Converting Between Component and Magnitude-Angle Notations

[Llama77]

So I am in a Calculus based Physics course and had some questions about converting to and from the component notation and the magnitude-angle notation.

The first is if I am in the magnitude-angle notation, in a Cartesian cordiante system, will it always be ax=a cos(theta) to get the x component and ay=a sin(theta) to get the 7 component. I am asking because I am a little confused, let's day if the vector is ib the negative quadrant 4, do we sill do the same process.


Secondly when going from components to magnitude-angle notation, we use arctan(ay/ax) to get the angle theta. But what I don't get is why, the book just says tan(theta)=(ay/ax) and though I know enough since I have been told to use the arctan, I don't know why we use it or why the book doesn't say this,

 

[cristo]

A vector can be written in component form as a=(a_x,a_y). The components of a vector can be found by constructing a right angled triangle and using trigonometry. As an example, consider the vector a at an angle θ from the horizontal. Now consider drawing a line from the end of the vector, down to the x axis. We now have a triangle involving the magnitude of a as the hypotenuse, and a_x and a_y, the magnitude of the components.

Using trigonometry on the triangle we can obtain a_x=acosθ (where a is the magnitude of a) and tanθ=a_y/a_x.

 

[Llama77]

I know that already, but why do we need to use a arctan instead of just the standard tan.

 

[cristo]

I know that already, but why do we need to use a arctan instead of just the standard tan.

arctan is the inverse operator of tan, otherwise written as tan^-1. So, given the equation tanθ=a_y/a_x, taking the inverse tangent of each side, we obtain arctan(tanθ)=arctan(a_y/a_x) => θ= arctan(a_y/a_x).