\theta = \pi + 2\pi$$ $$\theta = 3\pi$$ The second representation of the polar coordinates is: $$(r,\theta) = (1,3\pi)$$

Final Answer

The two different polar coordinates representations for the given Cartesian coordinates \((-1,0)\) are: $$ (1, \pi) \quad \text{and} \quad (1, 3\pi) $$