AP Precalc AB Unit 6: Periodic Functions Cont

Recall: Tangent of an angle was initially defined as the slope of the terminal ray or the ratio of sine to cosine values (Topic 3.2).
As angle $\theta$ increases from 0 to $\frac{\pi}{2}$ , the slope of the terminal ray increases.
When $\theta = 0$ , the slope is 0.
As $\theta$ approaches $\frac{\pi}{2}$ , the slope becomes steeper (more positive).
When $\theta = \frac{\pi}{2}$ , the terminal ray is vertical, and the slope is undefined.
The slope being undefined implies it approaches infinity: $\lim_{\theta \to \frac{\pi}{2}^-} tan(\theta) = +\infty$
From $\theta = \frac{\pi}{2}$ to $\theta = \pi$ , slopes transition from $-\infty$ to 0.
From $\theta = \pi$ to $\theta = \frac{3\pi}{2}$ , slopes are positive, increasing from 0 to approach $+\infty$ .
Key Concept: When the terminal ray is vertical, the slopes are undefined (approach $\pm \infty$ ).
This results in vertical asymptotes on the graph of the tangent function at $\theta = \frac{\pi}{2}$ and $\theta = \frac{3\pi}{2}$ .
Each half-turn creates another vertical asymptote.
Important Features for the Graph of the Tangent Function
- For $y = f(\theta) = atan(b(\theta + c)) + d$ :
  - Vertical dilation by a factor of $|a|$ .
  - Period = $|\frac{\pi}{b}|$ .
  - Phase shift of $-c$ units (horizontal translation).
  - Vertical translation of $d$ units.

Example 1: Determine the values of $a$ , $b$ , and $d$ for $f(x) = atan(b(x + c)) + d$ given a portion of the function $f$ .
Example 2: Find the period of the function $h(x) = 4tan(2x) + 5$ .
Example 3: Determine the vertical asymptotes of $g(x) = 2tan(\frac{x}{3}) - 1$ .
- A. $x = \frac{\pi}{2} + \pi k$ , where $k$ is an integer.
- B. $x = \frac{\pi}{6} + \frac{\pi}{3} k$ , where $k$ is an integer.
- C. $x = \frac{3\pi}{2} + 3\pi k$ , where $k$ is an integer.
- D. $x = \pi + 2\pi k$ , where $k$ is an integer.
Example 4: Given a portion of the function $k$ , determine which of the following could be the expression for $k$ .
- A. $tan(\frac{x}{2}) - 2$
- B. $tan(\frac{x}{2}) + 2$
- C. $tan(2x) - 2$
- D. $tan(2x) + 2$
Example 5: Evaluate the following using the unit circle.
- A. $tan(\frac{\pi}{6})$
- B. $tan(\frac{\pi}{4})$
- C. $tan(\frac{\pi}{3})$
- D. $tan(\frac{5\pi}{6})$
- E. $tan(\frac{4\pi}{3})$
- F. $tan(\frac{7\pi}{4})$
- G. $tan(\frac{3\pi}{2})$
- H. $tan(\pi)$

Inverse trig functions result from switching input (x) and output (y) values.
The output of an inverse trigonometric function is an angle measure.
Notation: Inverse trigonometric functions can be represented as $sin^{-1}(x)$ or $arcsin(x)$ (