Trigonometry
Topic outline

Welcome to Trigonometry [CMAT 1223]!
Course Introduction
This Trigonometry [CMAT 1223] course adheres to the scope and sequence of a onesemester Trigonometry course. The course description provided is the from the statewide common course information.Course Description: Trigonometric functions and graphs; inverse trigonometric functions; fundamental identities and angle formulas; solving equations; triangles with applications; polar coordinate system.Upon successful completion of this course, the student will be able to:
 Apply the definitions of angles, triangles, and the rectangular coordinate system to the six trigonometric functions.
 Solve problems using right triangle trigonometry.
 Extending the six trig functions on each quadrant by defining a reference angle with +  signs.
 Solve problems for arc length and area of a sector.
 Calculate linear and angular speed to solve related problems.
 Graph trigonometric functions and their transformations.
 Solve problems involving the inverses of trigonometric functions.
 Verify trigonometric identities and solve problems involving sum, difference, doubleangle, halfangle, sumtoproduct, and producttosum formulas.
 Solve trigonometric equations.
 Solve triangles and applications using the Laws of Sines, the Law of Cosines and area formulas.
 Plot points and rewrite equations between the rectangular and polar coordinate systems.
 Apply the De Moivre’s and n^{th} root Theorems
Adopting instructors can embed a welcome video or add additional text here.
Structure of the Course
The course includes 10 content Modules which covers each of the 10 chapters in the textbook (provided link). Each module includes a brief introduction text with module learning objectives, links to the corresponding Pressbook sections, Chapter Summary/Review, homework assignments in MyOpenMath, Section and Chapter Review Exercises from the Pressbook, a Quiz in MyOpenMath, and a Q&A discussion forum for that Module. There are additional modules for smaller Exams, a Midterm Exam, and/or Final Exam.
Navigating the Course
This course is set up in Modules covering various topics which may be accessed from the course navigation menu on the left or by scrolling below. Modules may be collapsed in the menu and it the body of the course to minimize scrolling. Many items are required and may be marked as completed automatically when the activity has been submitted (the broken check box), but others will marked as done by the student (the solid check box).
Please move through the items below and continue through the Learner Support and Getting Started modules before moving on to Module 1. Be sure to check for announcements and due dates to stay on track.Adopting instructors can embed a navigation video or add additional text here.
This course and its contents are licensed under a Creative Commons Attribution 4.0 International License by LOUIS: The Louisiana Library Network, except where otherwise noted.Adopting instructors should edit the About Your Instructor and Office Hours Information pages in this Module.

Adopting instructors should edit all pages in this Module  as per their own Institution's policies.

This module contains all the items you should review and complete before you begin Module 1. Before moving on, be sure to:
 Check the News and Announcements Forum
 Read the Course Syllabus
 Introduce yourself to the class
 Read the instructions for the Q & A Forum
Good luck in the course!
Use this forum to tell us a little about yourself and your interests. Some topic ideas:
 What is your field of study/research interest or concentration?
 What are you most interested in learning about in this class and why?
 Have you ever taken an online class before?
 Any other information you would like to share with your classmates, such as special interests or activities.
Post a picture! We look forward to meeting you.

Use this forum to ask your instructor any questions you have about the course. You may post at any time, and your instructor will respond here. Be as specific as possible.
Please keep in mind that others can see your posts, so do not post any personal information. If you have questions about your grade, please email your instructor directly. You can expect a response to posts and emails within [X] hours. [Recommendation is 24 hours MF, next business day on weekends.]
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 Check the News and Announcements Forum

Circles and triangles are the simplest geometric figures. A circle is the simplest sort of curve, and a triangle is the simplest polygon  the one with the fewest sides. Each has properties that makes it useful in many fields of endeavor. A triangle is the most stable of polygons, because once its sides are fixed in length, its angles cannot change. A triangular truss bridge is a stable structure. A threelegged stool will not wobble. A circle encloses more area then any other figure of the same length, or perimeter, and a sphere encloses more space.
A geodesic dome is a portion of a sphere constructed with triangles. It has been called "the strongest, lightest and most efficient means of enclosing space known to man.” Geodesic domes may also help us learn how to live on another planet.
In 2017, the United Arab Emirates began construction on Mars Science City, a series of interconnecting geodesic domes designed to be a realistic model for living on Mars. The city will cover 1.9 million square feet, and its walls will be 3Dprinted using sand from the desert. The city will contain laboratories to address food, sustainability, and energy issues all over the world. Finally, the project will implement an experiment wherein a team will spend a year living in the simulated planet for a year.
Image Caption: Many geometric designs are made of circles and triangles.
(Content & Image Source: Chapter 1 Introduction, Trigonometry, Katherine Yoshiwara, GNU Free Documentation License)
Upon completion of this module, you will be able to:1.1 Angles and Triangles
 Sketch a triangle with given properties
 Find an unknown angle in a triangle
 Find angles formed by parallel lines and a transversal
 Find exterior angles of a triangle
 Find angles in isosceles, equilateral, and right triangles
 State reasons for conclusions
1.2 Similar Triangles
 Identify congruent triangles and find unknown parts
 Identify similar triangles
 Find unknown parts of similar triangles
 Solve problems using proportions and similar triangles
 Use proportions to relate sides of similar triangles
1.3 Circles
Find the distance between two points

Distinguish between exact values and approximations

Graph a circle

Find and use the equation for a circle

Find the length of a fraction of a circle

Find the area of a sector of a circle
To achieve these objectives:
 Read the Module 1 Introduction (see above).
 Read Sections 1.11.3 of Chapter 1: Triangles and Circles in Trigonometry (links to each Section provided below)
 Note: The Algebra Refresher at the top of each Section might be beneficial before you begin
 At the end of each Section there is a list of Vocabulary, Concepts, Study Questions, and a SelfCheck H5P activity
 Complete the MyOpenMath Homework Assignments for each Section (links provided below)  These are graded!
 View the Chapter 1 Summary and Review (link provided below)
 Practice the problems on the Exercises Sections, checking the solutions provided (links to each Section provided below)
 View the Exercises: Chapter 1 Review Problems (link provided below)
 Complete the MyOpenMath Quiz for Chapter 1 (link provided below)  This is graded!
 Once you complete the Quiz, upload your work in the Quiz Work Upload Assignment using the submission link below.
 Post in the Chapter 1 Q&A Discussion Forum  link provided below.
Note the check boxes to the right that help you track your progress: some are automatic, and some are manual.
Module Pressbooks Resources and Activities
You will find the following resources and activities in this module at the Pressbooks website. Click on the links below to access or complete each item.
 Sketch a triangle with given properties

How would you measure the distance to an inaccessible object, such as a ship at sea? In the 6th century BC, the Greek philosopher Thales estimated the distances to ships at sea using triangulation, a method for calculating distances by forming triangles. Using trigonometry and the measured length of just one side, the lengths of the other sides can be calculated. Triangulation has been used to compute distances ever since. In the 16th century mapmakers began to use triangulation to position faraway places accurately. And as new methods in navigation and astronomy required greater precision, the idea of a survey using chains of triangles was developed.In 1802, the East India Company embarked on the Great Trigonometrical Survey of India. Its goal was to measure the entire Indian subcontinent with scientific precision. The surveyors began by measuring a baseline near Madras. The baseline was the only distance they measured; all other distances were calculated from it using measured angles. Each calculated distance became the base side of another triangle used to calculate the distance to another point, which in turn started another triangle. Eventually this process formed a chain of triangles connecting the origin point to other locations.Because of the size of the area to be surveyed, the surveyors did not triangulate the whole of India but instead created what they called a "gridiron" of triangulation chains running from North to South and East to West. You can see these chains in the map of the survey. The Survey was completed in 1871. Along the way it calculated the height of the Himalayan giants: Everest, K2, and Kanchenjunga, and provided one of the first accurate measurements of a section of an arc of longitude. Triangulation today is used for many purposes, including surveying, navigation, meteorology, astronomy, binocular vision, and location of earthquakes.Image Caption: The Pythagorean spiral is an image made with right triangles stacked by their hypotenuse and base.
(Content & Image Source: Chapter 2 Introduction, Trigonometry, Katherine Yoshiwara, GNU Free Documentation License)
Upon completion of this module, you will be able to:2.1 Side and Angle Relationships
Identify inconsistencies in figures

Use the triangle inequality to put bounds on the lengths of sides

Use the Pythagorean theorem to find the sides of a right triangle

Use the Pythagorean theorem to identify right triangles

Solve problems using the Pythagorean theorem

Use measurements to calculate the trigonometric ratios for acute angles

Use trigonometric ratios to find unknown sides of right triangles

Solve problems using trigonometric ratios

Use trig ratios to write equations relating the sides of a right triangle

Use relationships among the trigonometric ratios

Solve a right triangle

Use inverse trig ratio notation

Use trig ratios to find an angle

Solve problems involving right triangles

Know the trig ratios for the special angles
 Read the Module 2 Introduction (see above).
 Read Sections 2.12.3 of Chapter 2: The Trigonometric Ratios in Trigonometry (links to each Section provided below)
 Note: The Algebra Refresher at the top of each Section might be beneficial before you begin
 At the end of each Section there is a list of Vocabulary, Concepts, Study Questions, and a SelfCheck H5P activity
 Complete the MyOpenMath Homework Assignments for each Section (links provided below)  These are graded!
 View the Chapter 2 Summary and Review (link provided below)
 Practice the problems on the Exercises Sections, checking the solutions provided (links to each Section provided below)
 View the Exercises: Chapter 2 Review Problems (link provided below)
 Complete the MyOpenMath Quiz for Chapter 2 (link provided below)  This is graded!
 Once you complete the Quiz, upload your work in the Quiz Work Upload Assignment using the submission link below.
 Post in the Chapter 2 Q&A Discussion Forum  link provided below.
Note the check boxes to the right that help you track your progress: some are automatic, and some are manual.Module Pressbooks Resources and Activities
You will find the following resources and activities in this module at the Pressbooks website. Click on the links below to access or complete each item.


The first science developed by humans is probably astronomy. Before the invention of clocks and calendars, early people looked to the night sky to help them keep track of time. What is the best time to plant crops, and when will they ripen? On what day exactly do important religious festivals fall? By tracking the motions of the stars, early astronomers could identify the summer and winter solstices and the equinoxes. The rising and setting of certain stars marked the hours of the night. If we think of the stars as traveling on a dome above the Earth, we create the celestial sphere. Actually, of course, the Earth itself rotates among the stars, but for calculating the motions of heavenly objects, this model works very well.Babylonian astronomers kept detailed records on the motion of the planets, and were able to predict solar and lunar eclipses. All of this required familiarity with angular distances measured on the celestial sphere. To find angles and distances on this imaginary sphere, astronomers invented techniques that are now part of spherical trigonometry. The laws of sines and cosines were first stated in this context, in a slightly different form than the laws for plane trigonometry.
On a sphere, a greatcircle lies in a plane passing through the sphere's center. It gives the shortest distance between any two points on a sphere, and is the analogue of a straight line on a plane. A spherical angle is formed where two such arcs intersect, and a spherical triangle is made up of three arcs of great circles. The spherical law of sines was first introduced in Europe in 1464 by Johann Muller, also known as Regiomontus, who wrote: "You, who wish to study great and wondrous things, who wonder about the movement of the stars, must read these theorems about triangles. ... For no one can bypass the science of triangles and reach a satisfying knowledge of the stars."Image Caption: Representation of the declination and of the hour angle in a geocentric system. On this figure, the equator and the ecliptic are fixed. The apparent motion of the Sun corresponds to a complete revolution along the ecliptic in one year. The hour angle (and the location of the local meridian (brown circle) trough the North pole and the zenith) change as the Earth makes a complete rotation around its axis in 1 day. (credit: Universite catholique de Louvain, http://www.climate.be/textbook/chapter2_node5_2.xml, CC BYBC License)(Content & Image Source: Chapter 3 Introduction, Trigonometry, Katherine Yoshiwara, GNU Free Documentation License)
Upon completion of this module, you will be able to:3.1 Obtuse Angles
Use the coordinate definition of the trig ratios

Find the trig ratios of supplementary angles

Know the trig ratios of the special angles in the second quadrant

Find two solutions of the equation \( sin( \theta)=k \)

Find the area of a triangle
3.2 The Law of Sines
Use the Law of Sines to find a side

Use the Law of Sines to find an angle

Use the Law of Sines to solve an oblique triangle

Solve problems using the Law of Sines

Compute distances using parallax

Solve problems involving the ambiguous case
3.3 The Law of Cosines
Use the Law of Cosines to find the side opposite an angle

Use the Law of Cosines to find an angle

Use the Law of Cosines to find a side adjacent to an angle

Decide which law to use

Solve a triangle

Solve problems using the Law of Cosines
To achieve these objectives: Read the Module 3 Introduction (see above).
 Read Sections 3.13.3 of Chapter 3: Laws of Sines and Cosines in Trigonometry (links to each Section provided below)
 Note: The Algebra Refresher at the top of each Section might be beneficial before you begin
 At the end of each Section there is a list of Vocabulary, Concepts, Study Questions, and a SelfCheck H5P activity
 Complete the MyOpenMath Homework Assignments for each Section (links provided below)  These are graded!
 View the Chapter 3 Summary and Review (link provided below)
 Practice the problems on the Exercises Sections, checking the solutions provided (links to each Section provided below)
 View the Exercises: Chapter 3 Review Problems (link provided below)
 Complete the MyOpenMath Quiz for Chapter 3 (link provided below)  This is graded!
 Once you complete the Quiz, upload your work in the Quiz Work Upload Assignment using the submission link below.
 Post in the Chapter 3 Q&A Discussion Forum  link provided below.
Note the check boxes to the right that help you track your progress: some are automatic, and some are manual.Module Pressbooks Resources and Activities
You will find the following resources and activities in this module at the Pressbooks website. Click on the links below to access or complete each item.


In this module you will take your Exam 1 .
Read the Exam 1 Information and Instructions page carefully and take note of any special submission guidelines.
Upon completion of this module, you will have: Read and viewed the Exam 1 Information and Instructions page
 Scheduled your exam with the proctoring service [if applicable, delete if not needed]
 Post in the Exam 1 Q&A Discussion Forum  link provided below.
 Prepared for and submitted your Exam 1 [revise as needed]
 Uploaded your work in the Exam 1 Work Upload Assignment using the submission link below.
Attribution of image: ("Math, Numbers, Number image. Free for use." Pixabay.com. https://pixabay.com/photos/mathnumbersnumbercounting5247958/)
Adopting instructors: Edit the Exam 1 Information and Instructions page.

A Lissajous figure is a pattern produced by the intersection of two sinusoidal curves at right angles to each other. They are the curves you often see on oscilloscope screens depicting compound vibration. They were first studied by the American mathematician Nathaniel Bowditch in 1815, and later by the French mathematician JulesAntoine Lissajous, and today have applications in physics and astronomy, medicine, music, and many other fields.
In 1855 Lissajous invented a pair of tuning forks designed to visualize sound vibrations. Each tuning fork had a small mirror mounted at the end of one prong, and a light beam reflected from one mirror to the other was projected onto a screen, producing a Lissajous figure. Stable patterns appear only when the two forks vibrate at frequencies in simple ratios, such as 2:1 or 3:2, which correspond to the musical intervals of the octave and perfect fifth.So, by observing the Lissajous figures, people were able to make tuning adjustments more accurately than they could do by ear. In 1942 the Dadaist artist Max Ernst punched a small hole in a can of paint, attached it to a coupled pendulum, and set it swinging to create Lissajous figures. He then used the designs in some of his paintings.In 2001, NASA launched the Wilkinson Microwave Anisotropy Probe (WMAP) to make fundamental measurements of our universe as a whole. The probe was positioned near a gravitational balance point between Earth and the Sun and moved in a controlled Lissajous pattern around the point. This orbit isolated the spacecraft from radio emissions from Earth. The goal of WMAP was to map the relative cosmic microwave background (CMB) temperature over the full sky. CMB radiation is the radiant heat left over from the Big Bang. Tiny fluctuations in the CMB are the result of fluctuations in the density of matter in the early universe, so they carry information about the initial conditions for the formation of cosmic structures such as galaxies, clusters, and voids.From the WMAP data, scientists were able to:
estimate the age of the universe at 13.77 billion years old.

calculate the curvature of space to within 0.4% of "flat" Euclidean.

determine that ordinary atoms (also called baryons) make up only 4.6% of the universe.

find that dark matter (matter not made of atoms) is 24.0% of the universe.
Image Caption: A Lissajous knot is a knot defined by parametric equations that take on specific forms of trigonometric functions with phase shifts (credit: https://en.wikipedia.org/wiki/Lissajous_knot)
(Content & Image Source: Chapter 4 Introduction, Trigonometry, Katherine Yoshiwara, GNU Free Documentation License)
Upon completion of this module, you will be able to:4.1 Angles and Rotation
Use angles to represent rotations

Sketch angles in standard position

Find coterminal angles

Find and use reference angles

Find trigonometric ratios for the special angles
4.2 Graphs of Trigonometric Functions
Find coordinates

Use bearings to determine position

Sketch graphs of the sine and cosine functions

Find the coordinates of points on a sine or cosine graph

Use function notation

Find reference angles

Solve equations graphically

Graph the tangent function

Find and use the angle of inclination of a line
4.3 Using Trigonometric Functions
Solve trigonometric equations, graphically and algebraically

Find coordinates of points on circles

Use bearings to determine position

Find and use the angle of inclination of a line

Identify periodic functions and give their periods

Sketch periodic functions

Sketch graphs to model sinusoidal functions

Analyze periodic graphs
To achieve these objectives: Read the Module 4 Introduction (see above).
 Read Sections 4.14.3 of Chapter 4: Trigonometric Functions in Trigonometry (links to each Section provided below)
 Note: The Algebra Refresher at the top of each Section might be beneficial before you begin
 At the end of each Section there is a list of Vocabulary, Concepts, Study Questions, and a SelfCheck H5P activity
 Complete the MyOpenMath Homework Assignments for each Section (links provided below)  These are graded!
 View the Chapter 4 Summary and Review (link provided below)
 Practice the problems on the Exercises Sections, checking the solutions provided (links to each Section provided below)
 View the Exercises: Chapter 4 Review Problems (link provided below)
 Complete the MyOpenMath Quiz for Chapter 4 (link provided below)  This is graded!
 Once you complete the Quiz, upload your work in the Quiz Work Upload Assignment using the submission link below.
 Post in the Chapter 4 Q&A Discussion Forum  link provided below.
Note the check boxes to the right that help you track your progress: some are automatic, and some are manual.Module Pressbooks Resources and Activities
You will find the following resources and activities in this module at the Pressbooks website. Click on the links below to access or complete each item.


When we think of tennis as a game of angles, we may imagine players racing up to the net, creating options to deliver powerful cross shots that will leave their opponent stumbling toward the line.
This is an exciting and effective method of play, though it brings greater risk. But while the excitement of the game interplays with all types of geometry, some of the newest innovations make even more use of mathematics.
With balls traveling well over 100 miles per hour judges cannot always discern the centimeter or millimeters of difference between a ball that is in or out of bounds. Professional tennis was among the first sports to rely on an advanced tracking system called HawkEye to help make close calls. The system uses several highresolution cameras that are able to monitor and the ball's movement and its position on the court. Using the images from several cameras at once, the system's computers use trigonometric calculations to triangulate the ball's exact position and, essentially, turn a series of twodimensional images into a threedimensional one. Also, since the ball travels faster than the cameras' frame rate, the system also must make predictions to show where a ball is at all times. These technologies generally provide a more accurate game that builds more confidence and fairness. Similar technologies are used for baseball, and automated strikecalling is under discussion.
Image Caption: None.
(Content Source: Chapter 9 Introduction, Algebra and Trigonometry 2e, Jay Abramson, OpenStax, CC BY 4.0 License)
(Image Source: Chapter 5 Introduction, Trigonometry, Katherine Yoshiwara, GNU Free Documentation License)
Upon completion of this module, you will be able to:5.1 Algebra with Trigonometric Ratios
Evaluate trigonometric expressions

Simplify trigonometric expressions

Recognize equivalent expressions

Multiply or expand trigonometric expressions

Factor trigonometric expressions
5.2 Solving Equations
Use reference angles

Solve equations by trial and error

Use graphs to solve equations

Solve trigonometric equations for exact values

Use a calculator to solve trigonometric equations

Solve trigonometric equations that involve factoring
5.3 Trigonometric Identities
Recognize identities

Verify identities

Rewrite expressions using identities

Use identities to evaluate expressions

Solve trigonometric equations

Given one trig ratio, find the others
To achieve these objectives: Read the Module 5 Introduction (see above).
 Read Sections 5.15.3 of Chapter 5: Equations and Identities in Trigonometry (links to each Section provided below)
 Note: The Algebra Refresher at the top of each Section might be beneficial before you begin
 At the end of each Section there is a list of Vocabulary, Concepts, Study Questions, and a SelfCheck H5P activity
 Complete the MyOpenMath Homework Assignments for each Section (links provided below)  These are graded!
 View the Chapter 5 Summary and Review (link provided below)
 Practice the problems on the Exercises Sections, checking the solutions provided (links to each Section provided below)
 View the Exercises: Chapter 5 Review Problems (link provided below)
 Complete the MyOpenMath Quiz for Chapter 5 (link provided below)  This is graded!
 Once you complete the Quiz, upload your work in the Quiz Work Upload Assignment using the submission link below.
 Post in the Chapter 5 Q&A Discussion Forum  link provided below.
Note the check boxes to the right that help you track your progress: some are automatic, and some are manual.
Module Pressbooks Resources and Activities
You will find the following resources and activities in this module at the Pressbooks website. Click on the links below to access or complete each item.


In this module you will take your Midterm Exam or Exam 2.
Read the Midterm Exam or Exam 2 Information and Instructions page carefully and take note of any special submission guidelines.
Upon completion of this module, you will have: Read and viewed the Midterm Exam or Exam 2 Information and Instructions page
 Scheduled your exam with the proctoring service [if applicable, delete if not needed]
 Post in the Midterm Exam or Exam 2 Q&A Discussion Forum  link provided below.
 Prepared for and submitted your Midterm Exam or Exam 2 [revise as needed]
 Uploaded your work in the Midterm Exam or Exam 2 Work Upload Assignment using the submission link below.
Attribution of image: ("Math, Numbers, Number image. Free for use." Pixabay.com. https://pixabay.com/photos/mathnumbersnumbercounting5247958/)Adopting instructors: Edit the Midterm Exam or Exam 2 Information and Instructions page.

Have you ever wonderd why we divide the circle into 360 degrees? Nobody really knows the answer, but it may well have started around 600 BCE with the Babylonians. The Babylonians lived between the Tigris and Euphrates rivers in present day Turkey and Syria. They kept written records using a stylus to press cuneiform, or wedgeshaped, symbols into wet clay tablets, which were then baked in the sun. Thousands of these tablets have survived and give us detailed information about the mathematical practices of the time.The Babylonians used a base 60 number system because the number 60 has many factors. They did not invent decimal fractions, so they found it difficult to deal with remainders when doing division. But 60 can be divided evenly by 2, 3, 4, 5, and 6, which made calculations with common fractions much easier. We still see traces of their base 60 system in our own day: there are 60 seconds in a minute, and 60 minutes in an hour.In geometry, Babylonian mathematicians used the corner of an equilateral triangle as their basic unit of angular measure, and naturally divided that angle into 60 smaller angles. Now, if the corners of six equilateral triangles are placed together they form a complete circle, and that is why there are six times 60, or 360 degrees of arc in a circle. During the reign of Nebuchadnezzar, using the tools and technology available to them, Babylonian astronomers calculated that a complete year numbered 360 days. This made dividing the circle into 360 degrees even more useful. So the number 360 is not fundamental to the nature of a circle. If ancient civilizations had defined the full circle to be some other number of degrees, we'd probably be using that number today. But why do we need another, different way to measure angles? In this chapter we'll study radian measure, which at first may seem awkward and unnatural. As a hint, consider that although 360 is not fundamental to circles, the numberRadians connect the measure of an angle with the arclength it cuts out on a circle. Imagine a circle of radius 1unit rolling along a straight line. The circumference of a circle is \( 2 \pi \)The term radian first appeared in print on June 5, 1873, on an exam written by James Thomson, the brother of Lord Kelvin, at Queen's College in Belfast. In calculus and most other branches of mathematics beyond practical geometry, angles are nearly always measured in radians. Because radians arise naturally when dealing with circles, important relationships are expressed more concisely in radians. In particular, results involving trigonometric functions are simpler when the variables are expressed in radians.Image Caption: None
(Content & Image Source: Chapter 6 Introduction, Trigonometry, Katherine Yoshiwara, GNU Free Documentation License)
Upon completion of this module, you will be able to:6.1 Arclength and Radians
Express angles in degrees and radians

Sketch angles given in radians

Estimate angles in radians

Use the arclength formula

Find coordinates of a point on a unit circle

Calculate angular velocity and area of a sector
6.2 The Circular Functions
Know the trigonometric function values for the special angles in radians

Use a unit circle to find trig values

Find reference angles in radians

Evaluate trigonometric expressions

Find coordinates on a unit circle

Find an angle with a given terminal point on a unit circle

Use the tangent ratio to find slope

Find coordinates on a circle of radius
6.3 Graphs of the Circular Functions
Graph the trig functions of real numbers

Solve trigonometric equations graphically

Work with reference angles

Solve trigonometric equations algebraically

Evaluate trigonometric functions of real numbers

Use trigonometric models

Locate points on the graphs of the trigonometric functions

Find the domain and range of a function
To achieve these objectives: Read the Module 6 Introduction (see above).
 Read Sections 6.16.3 of Chapter 6: Radians in Trigonometry (links to each Section provided below)
 Note: The Algebra Refresher at the top of each Section might be beneficial before you begin
 At the end of each Section there is a list of Vocabulary, Concepts, Study Questions, and a SelfCheck H5P activity
 Complete the MyOpenMath Homework Assignments for each Section (links provided below)  These are graded!
 View the Chapter 6 Summary and Review (link provided below)
 Practice the problems on the Exercises Sections, checking the solutions provided (links to each Section provided below)
 View the Exercises: Chapter 6 Review Problems (link provided below)
 Complete the MyOpenMath Quiz for Chapter 6 (link provided below)  This is graded!
 Once you complete the Quiz, upload your work in the Quiz Work Upload Assignment using the submission link below.
 Post in the Chapter 6 Q&A Discussion Forum  link provided below.
Note the check boxes to the right that help you track your progress: some are automatic, and some are manual.Module Pressbooks Resources and Activities
You will find the following resources and activities in this module at the Pressbooks website. Click on the links below to access or complete each item.


Musical instruments produce sound by the vibration of a string on a violin, guitar, or piano, or a column of air in a brass or woodwind instrument. The vibration causes a periodic variation in air pressure that is heard as sound. Periodic vibrations create a pleasing, musical sound, while random vibrations sound like noise. There are three distinguishing characteristics of musical notes: loudness, pitch, and timbre.
 The loudness of a note is measured by the magnitude, or amplitude, of the changes in air pressure.
 The timbre of the note allows us to tell a piano note from a guitar note with the same loudness and pitch.
 The pitch of the note is determined by its frequency, the number of times its basic pattern is repeated each second. For example, a note with pitch 440 hertz repeats its pattern 440 times per second. Humans can hear frequencies that range roughly from 20 to 18,000 hertz.
The simplest pressure vibration is produced by a tuning fork. The graph of its pressure function resembles a sine curve, and the corresponding sound is called a pure tone. When a musical instrument produces a pure tone, not only is its frequency produced, but integer multiples of the fundamental frequency, called harmonics, are produced as well. Sine curves of different frequencies combine to create the timbre of the note. If two musical instruments play the same note, the notes have the same pitch, but they sound different because the amplitudes of each of the harmonics is different for the two instruments.The French mathematician Joseph Fourier, who lived from 1768 to 1830, discovered that any periodic wave can be written as the sum of a number of sines and cosines with differing amplitudes. Fourier created a method for determining the frequencies and amplitudes of the simpler waves that make up a more complicated periodic function. Applying the Fourier transform to a sampled musical note reveals which component frequencies are present in the note. The same sound can then be resynthesized by including those frequency components. Today, Fourier analysis is the foundation of signal processing, not only for audio waves, but for radio waves, light waves, seismic waves, and even images. By analyzing a compound waveform, the "noise" components can be isolated and removed in order to smooth out the signal. Fourier analysis is also used in Xray crystallography to reconstruct a crystal structure from its diffraction pattern, and in nuclear magnetic resonance spectroscopy to determine the mass of ions from the frequency of their motion in a magnetic field.Image Caption: Fourier Art also allows you to create beautiful animations by smoothly changing the Fourier coefficients of a curve. This change can be uniform for all the coefficients or modeled by a specific function for each one of them, which offers a great potential for expression. (credit: https://www.fourierart.com/)(Content & Image Source: Chapter 7 Introduction, Trigonometry, Katherine Yoshiwara, GNU Free Documentation License)
Upon completion of this module, you will be able to:7.1 Transformations of Graphs
Identify the amplitude, period, and midline of a circular function

Graph a circular function

Find a formula for the graph of a circular function

Model periodic phenomena with circular functions

Graph transformations of the tangent function

Solve trigonometric equations graphically
7.2 The General Sinusoidal Function Graph trigonometric functions using a table of values
 Find a formula for a transformation of a trigonometric function
 Solve trigonometric equations graphically
 Model periodic phenomena with trigonometric functions
 Fit a circular function to data
7.3 Solving Equations Find exact solutions to equations of the form \( sin(nx)=k \)
 Find all solutions between \( 0 \) and \( 2 \pi \)
 Use a substitution to solve trigonometric equations
 Write expressions for exact solutions
 Solve problems involving trigonometric models
To achieve these objectives:
 Read the Module 7 Introduction (see above).
 Read Sections 7.17.3 of Chapter 7: Circular Functions in Trigonometry (links to each Section provided below)
 Note: The Algebra Refresher at the top of each Section might be beneficial before you begin
 At the end of each Section there is a list of Vocabulary, Concepts, Study Questions, and a SelfCheck H5P activity
 Complete the MyOpenMath Homework Assignments for each Section (links provided below)  These are graded!
 View the Chapter 7 Summary and Review (link provided below)
 Practice the problems on the Exercises Sections, checking the solutions provided (links to each Section provided below)
 View the Exercises: Chapter 7 Review Problems (link provided below)
 Complete the MyOpenMath Quiz for Chapter 7 (link provided below)  This is graded!
 Once you complete the Quiz, upload your work in the Quiz Work Upload Assignment using the submission link below.
 Post in the Chapter 7 Q&A Discussion Forum  link provided below.
Note the check boxes to the right that help you track your progress: some are automatic, and some are manual.
Module Pressbooks Resources and Activities
You will find the following resources and activities in this module at the Pressbooks website. Click on the links below to access or complete each item.

In this module you will take your Exam 2 or Exam 3. [revise as needed]
Read the Exam 2 or Exam 3 Information and Instructions page carefully and take note of any special submission guidelines.
Upon completion of this module, you will have: Read and viewed the Exam 2 or Exam 3 Information and Instructions page
 Scheduled your exam with the proctoring service [if applicable, delete if not needed]
 Post in the Exam 2 or Exam 3 Q&A Discussion Forum  link provided below.
 Prepared for and submitted your Exam 2 or Exam 3 [revise as needed]
 Uploaded your work in the Exam 2 or Exam 3 Work Upload Assignment using the submission link below.
Attribution of image: ("Math, Numbers, Number image. Free for use." Pixabay.com. https://pixabay.com/photos/mathnumbersnumbercounting5247958/)
Adopting instructors: Edit the Exam 2 or Exam 3 Information and Instructions page.

Mapmakers have always faced an unavoidable challenge: It is impossible to translate the surface of a sphere onto a flat map without some form of distortion. Over the years, a variety of map projections have been developed to suit different uses.
The sixteenth century was an age of discovery, when explorers and merchants began sailing to distant and previously unknown lands. But at that time there was no reliable technology for navigation.Although more regions of the world were being mapped more accurately, a flat map by itself was not enough to help a sailor in the middle of the ocean. In 1569, the Flemish cartographer Gerardus Mercator published a new map using what is known as a cylindrical projection. To imagine how a Mercator projection works, picture shining a light through a glass globe onto a piece of paper rolled into a cylinder and wrapped around the globe. The cylinder is tangent to the globe at its equator. Notice how the latitude lines are farther apart the farther you get from the Equator. This projection distorts the size of objects as the latitude increases, so that Greenland and Antarctica appear much larger than they actually are.But the Mercator projection map is ideally suited for navigation, because any straight line on the map is a line of constant true bearing. If a navigator measures the bearing on the map from his location to his destination, he can set his ship's compass for the same bearing and maintain that course. However, the Mercator projection does not preserve distances. On a globe, circles of latitude (also known as parallels) get smaller as they move away from the Equator towards the poles. Thus, in the Mercator projection, when a globe is "unwrapped" on to a rectangular map, the parallels need to be stretched to the length of the Equator. Mercator had to increase the scale of his map gradually as it moved away from the equator, so that the latitude lines appear equal in length to the equator.The horizontal scale factor at any latitude must be inversely proportional to lengths on that latitude. Because the radius of the circle of latitude \( \theta \) is \( Rcos( \theta) \)Image Caption: All six of the trigonometric function graphs can be graphed on one set of axis, showing overall symmetry.(Content & Image Source: Chapter 8 Introduction, Trigonometry, Katherine Yoshiwara, GNU Free Documentation License)
Upon completion of this module, you will be able to:8.1 Sum and Difference Formulas
Find trig values for the negative of an angle

Verify or disprove possible formulas

Find exact values for trigonometric functions

Simplify or expand expressions

Solve equations

Prove standard identities
8.2 Inverse Trigonometric Functions
Decide whether a function has an inverse function

Evaluate the inverse trig functions

Model problems with inverse trig functions

Solve formulas

Simplify expressions involving the inverse trig functions

Graph the inverse trig functions
8.3 The Reciprocal Functions
Evaluate the reciprocal trig functions for angles in degrees or radians

Find values or expressions for the six trig ratios

Evaluate the reciprocal trig functions in applications

Given one trig ratio, find the others

Evaluate expressions exactly

Graph the secant, cosecant, and cotangent functions

Identify graphs of the reciprocal trig functions

Solve equations in secant, cosecant, and cotangent

Use identities to simplify or evaluate expressions
 Read the Module 8 Introduction (see above).
 Read Sections 8.18.3 of Chapter 8: More Functions and Identities in Trigonometry (links to each Section provided below)
 Note: The Algebra Refresher at the top of each Section might be beneficial before you begin
 At the end of each Section there is a list of Vocabulary, Concepts, Study Questions, and a SelfCheck H5P activity
 Complete the MyOpenMath Homework Assignments for each Section (links provided below)  These are graded!
 View the Chapter 8 Summary and Review (link provided below)
 Practice the problems on the Exercises Sections, checking the solutions provided (links to each Section provided below)
 View the Exercises: Chapter 8 Review Problems (link provided below)
 Complete the MyOpenMath Quiz for Chapter 8 (link provided below)  This is graded!
 Once you complete the Quiz, upload your work in the Quiz Work Upload Assignment using the submission link below.
 Post in the Chapter 8 Q&A Discussion Forum  link provided below.
Note the check boxes to the right that help you track your progress: some are automatic, and some are manual.
Module Pressbooks Resources and Activities
You will find the following resources and activities in this module at the Pressbooks website. Click on the links below to access or complete each item.
