Question

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 ?

Answer 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.