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 = 1.3333 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 ?

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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.

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