please check my work and explain, good explaination wil get 10 points for the best answer?

Question

i have a problem. a vector has an x-component of -25 units and y -component of 40.unit, find the magnitude and direction of this vector

i did this problem like
i drew from where they started to forward the left to get -25 units, then go up 40units
|
| 40 units
|
<-------------------
-25 units
magnitude is square root ((-25)^2 +40^2) = 47.2 units by pythagorean theorem

i have a diifficult to understand the angle , because i look the answer on the back i got the right answer but i dont know that is illegal method to do it

tan x = 40/-25
x = tan^-1 (40/-25) = -57.9

angle is 180+ - 57.9 = 122.1 degree
i dont know why i added 180 degree to that answer, please helps me out. thank you

Answer 1

Tan(40/-25)
The Right angle triangle in this case is in the second quadrant.
The Solution of angle i.e x=tan^-1(40/-25) returns the answer =-57.9,meaning that the angle is measured in the clockwise direction that is from the negative x axis to the Hypotenuse.
The 180 is added to this negative value only to obtain the anticlockwise measurement of the angle from the +tive X axis to the Hypotenuse( =180-57.9=122.1 degree) and the tangent of 122.1 is 40/-25=-1.6

Answer 2

There are 2 angles that have tan x = 40/-25.
One as in your case is in second quadrant, another is in forth. In latter case x is positive but y is negative.
Calculator gave you an answer for the second angle. For tan^-1 it will give you answers from -90 to +90. -57.9 is your case means that you have to measure angle clockwise from the x axis.
Now, if you look at your picture you will find that your vector is separated from negative x axis by the same angle so it is by 57.9 degrees less than 180, that is why you have to subtract 57.9 from 180 and get 180-57.9

Answer 3

It's because, to be able to handle periodic functions, your calculator gives you what is known as the 'principle value' for all of the trig (and hyperbolic) functions and their inverses. In this case, you were taking arctangent of -(40/25) and your calculator has no way of 'knowing' if the argument is negative because of the numerator or denominator being negative. So it assumes that the numerator is the negative quantity and gives you an answer that's in the 4'th quadrant. It's up to you to notice that the denominator is negative and add 180° to the result. (The same thing happens for a vector in the 3'rd quadrant. You'll get an angle in the 1'st quadrant.)
The general rule of thumb is that if the denominator (the x value) is negative, add 180° (or π rad) to the result.

HTH

Doug

Answer 4

YOu need to convert the angle into a second quadrant angle. If you drew the resultatnt vector (the sum of the two) you'd find it pointed up (positive y direction) and to the left (negative x direction). That put the angle in the second quadrant. Your calculator doesn't know anything of this, so it uses sin(-x) = -sin(x) and thus tan(-x) = -tan(x). You need to supply the smarts to get the angle inthe corerct quadrant - -- which you did!

Answer 5

The arctan function only gives values that are in the first and fourth quadrants. (+x,+y) and (+x,-y). When you plug in your numbers your calculator doesn't know if it is (-y/x) or (y/-x), and if its (-y/-x) that is the same as (y/x). I have heard in newer versions, such as the Ti-89, that this problem doesn't occur.

Theta = arctan(y/x) or opposite over adjacent.
Because your angle has a positive y value, and a negative x value, you will receive an answer that you must add an angle to (cause the function only gives values in the first and fourth quadrants).

Answer 6

The angle is measured with the positive direction of X-axis. The negative angle you got is with the negative direction of X-axis. When you add 180 degrees to it, you get the angle with the positive direction of X-axis which is the correct answer.

Answer 7

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