1. We will give (but not prove) Euler’s identity. From Euler’s equation we derive trigonometric functions. We postpone the proof for Euler’s equation until some future post, when we discuss Taylor series.
2. So we have a circle described by the equation
where we set the radius to unity (i.e., ). Great. What’s its solution in parametric form?
We can use trigonometric functions writing
(2) and .
This is taught to high school kids taking trigonometry. But I never took trigonometry! The only way I learned this stuff was through Euler’s equation. For a graphical idea of what’s going on, consider the following doodle:
Suppose we wrote any pair as a complex number . Then we can work with the complex plane (denoted ). So what? Well, we have this useful formula:
We should observe this traces out the unit circle in the complex plane. But we can recover all the trig identities this way!
Complex numbers have two components: the real part and the imaginary part. Ordered pairs have two components too. Thus this component-wise process relating ordered pairs and complex numbers…is kosher.
Moreover, the identification of with is a one-to-one correspondence. Each distinct complex number gives us an ordered pair, and each ordered pair gives us a distinct complex number. So this process is fairly “sane”.
3. We should first rewrite sine and cosine in terms of . Well, we see that
Thus we have
This is cosine.
Sine follows from similar reasoning. We simply have
From Equations (5a) and (5b) we can derive everything else in trigonometry.
Exercise 1. What is in terms of and ?
Exercise 2. What is using ?
Exercise 3. What is ?
Exercise 4. Calculate in terms of , , , .
Exercise 5. Calculate in terms of , , , .
Exercise 6. Using Equations (5a) and (5b), what is ???
Remark. Exercise 6 helps us a lot, because whenever we run into an integral of the form , we can use substitution and rewrite it with trigonometric functions!
Exercise 7 [Math.SE]. Prove