In this post, the derivative of the cosine function is found. To do this, the steps in reference 1 are followed. Start with a definition of a derivative, from this post: $\frac{df(x)}{dx}\bigg|_{a^+} \equiv \lim_{\Delta x \rightarrow 0^+} \frac{ f(a + \Delta x) – f(a) }{\Delta x} $. Since $f(x)$ and $\cos(x)$ are both functions of $x$, …
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, …
Now that several limit properties have been proven, it is possible for me to evaluate $ \lim_{\alpha \rightarrow 0} \frac{1 – \cos \alpha}{\alpha} $. To do this, I follow the steps in Reference [1]. However, I am going to constrain $\alpha$, in radians, to be greater than or equal to zero, so that I do not …
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:angle-addition/v/proof-angle-addition-sine [2] https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:trig/x9e81a4f98389efdf:angle-addition/v/proof-angle-addition-cosine