College Trigonometry: A Concise Guide

The creation of this open-source College Algebra textbook was a collaborative effort, involving not only the two main authors but also a significant number of other individuals and institutions. The authors explicitly thank their families for their support, highlighting the demanding nature of the project. Crucially, they acknowledge the invaluable contributions of numerous colleagues, including Rich Basich, Bill Previts, and Irina Lomonosov, who acted as early adopters and provided materials. Classroom feedback was also instrumental, with special thanks given to Katie Cimperman, Terry Dykstra, Frank LeMay, and Rich Hagen. The support extended beyond the academic community, with gratitude expressed to Don Anthan and Ken White for designing electric circuit applications included in the text, and to Drs. Wendy Marley and Marcia Ballinger for providing enrollment data from Lorain County Community College. The contribution of bookstore staff, Gwen Sevtis (Lakeland CC) and Chris Callahan (Lorain CCC), in providing affordable printed copies is also highlighted, along with the support from Lakeland personnel (Jeri Dickinson, Mary Ann Blakeley, Jessica Novak, and Corrie Bergeron) and the administrations of both Lakeland and Lorain County Community Colleges. This collaborative approach underscores the community aspect of creating an Open Educational Resource (OER). The book's open-source nature is emphasized, with all LaTeX files available under a Creative Commons License, allowing for adaptation and reuse by others. The authors' decision to reject several publishing offers to maintain complete creative control over the content is also noteworthy. The open-source nature of the project and the authors' dedication to keeping the text accessible and affordable are key themes.

A central motivation for creating this textbook was the prohibitive cost of traditional College Algebra textbooks, often exceeding $200. The authors directly address this issue in the preface, stating that their students' shared desire for a more affordable option was the primary inspiration for the project. This resonates with the broader movement towards Open Educational Resources (OER) and the goal of making educational materials more accessible to students. The authors' commitment to affordability is further demonstrated by their decision to provide the textbook electronically for free, thus directly combating the cycle of new editions released every 18 months to limit the used book market. The choice to use only black and white printing for the physical copies further reduces costs for students who prefer print versions. The availability of low-cost printed editions through Lulu.com, priced at under $19, is another example of their dedication to keeping the textbook accessible, and the authors' explicit renunciation of royalties from these sales reinforces this commitment. This focus on cost-effectiveness directly responds to the economic barriers faced by many college students, especially those classified as "non-traditional" students who are older and often balancing work, family, and education.

The authors clearly articulate their pedagogical approach and the rationale behind their content choices. They reject the traditional "drill and kill" method of extensive, repetitive exercises, opting instead for shorter, more thoughtful problem sets that encourage deeper understanding. The authors' rationale stems from their experience with students' ability to complete roughly 15-20 homework exercises per night, thus necessitating more focused and conceptually engaging questions. The inclusion of thought-provoking discussion questions without provided answers further supports this approach. This approach is in line with a broader shift towards more student-centered learning and a focus on conceptual comprehension over mere procedural skill. The careful selection of exercises and the emphasis on understanding are presented as central elements of the textbook's design and directly linked to their stated goal of helping students learn rather than merely profiting from textbook sales. The authors are transparent regarding their intentions in writing the book, showcasing their love of mathematics and passion for teaching, a common theme throughout the text.

The creation of the textbook leveraged open-source technologies, highlighting a commitment to accessibility and cost-effectiveness. The authors utilized TexNicCenter, an open-source text editor, along with MikTex 2.7, an open-source LaTeX distribution. Metapost within LaTeX was used for graph creation. This choice aligns with the open-source philosophy underpinning the project, ensuring that the methods used to create the textbook are also freely available and reproducible. The adoption of these tools facilitated the efficient creation and editing of the text, exercises, answers, and graphs. The authors completed Version 0.9 of the textbook in roughly eight months, showcasing their efficient workflow. The use of open-source software allows for greater accessibility for others seeking to create or adapt similar open educational resources. This transparency in the technological aspects of the project further emphasizes the commitment to making resources readily available to the broader educational community.

This section establishes a foundation for understanding angles and their measurement. It begins by defining degree measure, explaining that 1° represents 1/360th of a revolution. The text then introduces decimal degrees as a method for subdividing degrees, providing the example of 30.5° as halfway between 30° and 31°. The concept is extended beyond the range of 0° to 360°, introducing the idea of angles measuring more than one full revolution. The section then shifts to radian measure, defining one revolution as 2π radians, half a revolution as π radians, and so on. It clarifies that the distinction between an angle and its measure is often blurred in practice. Oriented angles are introduced, with positive measures indicating counter-clockwise rotation and negative measures indicating clockwise rotation. The concepts of supplementary and complementary angles (in radian measure) are reviewed, and the condition for coterminal angles (β = α + 2πk, where k is an integer) is stated. The section lays the groundwork for understanding angles as quantities associated with real numbers, moving beyond simply measuring rotations. This expansion prepares the reader for the application of algebraic techniques to geometric problems involving angles.

Building on the concept of angles, this section introduces circular motion and derives a formula for linear velocity. It begins by considering an angle θ in standard position and a point P on the unit circle. The x-coordinate of P is defined as cos(θ), and the y-coordinate as sin(θ). This sets the stage for connecting angles to the coordinates of points on a circle, a central idea throughout the section. The discussion then focuses on circular motion with a constant angular velocity (frequency) ω. The section introduces the concept of period (the time to complete one cycle) and its relationship to frequency. It is shown that linear velocity (v) and angular velocity (ω) are connected by the equation v = rω, where r is the radius of the circular path. This relationship shows that for a fixed angular frequency, points farther from the center of rotation must travel faster to complete one revolution in the same period. The section concludes by posing a question about determining the position of an object at a given time t, moving along a circular path with a fixed angular velocity, setting up the next stage of the mathematical exploration of circular motion. The concept of frequency and period in relation to angular velocity (ω) and linear speed is explained.

This section utilizes the concepts of circular motion and sinusoidal functions to model a real-world scenario: the Giant Wheel at Cedar Point amusement park. The Giant Wheel's physical characteristics are provided: a diameter of 128 feet, sitting on an 8-foot-tall platform, resulting in an overall height of 136 feet. The wheel completes two revolutions in 2 minutes and 7 seconds (127 seconds). The problem requires the derivation of a sinusoid which describes the height of a passenger above the ground as a function of time (in seconds) since passing the lowest point on the wheel. The radius is calculated as 64 feet and the period of the rotation is 127/2 seconds. The angular frequency (ω) is then determined as 4π/127 radians per second, combining the period and the concept of angular frequency. The fundamental equation for the y-coordinate in counterclockwise motion on a circle is given (y = r sin(ωt)), and the final equation describing the height (y) as a function of time (t) is derived based on the given parameters. This example demonstrates how mathematical models (specifically, sinusoidal functions) can be used to represent and predict motion in a real-world context.

This section introduces the cosine and sine functions using the unit circle as a foundational tool. An angle θ in standard position is considered, and the point P where the terminal side of θ intersects the unit circle is identified. The x-coordinate of this point P is defined as cos(θ), and the y-coordinate is defined as sin(θ). This geometric definition establishes the cosine and sine as functions, ensuring that for each angle θ, there is only one associated value for cos(θ) and sin(θ). The section then proceeds to illustrate how to find these values for various angles. For quadrantal angles (angles whose terminal sides lie on the coordinate axes), finding the cosine and sine values is relatively straightforward. However, for non-quadrantal angles, the process is more involved, often requiring the use of right triangles and the Pythagorean theorem within the unit circle context. Examples are provided to demonstrate the application of this method to specific angles, thereby building a practical understanding of how to determine cosine and sine values from the geometric properties of the unit circle.

The section introduces the Pythagorean Identity, a crucial trigonometric identity derived from the unit circle. Since the point (cos(θ), sin(θ)) always lies on the unit circle (x² + y² = 1), substituting the coordinates results in the identity cos²(θ) + sin²(θ) = 1. This identity is highlighted as one of the most important results in trigonometry, and its connection to the Pythagorean Theorem and the equation of a circle is explained. The utility of the Pythagorean Identity is demonstrated through examples, showcasing its use in determining one trigonometric function value when the other is known. The text acknowledges a common convention in writing (cos(θ))² as cos²(θ) and (sin(θ))² as sin²(θ). The crucial point emphasized is that this identity holds true regardless of the value of θ, highlighting its fundamental nature within trigonometry. The identity's role in removing ambiguity when determining the sign of a trigonometric function value is explained, based on the quadrant in which the terminal side of the angle lies. The section explains how to utilize the Pythagorean Identity for solving and manipulating trigonometric expressions.

This subsection expands the discussion of cosine and sine beyond the unit circle to circles with radius r centered at the origin. It demonstrates how the cosine and sine functions can be used to determine the coordinates of points on these larger circles. The method uses similar triangles formed by dropping perpendiculars from points on the unit circle and points on the larger circles. This establishes the coordinates of a point Q(x, y) on a circle with radius r as x = r cos(θ) and y = r sin(θ), where θ is the angle in standard position. This generalization is important because it extends the applicability of cosine and sine to a wider range of geometric problems involving circles of any radius, not only the unit circle. The formulas derived are presented as a generalization that holds true for all non-quadrantal angles. The text encourages the reader to verify that these formulas also hold true for quadrantal angles, extending the concepts even further. This broader application of cosine and sine lays the groundwork for further exploration and use within more complex trigonometric applications and proofs.

This section bridges the gap between angles and real numbers in the context of cosine and sine. It shows how cosine and sine can be viewed as functions of real numbers by identifying a real number t with the angle θ = t radians. The equivalence of viewing the input as either an angle in radian measure or as a real number is clarified. This connection extends the domain of cosine and sine from angles to the set of all real numbers. The correspondence is described in terms of an oriented arc: for each real number t, an oriented arc of length |t| is associated, starting at the point (1, 0) and ending at the point P(cos(t), sin(t)). This visualization aids in understanding the functional relationship between real numbers and the cosine and sine values. This conceptual shift is a crucial step in the development of the understanding of trigonometric functions and paves the way for more advanced applications of these functions within calculus and other areas of mathematics.

This section introduces the remaining four trigonometric functions: secant (sec), cosecant (csc), tangent (tan), and cotangent (cot). Instead of a purely geometric definition, these functions are defined in terms of cosine and sine, using reciprocal and quotient identities. Specifically, sec(t) = 1/cos(t), csc(t) = 1/sin(t), tan(t) = sin(t)/cos(t), and cot(t) = cos(t)/sin(t). This approach leverages the previously established understanding of cosine and sine, building upon those functions to define the new ones. The text mentions alternative approaches such as using the wrapping function or associating the real number t with the angle θ = t radians. However, the chosen method uses the reciprocal and quotient identities, demonstrating a clear and concise relationship with the already-defined cosine and sine functions. This approach streamlines the introduction of these four functions by building on existing knowledge, making it easier to understand their properties and relationships with each other.

Having defined the four trigonometric functions (secant, cosecant, tangent, and cotangent), the section focuses on determining their domains and ranges. The domain of each function is analyzed by considering where the denominator of the defining expressions (cosine and/or sine) becomes zero, leading to undefined values. For example, the function sec(t) = 1/cos(t) is undefined whenever cos(t) = 0. The text demonstrates how to find the values of t that satisfy this condition, identifying the domain in terms of the integer multiples of π/2. The process is repeated for cosecant, tangent, and cotangent, demonstrating how to determine the domain of each based on where their defining expressions are undefined. The ranges of these functions are then explored, which is also related to the behavior of cosine and sine functions. The text shows how to precisely describe the domain using set-builder notation, representing it as a set of real numbers. This detailed analysis of domains and ranges is crucial for understanding the behavior and limitations of these trigonometric functions. This knowledge is essential for their correct and accurate application in various mathematical contexts.

This section delves into the graphical representation of secant and cosecant functions. Since sec(x) = 1/cos(x), the graph of y = sec(x) is constructed by taking the reciprocals of the y-values of the graph of y = cos(x). The text highlights that because cos(x) = 0 at odd multiples of π/2, these values result in vertical asymptotes in the graph of y = sec(x). The behavior of sec(x) as x approaches these asymptotes from the left and right is examined (approaching positive or negative infinity). The same process is followed for cosecant (csc(x) = 1/sin(x)), establishing that vertical asymptotes occur where sin(x) = 0 (i.e., at integer multiples of π). The periodicity of sec(x) and csc(x), inherited from cosine and sine, is also explained. A fundamental cycle of each function is graphed, and the text explains how to generate the complete graph by copying and pasting this fundamental cycle across the x-axis. The graphical analysis demonstrates how the behavior of secant and cosecant is intricately linked to and directly influenced by the behavior of cosine and sine, which were previously defined and discussed. This section builds on the prior section and emphasizes the reciprocal relationship between functions and their graphical implications.

This section introduces the concept of inverse trigonometric functions, addressing the challenge that standard trigonometric functions (cosine, sine, tangent, etc.) are not one-to-one due to their periodic nature. To create invertible functions, the domains of these functions must be restricted. The example of the cosine function is used to illustrate this concept; restricting the domain to the interval [0, π] maintains a range of [-1, 1] while preserving smoothness and continuity. The text notes that the inverse functions reverse the process of the original function. For example, the function f(t) = cos(t) maps real number inputs to x-coordinates on the unit circle. The inverse function, arccos(x), would take x-coordinates and return oriented arcs, explaining the "arc" terminology. The graphs of f(x) = cos(x) and its inverse, f⁻¹(x) = arccos(x), are shown, highlighting that the inverse is obtained by reflecting the original graph across the line y = x. This section lays the foundation for understanding inverse trigonometric functions and their significance in solving trigonometric equations.

The section demonstrates how to use inverse trigonometric functions to solve equations that cannot be easily solved using the unit circle alone. The text presents the problem of solving equations like sin(θ) = 1/3 or tan(t) = -2, emphasizing that direct solutions using only familiar angles are not possible. The inverse trigonometric functions are introduced as a tool to solve these equations, drawing a parallel to how the square root function is used to solve certain quadratic equations. Examples are provided, illustrating the use of inverse trigonometric functions to find solutions to equations that involve angles or real numbers. The text highlights that there isn't a universally accepted range for all inverse trigonometric functions, which can cause complications. Strategies are introduced for dealing with these ambiguities. The section shows how the inverse functions provide a systematic approach for solving trigonometric equations, expanding the range of solvable equations beyond those involving only the 'common angles' found on the unit circle.

The section expands on the techniques for solving trigonometric equations using inverse functions, focusing on more complex scenarios. It addresses the challenge presented by equations like sec²(x) = 4, where the trigonometric function is squared. A step-by-step procedure is given, involving taking square roots to obtain sec(x) = ±2. The equation is then converted to an equivalent cosine problem (cos(x) = ±1/2). The solutions are expressed using the inverse cosine function and general forms that represent all possible solutions. The text emphasizes the importance of interpreting the solutions geometrically to combine similar solution sets. The use of the unit circle and reference angles is integral to this process, and careful consideration is given to ensuring that all solutions are identified. The section demonstrates how to deal with equations containing squared trigonometric functions and emphasizes the use of both algebraic manipulation and the geometric understanding of the unit circle to find a complete solution set. The analytical approach in solving equations using inverse functions is further illustrated.

This section introduces sinusoidal motion, building upon the previously established concepts of circular motion and trigonometric functions. It connects the concepts of period, frequency, and angular velocity to sinusoidal motion. The period is defined as the time it takes for one complete cycle of the motion, while frequency relates to the number of cycles completed per unit time. Angular frequency (ω) is introduced as the number of cycles completed over a 2π interval. The equation v = rω (where v is linear velocity, r is radius, and ω is angular velocity) is revisited, emphasizing that points farther from the center of rotation must travel faster to maintain the same angular frequency. The section sets the stage for understanding how trigonometric functions can be used to model oscillatory or repetitive motion, laying the groundwork for more detailed analysis of sinusoidal functions and their properties.

This section applies the concepts of sinusoidal motion to a real-world example: the Giant Wheel at Cedar Point amusement park. The physical characteristics of the wheel (diameter 128 feet, platform height 8 feet, total height 136 feet, two revolutions in 127 seconds) are provided to create a mathematical model. The goal is to derive a sinusoidal function that describes the height of a passenger above the ground as a function of time (in seconds), starting from the point where the passenger is closest to the ground. The radius (r = 64 feet) and the period (T = 127/2 seconds) are calculated. The angular frequency (ω = 4π/127 radians per second) is derived. The general equation for the y-coordinate in counterclockwise circular motion (y = r sin(ωt)) is used as the foundation for the model. The final sinusoidal equation representing the passenger's height as a function of time is derived and explained, directly demonstrating how sinusoidal functions model periodic real-world phenomena.

The section explains parameters crucial to describing sinusoidal functions: amplitude, period, angular frequency, phase shift, and vertical shift. The amplitude is defined as the maximum displacement from the baseline (determined by the vertical shift). The period is the time required for one complete cycle, while the angular frequency indicates the number of cycles over a 2π interval. The phase shift represents a horizontal shift of the graph, and the vertical shift adjusts the baseline of the function. The text notes that, in some fields, the phase shift is referred to as the 'phase' of the sinusoid. A theorem is mentioned (Theorem 10.23) that provides formulas for determining these parameters given a sinusoidal function in a specific form. It clarifies that the angular frequency (ω) is always positive. This section builds on the previous by clarifying how these parameters affect the shape and position of the sinusoidal graph, allowing for a more thorough analysis and manipulation of these important trigonometric functions.