Trigonometry

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, …

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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.  

Double Angle Formulas

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 …

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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| < …

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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 …

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De Moivre’s Theorem

This is a proof of de Moivre’s Theorem. This was originally presented by reference [1]. References [1] https://ccrma.stanford.edu/~jos/st/Direct_Proof_De_Moivre_s.html