Finding the Angle with Arctan: A Manual Approach

[carlodelmundo]

Homework Statement

http://carlodm.com/images/mth.png


** Note: The above is from a PRACTICE question for my course.

Homework Equations

arctan x = (theta)

The Attempt at a Solution

So. (-3,6) is in Quadrant 2. To solve for this angle we use:

(pi) + arctan (-6/2) = (pi) + arctan (-2) = 2.034.

My question is:

How can I solve for arctan(-2) if I don't have a calculator? aka... how do I do it by hand?

Thanks

 

[rock.freak667]

well you can't really do it by hand, you can approximate it, but you'd need a calculator for that.

 

[carlodelmundo]

I only have a four function calculator to solve these types of problems in class. Is it possible with a 4 function? Any other method?

 

[Mark44]

If you're taking a class involving trig, you really should have a scientific calculator. A four-function calculator is not much use at this level.

 

[Hurkyl]

Finding the arctangent is way overkill.

Just looking at the quadrant narrows things down to two possibilities. The right answer can be determined by splitting the quadrant into two parts.

 

[symbolipoint]

Try drawing the triangle on a cartesian coordinate system. You can relate it to a similar triangle with bottom leg 1, left leg of 2, and by pythagorean theorem, hypotenuse of sqrt(5).

In this arrangement, there is reference angle for which sine of this reference angle is same as sine of pi minus reference angle.

My first result for this seems arcsin(ref.angle) = (2/5)*sqrt(5)

I would either use scientific calculator or table of trigonom functions to find the reference angle; then find actual requested angle by pi minus reference angle.

 

[njama]

Here is diagram that I draw recently
http://img196.imageshack.us/img196/8480/arctan.png

Here is what I do. First draw the line between (-3,6) and (0,0). You can see what the angle is, but still can not determine it. The next step is to divide the quadrant on half, (180+90)/2 = 135. You can still see that it is still not too close. Again find the half between 135 and 90 (because the angle is between these ones), (135+90)/2 = 112.5. Now it is close to the right angle. So you can just continue doing the same process over and over again until you are satisfied with the result.

I hope I was helpful.