Proofs of trig identities
Contents
[hide]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.
Symmetric identities
If we draw a few copies of the triangle, we get:
The other three can be derived by taking the reciprocals of these three.
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:
Double angle formulas
This is a breeze. Just sub in for sum:
Sum to Product to Sum
Our angle addition formulas look nasty. Let's try to cancel something.
Chapter 1
Let's start with the formula . Both terms are symmetrical, let's try to cancel out the first one.
and might give us an idea.
Therefore, .
To put this in a nicer form, where:
Chapter 2
We did all we could. Now let's try doing something to the other formula: . Let's try to cancel the first one first. is a lot easier to cancel than so we have to subtract.
So
Or, (same conversions of )
Chapter 3
Next: . Now let's cancel the second one.
So
Or, (same conversions of )
Bonus: Product identity
This is a special identity most people don't know about. I hope this helps you.