In this post, I find an expression for the radial unit vector, $\vec{e}_r$. The three unit vectors in the following digram form a right-handed spherical coordinate system. This unit vector is easier to find than the other two unit vectors because all that is needed is vector addition. The Radial Unit Vector in Terms of Spherical Coordinates Suppose $r=1$. Using vector addition, $\vec{r} = r’ \cos\phi \vec{e}_x + r’ \sin\phi \vec{e}_y + \cos \theta \vec{e}_z$. Since $r=1$, the expression on the right is equal to $\vec{e}_r$: $\vec{e}_r = r’ \cos\phi \vec{e}_x + r’ \sin\phi \vec{e}_y + \cos \theta \vec{e}_z$. […]

The triangle inequality for real numbers is $ |a+b| \le |a| + |b|$ in which $a$ is a variable for a real number, and $b$ is a variable for a real number. Proof: I use four cases. Case 1: If $a=0$ and $b$ is any real number, then the left side of the triangle inequality is $|b|$. And the right side is $|b|$. So the left side is equal to the right side. Similarly, if $b=0$ and $a$ is any real number, then the left side is $|a|$ and the right side is $|a|$. Case 2: If $a$ and

Proof of the Triangle Inequality for Real NumbersRead More »

The Angle Addition Identities were derived using $\sin\theta \equiv \frac{opp}{hyp}$ and $\cos\theta \equiv \frac{adj}{hyp}$. In addition, $\theta$ was constrained to the interval of $[0, \frac{\pi}{2})$. The question addressed in this post is, do the Angle Addition Identities work if the arguments of the sine function and cosine functions are generalized to include negative numbers? In the following, $\theta$ will still be constrained to the interval of $[0, \frac{\pi}{2})$. The generalization of choice is $ \displaystyle \sin(-\theta) = -\sin\theta$ and $ \displaystyle \cos(-\theta) = \cos\theta$ This generalization is applicable to the unit circle, specifically in the 1st and 4th quadrants. The 2nd

Do the angle addition identities only work for positive angles?Read More »

This following diagram shows that $\displaystyle \sin^2 \theta + \cos^2 \theta = 1$, given $ \displaystyle \cos \theta \equiv \frac{adj}{hyp}$ and $ \displaystyle \sin \theta \equiv \frac{opp}{hyp}$ and the Pythagorean theorem. Here, $opp$, $hyp$, and $adj$ are variables for lengths. The values corresponding to these lengths are positive numbers or zero.  

In the Angle Addition Identities post it was shown that $ \displaystyle \sin(x+y) = \sin x \cos y + \sin y \cos x $ and $ \displaystyle \cos(x+y) = \cos x \cos y – \sin x \sin y $. These identities are valid if $\sin \theta \equiv \frac{opp}{hyp}$ and $\cos \theta \equiv \frac{adj}{hyp}$, which implies that $\theta \in [0,\frac{\pi}{2})$ in order for $\theta$ to be an acute angle of a right triangle. These constraints are imposed because the two Angle Addition Identities were derived using these constraints. The next step, following the steps in reference [1], is setting $y$ equal to

Double Angle FormulasRead More »

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 < |x-a| < \delta $ we have $|f(x) – A| < \epsilon$. That is, when $x$ is near $a$ (within a distance $\delta$ from it), $f(x)$ is near $A$ (within a distance $\epsilon$ from it). In symbols we write $\lim_{x \rightarrow a} f(x) = A$. Using this definition, $a$ is $0$ in this case. Now

The sine function divided by its angle, and a certain limitRead More »

The Radian

In Chapter 2 of Reference [1], one of the footnotes has a definition of a radian. For a unit circle with a radius of one unit of length called $ u$, one radian is the angle corresponding to an arc-length of $ u$. In degrees, one radian is approximately 57 degrees, which is perhaps easier to visualize. A diagram of one radian is included here for reference. References [1] Konrad Knopp. Theory and Application of Infinite Series. Dover Publications. 1990.