Question

Determine two pairs of polar coordinates for the point (3, 3) with 0° ≤ θ < 360

Answer 1

Work shown above! Answers are (3√2, π/4) and (3√2, 5π/4)

Answer 2

Answer: The answer is  (3√2, 45°) and (3√2, 225°).

Step-by-step explanation:  We are given to determine two pairs of polar co-ordinates for the point (3, 3), where 0° ≤ θ < 360.

We know that the relation between Cartesian Coordinates (x,y) and Polar Coordinates (r,θ) is given by the following:

$r=\sqrt{x^2+y^2},~~~~~~~~~~~\theta=\tan^{-1}\dfrac{y}{x}.$

We have,

(x, y) = (3, 3).

Therefore,

$r=\sqrt{x^2+y^2}=\sqrt{3^2+3^2}=\sqrt{18}=3\sqrt2,$

and

$\theta=\tan^{-1}\dfrac{y}{x}=\tan^{-1}\dfrac{3}{3}=\tan^{-1}(1)=\tan^{-1}\tan\dfrac{\pi}{4}\\\\\\\Rightarrow \theta=\tan^{-1}\tan\left(n\pi+\dfrac{\pi}{4}\right),~~\textup{['n' is an integer and the period of tan function is }\pi]\\\\\\\Rightarrow \theta=n\pi+\dfrac{\pi}{4}=180^\circ n+45^\circ.$

If n = -1, then  θ = -180° + 45° = -135°,

If n = 0, then  θ = 0° + 45° = 45°,

If n = 1, then  θ = 180° + 45° = 225°,

If n = 2, then  θ = 360° + 45° = 405°, etc.

Since  0° ≤ θ < 360, therefore the value of  θ is 45° and 225°.

Thus, the two pairs of polar co-ordinates are (3√2, 45°) and (3√2, 225°).