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Math Help - Basic concept I need explained for my calculus class

  1. #1
    TWN
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    Basic concept I need explained for my calculus class

    This is probably a basic concept that I just never really grasped...now I'm in Calc III and it's biting me in the butt.

    I need to find theta = arctan(-1/2)

    I typed it into my calculator to receive the answer -.464 radians

    But calculators are limited when it comes to solving trig, and as I understand there are other answers other than what I received. How do I go about finding the other answer(s)?
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  2. #2
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    Re: Basic concept I need explained for my calculus class

    Hey TWN.

    What kind of value are you looking for? Are you looking for something in terms of pi?

    I used the R package and I got the following answer:

    > atan(-1/2)
    [1] -0.46364760900080609352

    which is also in radians.
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    Re: Basic concept I need explained for my calculus class

    Quote Originally Posted by chiro View Post
    Hey TWN.

    What kind of value are you looking for? Are you looking for something in terms of pi?

    I used the R package and I got the following answer:

    > atan(-1/2)
    [1] -0.46364760900080609352

    which is also in radians.
    No, the OP wants to know how to get all the other answers. I answered this in his other thread.
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    Re: Basic concept I need explained for my calculus class

    Just for future reference, do you mean the other duplicates for the unit circle (or other periodic translations)?
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    Re: Basic concept I need explained for my calculus class

    I think he means that since the tangent function is periodic with period \pi, the general solution to \tan{x} = -\frac{1}{2} is x = -0.464 +k\pi, where k is any integer.
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    Re: Basic concept I need explained for my calculus class

    One way to thinking about this is to draw a line from the origin of a coordinate system to (2, 0) then down to (2, -1) then back to (0, 0). That gives you a right triangle with opposite side "-1" and "near side" 2 so that the tangent of the angle at the origin is -1/2. If you continue that line from the fourth quadrant to the second quadrant, you will get a line to (-2, 1) which gives a right triangle that has "near side" -2 and "opposite side" 1 so that we still have tangent equal to -1/2. That is why "tangent is periodic with period \pi". And of course, adding any multiple of 2\pi to either of those will take you around in a complete circle and back to the same position.
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    TWN
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    Re: Basic concept I need explained for my calculus class

    The question I am applying this to is as follows:

    The rectangular coordinates of a point are (4, -2). Plot the point and find two sets of polar coordinates for the point for 0 > theta > 2pi.

    So adding multiples of 2pi would not be correct in this situation. I'm either not looking at this concept correctly or I am just not doing this problem correctly. Can someone walk me through it?
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  8. #8
    TWN
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    Re: Basic concept I need explained for my calculus class

    Someone please answer the question in my last post as soon as possible. I am going to need to understand it within the next few hours.
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    Re: Basic concept I need explained for my calculus class

    Well, if we just wanted to convert to polar coordinates, I would say:

    r=\sqrt{x^2+y^2}=\sqrt{4^2+(-2)^2}=\sqrt{20}=2\sqrt{5}=4.472
    \theta=\tan^{-1}\frac{y}{x}=\tan^{-1}\frac{-2}{4}=\tan^{-1}-\frac{1}{2}=-0.464

    (approximately). And for these conversions, you need to double-check: the point x=4, y=-2 is in quadrant 4, and r=4.472, theta=-0.464 is also in quadrant 4, and roughly the same place. I always draw a graph - that's hard to do here.

    You might be aware that the polar representation of a given point is not unique. If (r,\theta) is a point in the plane, then (r,\theta+2\pi), (r,\theta+4\pi), etc. represent the same point. And also (-r,\theta+\pi), (-r,\theta+3\pi), etc.

    In the problem, you are asked to find the two representations with angle between 0 and 2\pi. Since -0.464 is negative, that representation won't do. But if you add 2\pi, you get r=4.472, theta=5.820 which is good, and the other one has r negative: r=-4.472, theta=2.678.

    - Hollywood
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