Proofs of trig identities
Contents
Introduction
and
are easy to define. I prefer the unit circle definition as it makes these proofs easier to understand.
Next, we define some other functions:
Note: I've omitted because it's unnecessary and might clog things up a little.
With a bit of ingenuity, we can create the following diagram:
We can note that the functions are correct by similar triangles.
Pythagorean identities
Pythagorean identities are easy and there's no algebra involved. In fact, the name Pythagorean is a giveaway of what we should do!
The proof here is very straightforward. We use the pythagorean theorem on giving us
or
.
Same story here. Applying pythagorean to gives us
or
.
Same. Pythagorean on gives
or
.
Conclusion
Even though with the first one and the definitions, we can make the rest from algebra, having a geometric meaning is nice when we want to know what it actually means.
Angle addition and subtraction
where
The diagram illustrates the identities nicely.
The diagram shows the height of point is
.
However, the length of
is
. To compensate, we must divide by
to make it the sine. After some *easy* algebra, we arrive at
.
The diagram says that it is , but we need to divide by
again. We arrive at
.
This time we can't get it from our diagram. We need to go back to the original definition of tangent. This is the summary of my algebra: