# Trigonometry

## Proof of the Triangle Inequality for Real Numbers

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 …

## Do the angle addition identities only work for positive angles?

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, …

## The Pythagorean Theorem and the Unit Circle

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 $\theta … ## The sine function divided by its angle, and a certain limit What is$ \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 < |x-a| < \delta $we have$|f(x) – A| < …

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 …