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:
[tex]r=\sqrt{x^2+y^2},~~~~~~~~~~~\theta=\tan^{-1}\dfrac{y}{x}.[/tex]
We have,
(x, y) = (3, 3).
Therefore,
[tex]r=\sqrt{x^2+y^2}=\sqrt{3^2+3^2}=\sqrt{18}=3\sqrt2,[/tex]
and
[tex]\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.[/tex]
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°).
The two pairs of polar coordinates for the point (3, 3) with 0° ≤ θ < 360 are (3√2, 45) and (3√2, 225)
The polar form of the coordinate (x, y) is expressed as (r, theta)
Determine the modulus "r"
r = √x^2 + y^2
r = √3^2+3^2
r = √18
r = 3√2
Determine the modulus
theta = tan^-1(3/3)
theta = tan^-1(1)
theta = 45 degrees
Hence one of the polar pair is (3√2, 45)
Since tan is positive in the third quadrant hence the other polar coordinate is (3√2, 225)
Learn more on polar coordinates here: https://brainly.com/question/14965899
