section 4.5

Trigonometric Functions and Their Graphs

Tangent and Cotangent Functions

The tangent and cotangent functions are fundamental trigonometric functions with unique properties and graphs. Understanding their behavior, periods, and asymptotes is essential for analyzing periodic phenomena and solving trigonometric equations.

• Tangent Function: Defined as $\tan(x) = \frac{\sin(x)}{\cos(x)}$.

• Period of Tangent: The period of $\tan(x)$ is $\pi$.

• Vertical Asymptotes: Occur where $\cos(x) = 0$, i.e., at $x = \frac{\pi}{2} + k\pi$ for integer $k$.

• Key Points: $\tan(0) = 0$, $\tan(\frac{\pi}{4}) = 1$, $\tan(-\frac{\pi}{4}) = -1$.

• Graph Features: The graph passes through the origin and repeats every $\pi$ units.

x	$\tan(x)$
0	0
$\frac{\pi}{4}$	1
$\frac{\pi}{2}$	DNE (asymptote)
$\pi$	0
$-\frac{\pi}{4}$	-1
$-\frac{\pi}{2}$	DNE (asymptote)

• Cotangent Function: Defined as $\cot(x) = \frac{\cos(x)}{\sin(x)}$.

• Period of Cotangent: The period of $\cot(x)$ is $\pi$.

• Vertical Asymptotes: Occur where $\sin(x) = 0$, i.e., at $x = k\pi$ for integer $k$.

• Graph Features: The graph has vertical asymptotes at integer multiples of $\pi$ and repeats every $\pi$ units.

Transformations of Tangent and Cotangent

Transformations allow us to shift, stretch, or compress the graphs of tangent and cotangent functions. The general form is:

• Tangent Transformation: $y = a \tan(b(x - c)) + d$

• Period: $\text{Period} = \frac{\pi}{|b|}$

• Phase Shift: $c$ shifts the graph horizontally.

• Vertical Shift: $d$ shifts the graph vertically.

• Amplitude: Not defined for tangent/cotangent since their range is infinite.

Example: For $y = 2 \tan(3(x - \frac{\pi}{4})) + 1$:

• Period: $\frac{\pi}{3}$

• Phase Shift: $\frac{\pi}{4}$ to the right

• Vertical Shift: Up 1 unit

Secant and Cosecant Functions

The secant and cosecant functions are the reciprocals of cosine and sine, respectively. Their graphs feature repeating patterns and vertical asymptotes.

• Secant Function: $\sec(x) = \frac{1}{\cos(x)}$

• Cosecant Function: $\csc(x) = \frac{1}{\sin(x)}$

• Period: Both $\sec(x)$ and $\csc(x)$ have a period of $2\pi$.

• Vertical Asymptotes: For $\sec(x)$, at $x = \frac{\pi}{2} + k\pi$; for $\csc(x)$, at $x = k\pi$.

• Graph Features: The graphs consist of repeating U-shaped curves between asymptotes.

Function	Period	Vertical Asymptotes
$\sec(x)$	$2\pi$	$x = \frac{\pi}{2} + k\pi$
$\csc(x)$	$2\pi$	$x = k\pi$

Graphing Trigonometric Functions

To graph tangent, cotangent, secant, and cosecant functions, identify their periods, asymptotes, and key points. Use transformations to adjust the graph as needed.

• Mark vertical asymptotes at points where the function is undefined.

• Plot key points between asymptotes.

• Apply transformations for period, phase shift, and vertical shift.

Example: The graph of $y = \tan(x)$ has vertical asymptotes at $x = \frac{\pi}{2} + k\pi$ and passes through the origin.

Summary Table: Periods of Trigonometric Functions

Function	Period
$\tan(x)$	$\pi$
$\cot(x)$	$\pi$
$\sec(x)$	$2\pi$
$\csc(x)$	$2\pi$

Additional info: These notes also include graphical sketches and transformation examples, which are essential for mastering the analysis and graphing of trigonometric functions in Precalculus.