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)?

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.

Re: Basic concept I need explained for my calculus class

Quote:

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.

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)?

Re: Basic concept I need explained for my calculus class

I think he means that since the tangent function is periodic with period $\displaystyle \pi$, the general solution to $\displaystyle \tan{x} = -\frac{1}{2}$ is $\displaystyle x = -0.464 +k\pi$, where k is any integer.

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 $\displaystyle \pi$". And of course, adding any multiple of $\displaystyle 2\pi$ to either of those will take you around in a complete circle and back to the same position.

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?

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.

Re: Basic concept I need explained for my calculus class

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

$\displaystyle r=\sqrt{x^2+y^2}=\sqrt{4^2+(-2)^2}=\sqrt{20}=2\sqrt{5}=4.472$

$\displaystyle \theta=\tan^{-1}\frac{y}{x}=\tan^{-1}\frac{-2}{4}=\tan^{-1}-\frac{1}{2}=-0.464$

(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 $\displaystyle (r,\theta)$ is a point in the plane, then $\displaystyle (r,\theta+2\pi)$, $\displaystyle (r,\theta+4\pi)$, etc. represent the same point. And also $\displaystyle (-r,\theta+\pi)$, $\displaystyle (-r,\theta+3\pi)$, etc.

In the problem, you are asked to find the two representations with angle between $\displaystyle 0$ and $\displaystyle 2\pi$. Since -0.464 is negative, that representation won't do. But if you add $\displaystyle 2\pi$, 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