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 .
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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.
