Some Trigonometry (Thinking Assignment)

Thomas J. Kennedy

Contents:

So far we have focused on relatively benign functions (polynomials, monomials, logarithms, and exponentials). For these functions we generally examined behavior as:

$x \to \infty$
$x \to 0$
$x \to -\infty$

Let us select two relatively simple trig functions: $sin(x)$ and $cos(x)$ . Our normal analysis method falls short. Trigonometric (trig) functions are periodic. We must:

identify the size of one period (cycle).
examine the behavior (by looking for minima, maxima, and asymptotes).
generalize our analysis to all periods.

1 Tools of the Trigonometry Trade

Anytime I work with trigonometry (real numbers or complex numbers), I draw/write the unit circle. This time… I will draw it as a table:

$\theta$	$x$	$y$	$cos(\theta)$	$sin(\theta)$	$tan(\theta)$
$0$	1	0	1	0	0
$\frac{\pi}{2}$	0	1	0	1	–
$\pi$	-1	0	-1	0	0
$\frac{3\pi}{2}$	0	-1	0	-1	–
$2\pi$	1	0	1	0	0

Once we reach $\theta = 2\pi$ the table starts to repeat (i.e., we add $2\pi$ to each of our previous values).

2 Conditioning of sin(x)

Let us examine

$f(x) = cos(x)$

We know that the derivative is

$f’(x) = -sin(x)$

Let us take a quick look at $cos(x)$ and $-sin(x)$ .

Let us examine the period $x \in [0, 2\pi]$ and use the general form of the condition number

$(cond \, f)(x) = \Bigg|\frac{xf’(x)}{f(x)}\Bigg| = \Big| -xtan(x) \Big|$

Using this general form we must restrict our domains of analysis to those where $x \neq 0$ and $cos(x) \neq 0$ . Let us take a quick moment to sketch, and examine the behavior, of both $tan(x)$ and $-xtan(x)$

Now we need to carefully partition our domain; ( $-xtan(x)$ ) is not exactly periodic.

3 Thinking Assignments

As thinking assignments:

Complete the analysis started in the previous section (i.e., define subdomains and bound the condition number)
Complete the analysis for cases where $x = 0$ .
Complete the analysis for cases where $f(x) = 0$ .