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Differentiation

### Differentiating the Cosine Function

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 $x$, replace $f$ with $\cos$: $\frac{d\cos(x)}{dx}\bigg|_{a^+} = \lim_{\Delta x \rightarrow 0^+} \frac{ \cos(a + \Delta x) – \cos(a) }{\Delta x}$. Recall the angle addition identities. Specifically, $\cos(A+B) = \cos A \cos B – \sin A \sin B$. Using this identity, the numerator of the previous …

Differentiation

### 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, I find the derivative of $\sin x$ using the previous definition of a derivative. I also approximately follow the steps in reference [1]. Using $f(x) = \sin x$, the derivative is $\frac{d \sin(x)}{dx} \big|_{a^+} = \lim_{\Delta x \rightarrow 0^+} \frac{\sin(a+\Delta x) – \sin(a)}{\Delta x}$. …

Limit

### A Limit Involving the Cosine Function

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 need to deal with the issue of negative angles and how they are used in trigonometric identities proven with right triangles and nonnegative angles. I know that the limit definition requires negative and positive numbers if $x \rightarrow 0$, but I am going to ignore …

Trigonometry

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
The following drawing shows how to convert from Cartesian coordinates to spherical coordinates. This is a short post, but these three equations are pretty useful. I am going to use the end of this post to define the cosine and sine functions: $\sin\phi \equiv \frac{opp}{hyp}$ $\cos\phi \equiv \frac{adj}{hyp}$. Here, $hyp$ is the length of the hypotenuse of a right triangle, $opp$ is the length of the side that is opposite to the angle $\phi$, and $adj$ is the length of the side that is adjacent to the angle $\phi$. Note that $\phi$ …