In the ap physics textbook it says that if 0 denotes the angle that the vector a=3i+4j makes with the +x axis, then tan 0 =4/3, so 0 =tan^-1(4/3) =53.1 degrees i don’t understand how did they use (4/3) to get the result 53.1 degrees ? 4/3 = 1. is not 53.1 and even if i multiplied it with the angle i don’t get 53.1 4/3 *45=59.9985 if i use tan in the calculator i get 0.0232 so how did they get 53.1 ?

If you can, you should sketch this as I describe it:

-- The vector 'a' starts at the origin.  The end point is 3 units along the x-axis (3i)
and 4 units along the y-axis (4j).

-- The vector makes a right triangle with its components on the 'x' and 'y' axes.
The angle that the vector makes with the x-axis is one of the acute angles in
the right triangle.

-- The 'y' component ... the 4 units standing up ... is the side opposite that angle.

-- The x-component ... the 3 units along the x-axis ... is the side adjacent to it.

-- The tangent of any acute angle in a right triangle is
                          
                                         tan = (opposite leg) / (adjacent leg).

-- The tangent of THIS angle is (4 units) / (3 units)  =  4/3 .


Now, let's review some notation that I'm sure you've learned by now:

How do you write "The angle that has a tangent of  4/3" ?

There are two popular ways to write that in math:

                      One is        arctan(4/3) .

           The other one is      tan⁻¹(4/3) .

These are both ANGLES.  Whenever you see ARCtrigfunction(N)
or trigfunction⁻¹(N), those are ANGLES.  They mean "the angle that
has a trigfunction of N" .

In the example you're working on now, " tan⁻¹(4/3) " is an angle.
It means "the angle that has a tangent of 4/3".

You can't calculate what the angle is. You have to use a calculator, or else
look it up in a real book.
 
Somewhere on your calculator you'll find a button marked "tan⁻¹ ".  You put
a number into the calculator and hit that button, and the calculator tells you
the ANGLE that has that number for its tangent.

The angle that has  4/3  for a tangent is about  53.1 degrees.