The tangent function is a powerful tool in trigonometry, repeating every π units and stretching from negative to positive infinity. It's unique because it has vertical asymptotes at odd multiples of π/2, creating a distinctive graph that crosses the x-axis at multiples of π.

Transformations can alter the tangent function's shape and position. By shifting, stretching, or compressing the graph, we can create variations that model real-world phenomena. Understanding these transformations is key to mastering trigonometric functions and their applications.

Graph of tangent function

• The tangent function $y = \tan x$ has a period of $\pi$ repeats every $\pi$ units along the x-axis
• Domain of tangent function includes all real numbers except odd multiples of $\frac{\pi}{2}$ vertical asymptotes occur at these values
• Range of tangent function encompasses all real numbers
• Graph of $y = \tan x$ has vertical asymptotes at $x = \frac{\pi}{2}$ and $x = -\frac{\pi}{2}$
  • Function approaches positive infinity as $x$ approaches $\frac{\pi}{2}$ from the left
  • Function approaches negative infinity as $x$ approaches $\frac{\pi}{2}$ from the right
• Graph of $y = \tan x$ crosses x-axis at integer multiples of $\pi$ called zeros or roots of the function
• The tangent function can be derived from the unit circle, which is a fundamental tool for understanding trigonometric functions

Variations of tangent function

• Vertical shifts $y = \tan x + k$ shift graph of $y = \tan x$ up by $k$ units if $k > 0$ and down by $|k|$ units if $k < 0$
• Horizontal shifts $y = \tan(x - h)$ shift graph of $y = \tan x$ right by $h$ units if $h > 0$ and left by $|h|$ units if $h < 0$
• Vertical stretches and compressions $y = a \tan x$ stretch graph of $y = \tan x$ vertically by factor of $|a|$ if $|a| > 1$ and compress it if $0 < |a| < 1$
  • If $a < 0$, graph is also reflected across x-axis
• Horizontal stretches and compressions $y = \tan(bx)$ compress graph of $y = \tan x$ horizontally by factor of $|b|$ if $|b| > 1$ and stretch it if $0 < |b| < 1$
• Combinations of transformations can be applied to create more complex variations of tangent function

Secant vs cosecant graphs

• Both secant and cosecant functions have a period of $2\pi$
• Domain of secant function includes all real numbers except odd multiples of $\frac{\pi}{2}$ vertical asymptotes occur at these values
• Domain of cosecant function includes all real numbers except integer multiples of $\pi$ vertical asymptotes occur at these values
• Range of both secant and cosecant functions is $(-\infty, -1] \cup [1, \infty)$
• Graph of $y = \sec x$ has vertical asymptotes at $x = \frac{\pi}{2}$ and $x = -\frac{\pi}{2}$
• Graph of $y = \csc x$ has vertical asymptotes at $x = 0$ and $x = \pm\pi$
• Both graphs have minimum value of 1 and maximum value of -1

Cotangent function analysis

• Cotangent function $y = \cot x$ is reciprocal of tangent function $\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}$
• Graph of $y = \cot x$ has period of $\pi$, just like tangent function
• Domain of cotangent function includes all real numbers except integer multiples of $\pi$ vertical asymptotes occur at these values
• Range of cotangent function encompasses all real numbers
• Graph of $y = \cot x$ has vertical asymptotes at $x = 0$ and $x = \pm\pi$
• Graph of $y = \cot x$ crosses x-axis at odd multiples of $\frac{\pi}{2}$

Transformations of reciprocal functions

• Vertical shifts, horizontal shifts, vertical stretches and compressions, and horizontal stretches and compressions can be applied to secant, cosecant, and cotangent functions in same manner as tangent functions
• General forms of transformed functions:
  • Secant: $y = a \sec(b(x - h)) + k$
  • Cosecant: $y = a \csc(b(x - h)) + k$
  • Cotangent: $y = a \cot(b(x - h)) + k$
• Combinations of transformations can be applied to create more complex variations of these functions

Properties of trigonometric functions

• Sine and cosine functions:
  • Domain: all real numbers
  • Range: $[-1, 1]$
  • Period: $2\pi$
  • No asymptotes
• Tangent function:
  • Domain: all real numbers except odd multiples of $\frac{\pi}{2}$
  • Range: all real numbers
  • Period: $\pi$
  • Vertical asymptotes at odd multiples of $\frac{\pi}{2}$
• Cotangent function:
  • Domain: all real numbers except integer multiples of $\pi$
  • Range: all real numbers
  • Period: $\pi$
  • Vertical asymptotes at integer multiples of $\pi$
• Secant function:
  • Domain: all real numbers except odd multiples of $\frac{\pi}{2}$
  • Range: $(-\infty, -1] \cup [1, \infty)$
  • Period: $2\pi$
  • Vertical asymptotes at odd multiples of $\frac{\pi}{2}$
• Cosecant function:
  • Domain: all real numbers except integer multiples of $\pi$
  • Range: $(-\infty, -1] \cup [1, \infty)$
  • Period: $2\pi$
  • Vertical asymptotes at integer multiples of $\pi$
• All trigonometric functions are periodic functions, repeating their values at regular intervals

Trigonometric Identities and Radian Measure

• Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables
• These identities are often used to simplify expressions or solve equations
• Radian measure is an alternative way to measure angles, where one radian is the angle subtended by an arc length equal to the radius of the circle
• Radian measure is particularly useful when working with trigonometric functions, as it simplifies many formulas and calculations