
One Cycle of a Trigonometric Function
Integral of a Complex Exponential Consider the integral $ \int_0^{2 \pi} d\theta e^{i \theta} $. Upper Limit of $2 \pi$ Radians The upper limit is the number, $2 \pi$. For a circle with radius $r=1$, the circumference of the circle is $2 \pi r = 2 \pi$. Even though $2 \pi$ is a number, it…

Differentiating the Sine Function
From this post, one type of derivative is $\lim_{\Delta x\rightarrow0^+}\frac{f(a+\Delta x)f(a)}{\Delta x}\equiv\frac{df(x)}{dx}\big_{a^+}$ To be consistent with my previous interpretation of $0^+$ in this post, $\Delta x \rightarrow 0^+$ means constraining $\Delta x$ to positive numbers. Next, define $x$ and $a$ as variables for nonnegative real numbers, to avoid having a negative angle for the sine function. In this post,…

The sine function divided by its angle, and a certain limit
What is $ \lim_{x\rightarrow0} \frac{\sin x}{x}$ ? Recall the definition of a limit, repeated here for reference [2]. A function $f(x)$ approaches a limit $A$ as $x$ approaches $a$ if, and only if, for each positive number $\epsilon$ there is another, $\delta$, such that whenever $0 < xa < \delta $ we have $f(x) – A <…

Angle Addition Identities
This post includes proofs of two angle addition identities. $ \sin(x+y) = \sin x \cos y + \sin y \cos x$ $ \cos(x+y) = \cos x \cos y – \sin x \sin y$ References [1] https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:trig/x9e81a4f98389efdf:angleaddition/v/proofangleadditionsine [2] https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:trig/x9e81a4f98389efdf:angleaddition/v/proofangleadditioncosine