Glossary

8.2 Graphs of the Other Trigonometric Functions

3 min read•june 24, 2024

Trigonometric functions beyond sine and cosine offer unique patterns and behaviors. , , , and each have distinct graphs with their own periods, asymptotes, and symmetries.

These functions can be transformed through shifts, stretches, and reflections. Understanding their graphs and helps in modeling periodic phenomena and solving complex trigonometric equations.

Graphs of Tangent, Secant, Cosecant, and Cotangent Functions

Graph of tangent function

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  • repeats its pattern every π\pi units
  • Vertical asymptotes occur at x=π2+nπx = \frac{\pi}{2} + n\pi, where nn is any integer
    • Graph approaches positive or negative infinity as xx approaches these values from either side
  • xx-intercepts located at x=nπx = n\pi, where nn is any integer
  • Graph passes through the (0,0)(0, 0)
  • tan⁡(−x)=−tan⁡(x)\tan(-x) = -\tan(x) exhibits about the
  • is all real numbers except where vertical asymptotes occur

Variations of tangent functions

  • Vertical shifts y=tan⁡x+ky = \tan x + k move the graph up or down by kk units
    • Positive kk values shift the graph upward
    • Negative kk values shift the graph downward
  • Horizontal shifts y=tan⁡(x−h)y = \tan(x - h) move the graph left or right by hh units
    • Positive hh values shift the graph to the right
    • Negative hh values shift the graph to the left
  • changes y=atan⁡xy = a \tan x stretch or compress the graph vertically
    • ∣a∣>1|a| > 1 vertically stretches the graph by a factor of ∣a∣|a|
    • 0<∣a∣<10 < |a| < 1 vertically compresses the graph by a factor of ∣a∣|a|
    • Negative aa values reflect the graph across the xx-axis
  • changes y=tan⁡(bx)y = \tan(bx) alter the repetition interval of the function
    • Period becomes π∣b∣\frac{\pi}{|b|}
    • ∣b∣>1|b| > 1 horizontally compresses the graph
    • 0<∣b∣<10 < |b| < 1 horizontally stretches the graph

Secant vs cosecant graphs

  • Both functions have a period of 2π2\pi and lack xx-intercepts
  • Vertical asymptotes differ in location
    • y=sec⁡xy = \sec x has vertical asymptotes at x=π2+nπx = \frac{\pi}{2} + n\pi, where nn is any integer
    • y=csc⁡xy = \csc x has vertical asymptotes at x=nπx = n\pi, where nn is any integer
  • y=sec⁡xy = \sec x is an symmetric about the yy-axis
  • y=csc⁡xy = \csc x is an symmetric about the origin
  • Both are of cosine and sine, respectively

Transformations of secant and cosecant

  • Vertical shifts y=sec⁡x+ky = \sec x + k and y=csc⁡x+ky = \csc x + k move the graph up or down by kk units
  • Horizontal shifts y=sec⁡(x−h)y = \sec(x - h) and y=csc⁡(x−h)y = \csc(x - h) move the graph left or right by hh units
  • Amplitude changes y=asec⁡xy = a \sec x and y=acsc⁡xy = a \csc x stretch or compress the graph vertically by a factor of ∣a∣|a|
    • Negative aa values reflect the graph across the xx-axis
  • Period changes y=sec⁡(bx)y = \sec(bx) and y=csc⁡(bx)y = \csc(bx) alter the repetition interval to 2π∣b∣\frac{2\pi}{|b|}

Characteristics of cotangent graph

  • Periodic function with a period of π\pi
  • Vertical asymptotes occur at x=nπx = n\pi, where nn is any integer
  • xx-intercepts located at x=π2+nπx = \frac{\pi}{2} + n\pi, where nn is any integer
  • Odd function cot⁡(−x)=−cot⁡(x)\cot(-x) = -\cot(x) exhibits symmetry about the origin
  • of cotangent function is all real numbers

Transformations of cotangent functions

  • Vertical shifts y=cot⁡x+ky = \cot x + k move the graph up or down by kk units
  • Horizontal shifts y=cot⁡(x−h)y = \cot(x - h) move the graph left or right by hh units
  • Amplitude changes y=acot⁡xy = a \cot x stretch or compress the graph vertically by a factor of ∣a∣|a|
    • Negative aa values reflect the graph across the xx-axis
  • Period changes y=cot⁡(bx)y = \cot(bx) alter the repetition interval to π∣b∣\frac{\pi}{|b|}

Additional Concepts

  • and : Trigonometric functions are continuous except at points of discontinuity (vertical asymptotes)
  • : Combining trigonometric functions with other functions can create new, complex relationships
  • : Arcsine, arccosine, and arctangent are examples that "undo" their corresponding trigonometric functions

Key Terms to Review (37)

Amplitude: Amplitude refers to the maximum displacement or the maximum value of a periodic function, such as a sine or cosine wave, from its mean or average value. It represents the magnitude or size of the oscillation or variation in the function.
Axes of symmetry: Axes of symmetry are lines that divide a figure into two mirror-image halves. In hyperbolas, these axes typically refer to the transverse and conjugate axes.
Continuity: Continuity is a fundamental concept in mathematics that describes the smooth and uninterrupted behavior of a function or graph. It is a crucial property that ensures a function's values change gradually without any abrupt jumps or breaks.
Cosecant: The cosecant, often abbreviated as csc, is one of the fundamental trigonometric functions. It represents the reciprocal of the sine function, meaning it is the ratio of the hypotenuse to the opposite side of a right triangle. The cosecant function is particularly useful in understanding the properties and applications of trigonometry, which are essential in various mathematical and scientific disciplines.
Cotangent: The cotangent is one of the fundamental trigonometric functions, defined as the reciprocal of the tangent function. It represents the ratio of the adjacent side to the opposite side of a right triangle, providing a way to describe the relationship between the sides of a right triangle and the angles formed within it.
Discontinuity: Discontinuity refers to a break or interruption in the continuity of a function. It occurs when a function is not defined at a particular point or when the function exhibits a sudden jump or change in its value at a specific point within its domain.
Domain: The domain of a function is the complete set of possible input values (x-values) that allow the function to work within its constraints. It specifies the range of x-values for which the function is defined.
Domain: The domain of a function refers to the set of input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is the set of all possible values that can be plugged into the function to produce a meaningful output.
Even function: An even function is a function $f(x)$ where $f(x) = f(-x)$ for all $x$ in its domain. This symmetry means the graph of an even function is mirrored across the y-axis.
Even Function: An even function is a mathematical function where the output value is the same regardless of whether the input value is positive or negative. In other words, the function satisfies the equation f(x) = f(-x) for all values of x in the function's domain.
Factor by grouping: Factor by grouping is a method used to factor polynomials that involves rearranging and combining terms into groups that have a common factor. This technique is particularly useful for polynomials with four or more terms.
Function Composition: Function composition is the process of combining two or more functions to create a new function. The output of one function becomes the input of the next function, allowing for the creation of more complex mathematical relationships.
Horizontal Shift: A horizontal shift is a transformation of a function that moves the graph of the function left or right along the x-axis, without changing the shape or orientation of the graph. This concept is applicable to various types of functions, including transformations of functions, absolute value functions, exponential functions, trigonometric functions, and the parabola.
Inverse Trigonometric Functions: Inverse trigonometric functions are the inverse operations of the standard trigonometric functions, allowing one to find the angle given the value of a trigonometric ratio. They are essential for solving trigonometric equations and understanding the behavior of periodic functions.
Multiplicative inverse: The multiplicative inverse of a number is another number that, when multiplied together, yield the product 1. For any nonzero number $a$, its multiplicative inverse is $\frac{1}{a}$.
Multiplicative inverse of a matrix: The multiplicative inverse of a matrix $A$ is another matrix $A^{-1}$ such that when $A$ is multiplied by $A^{-1}$, the result is the identity matrix. Not all matrices have a multiplicative inverse; only square matrices with non-zero determinants do.
Odd function: An odd function is a function $f(x)$ that satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain. Graphically, odd functions exhibit symmetry about the origin.
Odd Function: An odd function is a function that satisfies the property $f(-x) = -f(x)$ for all $x$ in the domain of the function. This means that the graph of an odd function is symmetric about the origin, with the graph being a reflection across both the $x$-axis and the $y$-axis.
Origin: In the rectangular coordinate system, the origin is the point where the x-axis and y-axis intersect. It is denoted by the coordinates (0, 0).
Origin: The origin is a specific point in a coordinate system that serves as the reference point for all other points. It is the intersection of the x-axis and y-axis, and is typically denoted as the point (0, 0). The origin is a fundamental concept in various mathematical and scientific contexts, as it provides a common starting point for measurement and analysis.
Period: The period of a function is the distance or interval along the independent variable axis over which the function's shape or pattern repeats itself. It is a fundamental concept in the study of periodic functions, such as trigonometric functions, and is essential for understanding their properties and graphs.
Periodic Function: A periodic function is a function that repeats its values at regular intervals. This means that the function's graph consists of identical copies of a specific pattern or shape that are repeated at fixed intervals along the x-axis.
Pi (π): Pi (π) is a fundamental mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number, meaning its decimal representation never ends or repeats, and it is widely used in various mathematical and scientific applications, including the study of real numbers, angles, trigonometry, and the graphing of trigonometric functions.
Range: In mathematics, the range refers to the set of all possible output values (dependent variable values) that a function can produce based on its input values (independent variable values). Understanding the range helps in analyzing how a function behaves and what values it can take, connecting it to various concepts like transformations, compositions, and types of functions.
Reciprocal functions: Reciprocal functions are mathematical functions defined as the multiplicative inverse of a given function. Specifically, if a function is represented as $$f(x)$$, its reciprocal function is given by $$g(x) = \frac{1}{f(x)}$$. These functions exhibit unique properties in the context of trigonometric functions, particularly with the sine, cosine, and tangent functions, as they help illustrate relationships among various trigonometric identities.
Removable discontinuity: A removable discontinuity occurs at a point in a function where the limit exists, but the function is either not defined or defined differently. This type of discontinuity can be 'removed' by appropriately defining or redefining the function at that point.
Secant: A secant is a line that intersects a circle at two distinct points. It is one of the fundamental trigonometric functions, along with sine, cosine, tangent, and others, that describe the relationships between the sides and angles of a right triangle.
Symmetry: Symmetry is the quality of being made up of exactly similar parts facing each other or around an axis. It is a fundamental concept in mathematics and geometry that describes the balanced and harmonious arrangement of elements in an object or function.
Tangent: A tangent is a straight line that touches a curve at a single point, forming a right angle with the curve at that point. It is a fundamental concept in trigonometry and geometry, with applications across various mathematical disciplines.
Transformations: Transformations refer to the processes of altering the position, size, shape, or orientation of a graph in a coordinate plane. They are crucial for understanding how different functions behave when subjected to changes such as translations, reflections, stretches, and compressions. By applying these transformations, one can gain insight into the properties of various types of functions and how they can be manipulated to produce new graphs.
Vertical asymptote: A vertical asymptote is a line $x = a$ where a rational function $f(x)$ approaches positive or negative infinity as $x$ approaches $a$. It represents values that $x$ cannot take, causing the function to become unbounded.
Vertical Asymptote: A vertical asymptote is a vertical line that a graph of a function approaches but never touches. It represents the vertical limit of the function's behavior, indicating where the function's value becomes arbitrarily large or small.
Vertical shift: A vertical shift is a transformation that moves a graph up or down in the coordinate plane by adding or subtracting a constant to the function's output. It does not affect the shape of the graph, only its position.
Vertical Shift: Vertical shift refers to the movement of a graph or function up or down the y-axis, without affecting the shape or orientation of the graph. This transformation changes the y-intercept of the function, but leaves the x-intercepts and the overall shape unchanged.
X-intercept: The x-intercept is the point where a graph crosses the x-axis, where the y-coordinate is zero. It represents the solution(s) to an equation when $y = 0$.
X-Intercept: The x-intercept of a graph is the point where the graph of a function or equation intersects the x-axis, indicating the value of x when the function's output or the equation's value is zero. The x-intercept is a crucial concept in understanding the behavior and properties of various mathematical functions and equations.
Y-axis: The y-axis is the vertical axis in a rectangular coordinate system, which represents the dependent variable and is typically used to plot the values or outcomes of a function. It is perpendicular to the x-axis and provides a visual reference for the range of values a function can take on.
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