Chapter 4 - Trigonometric Functions
"The trigonometric functions arose from the consideration of ratios within right triangles, the ultimate computational tool for engineers in the ancient world. As the great mysteries of civilization progressed from a flat earth to a world of circles and spheres, trigonometry was soon seen to be the secret to understanding circular phenomena as well. Then circular motion led to harmonic motion and waves, and suddenly trigonometry was the indispensable tool for understanding everything from electrical current to modern telecommunications.
The advent of calculus made the trigonometric functions more important than ever. It turns out that every kind of periodic (recurring) behavior can be modeled to any degree of accuracy by simply combining sine functions. The modeling aspect of trigonometric functions is another focus of our study."
(Demana, Franklin D. "Chapter 4." Precalculus: Graphical, Numerical, Algebraic. 7th ed. Boston: Addison-Wesley, 2007. 350. Print.)
Hipparchus of Nicaea (190-120 B.C.)
"Hipparchus of Nicaea, the “father of trigonometry,” compiled the first trigono- metric tables to simplify the study of astronomy more than 2000 years ago. Today, that same mathematics enables us to store sound waves digitally on a com- pact disc. Hipparchus wrote during the second century B.C., but he was not the first mathematician to “do” trigonometry. Greek mathematicians like Hippocrates of Chois (470–410 B.C.) and Eratosthenes of Cyrene (276–194 B.C.) had paved the way for using triangle ratios in astronomy, and those same triangle ratios had been used by Egyptian and Babylonian engineers at least 4000 years earlier. The term “trigonometry” itself emerged in the 16th century, although it derives from ancient Greek roots: “tri” (three), “gonos” (side), and “metros” (measure)."
(Demana, Franklin D. "Chapter 4 interesting fact." Precalculus: Graphical, Numerical, Algebraic. 7th ed. Boston: Addison-Wesley, 2007. 350. Print.)
"When the motion of an object causes air molecules to vibrate, we hear a sound. We measure sound according to its pitch and loudness, which are attributes associated with the frequency and amplitude of sound waves. As we shall see, it is the branch of mathematics called trigonometry that enables us to analyze waves of all kinds; indeed, that is only one appli- cation of this powerful analytical tool. See page 431 for an application of trigonometry to sound waves."
(Demana, Franklin D. "Chapter 4." Precalculus: Graphical, Numerical, Algebraic. 7th ed. Boston: Addison-Wesley, 2007. 349. Print.)
Lesson 4.1 - Angles and their Measures
This lesson will cover the problem of angular measure, degrees and radians, circular arc length, and angular and linear motion.
Lesson 4.2 - Trigonometric Functions of Acute Angles
This lesson will cover right triangle trigonometry, two famous triangles, evaluating trigonometric functions with a calculator, and applications of right triangle trigonometry.
Lesson 4.3 - Trigonometry Extended: The Circular Functions
This lesson will cover trigonometric functions of any angle, trigonometric functions of real numbers, periodic functions, and the 16-point unit circle.
Lesson 4.4 - Graphs of Sine and Cosine: Sinusoids
This lesson will revisit the basic waves, cover sinusoids and transformations, and modeling periodic behavior with sinusoids.
Lesson 4.5 - Graphs of Tangent, Cotangent, Secant, and Cosecant
This lesson will cover the tangent, cotangent, secant and cosecant functions.
Lesson 4.6 - Graphs of Composite Trigonometric Functions
This lesson will cover combining trigonometric and algebraic functions, sums and differences of sinusoids, and damped oscillation.
Lesson 4.7 - Inverse Trigonometric Functions
This lesson will cover inverse sine, cosine and tangent functions. Additionally, composing trigonometric and inverse trigonometric functions. Finally, applications of inverse trigonometric functions.
Lesson 4.8 - Solving Problems with Trigonometry
This lesson will have more right triangle problems, and simple harmonic motion.