Thread: 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)?

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.

Originally Posted by chiro

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.

Just for future reference, do you mean the other duplicates for the unit circle (or other periodic translations)?

I think he means that since the tangent function is periodic with period , the general solution to is , where k is any integer.

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 ". And of course, adding any multiple of to either of those will take you around in a complete circle and back to the same position.

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?

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.

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




(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 is a point in the plane, then , , etc. represent the same point. And also , , etc.

In the problem, you are asked to find the two representations with angle between and . Since -0.464 is negative, that representation won't do. But if you add , 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