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Points A (-5,6), B (2,-2), and C (-6,-3) are placed in three different quadrants of a Cartesian coordinate system. Convert each set of Cartesian coordinates to polar coordinates. The angle should be reported as a positive angle, in degrees, from the positive x-axis. (0° < θ < 360°). What is the θ coordinate for point A? Recall this should be in units of degrees.

Respuesta :

Answer: A ([tex]\sqrt{61}[/tex],309.8°)

              B (2[tex]\sqrt{2}[/tex], 315°)

             C ([tex]3\sqrt{5}[/tex], 26.56°)

Explanation: To transform rectangular coordinates into polar coordinates use:

[tex]r=\sqrt{x^{2}+y^{2}}[/tex] and [tex]\theta=tan^{-1}(\frac{y}{x})[/tex]

For point A:

[tex]r=\sqrt{(-5)^{2}+6^{2}}[/tex]

[tex]r=\sqrt{61}[/tex]

[tex]\theta=tan^{-1}(\frac{6}{-5})[/tex]

[tex]\theta=tan^{-1}(-1.2)[/tex]

[tex]\theta=-50.2[/tex]°

Point A is in the II quadrant, so we substract the angle for 360° since it is in degrees:

[tex]\theta=360-50.2[/tex]

[tex]\theta=[/tex] 309.8°

Polar coordinates for point A is ([tex]\sqrt{61}[/tex], 309.8°)

For point B:

[tex]r=\sqrt{2^{2}+(-2)^{2}}[/tex]

[tex]r=\sqrt{8}[/tex]

[tex]r=2\sqrt{2}[/tex]

[tex]\theta=tan^{-1}(\frac{-2}{2} )[/tex]

[tex]\theta=tan^{-1}(1)[/tex]

[tex]\theta=-45[/tex]°

Point B is in IV quadrant, so:

[tex]\theta=360-45[/tex]

[tex]\theta=[/tex] 315°

Polar coordinates for point B is ([tex]2\sqrt{2}[/tex], 315°)

For point C:

[tex]r=\sqrt{(-6)^{2}+(-3)^{2}}[/tex]

[tex]r=\sqrt{45}[/tex]

[tex]r=3\sqrt{5}[/tex]

[tex]\theta=tan^{-1}(\frac{-3}{-6} )[/tex]

[tex]\theta=tan^{-1}(0.5)[/tex]

[tex]\theta=[/tex] 26.56°

Polar coordinates for point C is ([tex]3\sqrt{5}[/tex], 26.56°)

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