Precalculus II continues the in-depth study of functions addressed in Precalculus I by adding the trigonometric functions to your function toolkit. In this course, you will cover families of trigonometric functions, as well as their inverses, properties, graphs, and applications. Additionally, you will study trigonometric equations and identities, the laws of sines and cosines, polar coordinates and graphs, parametric equations and elementary vector operations.
You might be curious how the study of trigonometry, or “trig,” as it is more often referred to, came about and why it is important to your studies still. Trigonometry, from the Greek for “triangle measure,” studies the relationships between the angles of a triangle and its sides and defines the trigonometric functions used to describe those relationships. Trigonometric functions are particularly useful when describing cyclical phenomena and have applications in numerous fields, including astronomy, navigation, music theory, physics, chemistry, and – perhaps most importantly, to the mathematics student – calculus.
In this course, you will begin by establishing the definitions of the basic trig functions and exploring their properties and then proceed to use the basic definitions of the functions to study the properties of their graphs, including domain and range, and to define the inverses of these functions and establish the properties of these. Through the language of transformation, you will explore the ideas of period and amplitude and learn how these graphical differences relate to algebraic changes in the function formulas. You will also learn to solve equations, prove identities using the trig functions, and study several applications of these functions.
Upon successful completion of this course, the student will be able to:
- measure angles in degrees and radians, and relate them to arc length;
- solve problems involving right triangles and unit circles using the definitions of the trigonometric functions;
- solve problems involving non-right triangles;
- relate the equation of a trigonometric function to its graph;
- solve trigonometric equations using inverse trig functions;
- prove trigonometric identities;
- solve trig equations involving identities;
- relate coordinates and equations in Polar form to coordinates and equations in Cartesian form;
- perform operations with vectors and use them to solve problems;
- relate equations and graphs in Parametric form to equations and graphs in Cartesian form;
- link graphical, numeric, and symbolic approaches when interpreting situations and analyzing problems;
- write clear, correct, and complete solutions to mathematical problems using proper mathematical notation and appropriate language; and
- communicate the difference between an exact and an approximate solution and determine which is more appropriate for a given problem.
