SOLUTION: Find two polar coordinate representations for the rectangular coordinate point (&#8722;6, 2 sqrt 3), one with r > 0 and one with r < 0 and both with 0 &#8804; &#952;

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Question 1009209: Find two polar coordinate representations for the rectangular coordinate point
(−6, 2 sqrt 3), one with r > 0 and one with r < 0 and both with 0 ≤ θ < 2π.

Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
the rectangular coordinate is (-6,2sqrt(3))

this point is in the second quadrant where the x-coordinate is -6 and the y-coordinate is 2sqrt(3).

this creates a triangle where the length of the adjacent side is 6 units in length and the opposite side is 2sqrt(3) units in length.

to find the angle in the triangle we find tan^-1(2sqrt(3)/6) which is equal to 30 degrees.

that angle is the reference angle.

the reference angle is the equivalent angle in the first quadrant.

to find the angle in the second quadrant, we take 180 - 30 = 150 degrees.

the angle we want is 150 degrees.

the length of the hypotenuse of this triangle is equal to sqrt(48).

that's by pythagorus that says that x^2 + y^2 = h^2.

we get (2sqrt(3))^2 + (6)^2 = 12 + 36 = 48 which makes h = sqrt(48).

h = hypotenuse of the right triangle and also equals r.

the point (-6,2sqrt(3)) in rectnagular coordinates is the same as the point (sqrt(48),150 degrees) in polar coordinates.

so one way to describe the point is (sqrt(48),150 degrees)

this is the form of (r,angle), where r = sqrt(48) and angle = 150 degrees.

another way to describe the point is (-sqrt(48),330 degrees).

we get this by adding 180 to 150 to get 330 and then taking -r rather than r.

it's another way of describing the exact same point.

the point (-6,2sqrt(3)) in rectangular coordinates is equivalent to (sqrt(48),150 degrees) or (-sqrt(48),330 degrees) in polar coordinates.

to convert from degrees to radians, multiply degrees by pi and then divide by 180 to get radians.

150 degrees * pi / 180 is equal to 5/6 * pi radians.

330 degrees * pi / 180 is equal to 11/6 * pi radians.

in radians, then:

(-6,2sqrt(3)) equals (sqrt(48),5/6*pi radians) and (-sqrt(48),11/6*pi radians).

visually, this looks like this:

both of these diagrams point to the same point of (-6,2sqrt(3)).

when r is minus, you go the indicated number of units in the opposite direction shown.

here's a reference to some good lessons on polar cooraindates by professor robbob.

there are 3 videos in the series.

take a look at the first and it should then point to the second and third if you wish to learn further.

https://www.youtube.com/watch?v=tKi05dfUhAA

SOLUTION: Find two polar coordinate representations for the rectangular coordinate point (&#8722;6, 2 sqrt 3), one with r > 0 and one with r < 0 and both with 0 &#8804; &#952; < 2&#960;.

SOLUTION: Find two polar coordinate representations for the rectangular coordinate point (−6, 2 sqrt 3), one with r > 0 and one with r < 0 and both with 0 ≤ θ < 2π.