# Pre-Calculus

## Course Overview

Pre-calculus provides the foundation for a deeper investigation into differential calculus. We start with a deep dive into trigonometry, including ratio definitions, the unit circle, graphs, identities and equations, along with the Law of Sines and Law of Cosines. We extend the study of trigonometry by investigating polar coordinates and graphs and vectors. In preparation for more complex topics in calculus, the course also covers parametric equations, partial fractions, systems of inequalities, and sequences and series. The last unit begins the study of limits and derivatives, which provides an easy segue to the first topics of calculus.

## Course Content

### Unit 1: Trigonometric Functions

This unit is the foundation upon which all of Trigonometry is built. Students must be not only proficient, but fluent in determining values of the six trigonometric functions at the common angle measures. This unit provides students with ample practice at both unit circle trigonometry and graphical trigonometry. This unit also extends student understanding of graph theory as it pertains to the trigonometric functions. This unit covers the following topics:

- Convert angles from radians to degrees and degrees to radians
- Find the length of the arc of a circle with known radius and central angle
- Find the area of the sector of a circle with known radius and central angle
- Find the linear speed of an object traveling in a circular motion
- Use Unit Circle to evaluate the six trigonometric function values at theta =0, π/6, π/4, π/3, π/2
- Generalize function values to the other three quadrants of the Unit Circle using “All Students Take Calculus”
- Use Calculator to evaluate trig values
- Evaluate trig values for a circle of radius r
- Determine domain, range, and period of trig functions
- Determine signs of trig functions based on quadrants
- Find values of trig functions using identities/definitions
- Use even-odd properties to find values of trig functions
- Graph functions of the form y = Asin(Bx) and y = Acos(bx)
- Determine amplitude and period of sinusoidal functions and graph those functions
- Find an equation from a information about amplitude, period and initial points
- Graph functions of the form y=Atan(Bx)+C, y=Acot(Bx)+C, y=Asec(Bx)+C, y=Acsc(Bx)+C
- Graph sinusoidal functions of the form y=Asin[B(x-C)]+D
- Build sinusoidal functions from data

### Unit 2: Analytic Trigonometry

This unit takes all of the fundamentals learned in Unit 1 and develops new ideas around them. Inverse trigonometric values and their functions, trigonometric identities, and trigonometric equations are the major themes of this unit. As always, applications and deeper levels of problem solving should be the primary goals of the instructor in this unit, as it is the content of this unit that will follow the student into Calculus and beyond. This unit covers:

- Find exact values of inverse sine function
- Find approximate values of inverse sine function
- Find exact values of composite inverse trig functions
- Solve equations using inverse trig functions
- Find exact values of expressions using inverse Sine, Cosine, and Tangent Functions
- Define inverse Secant, Cosecant and Cotangent
- Use Calculator to Evaluate Invers Secant, Cosecant, and Cotangent
- Solve equations involving trig functions
- Solve trig equations using calculators, quadratic forms, identities, and graphing utilities
- Use Algebra to simplify trig expressions
- Establish identities
- Use Sum and Difference Formulae to find exact values, establish identities, and involving inverse trig functions
- Solve trig equations using Sum and Difference Formulae
- Use Double-Angle Formulae to find exact values and establish identities
- Use Half-Angle Formulae to find exact values
- Express products and sums and sums as products

### Unit 3: Applications of Trigonometric Functions

This unit covers all of the mathematical applications of the six trigonometric functions, prior to the introduction of vectors. Right triangle trigonometry, with the SOH-CAH-TOA mnemonic, Law of Sines with the Ambiguous Case, Law of Cosines, and areas of triangles using trigonometry and Huron’s formula are the major content area of this unit. This unit includes the following topics:

- Find the value of trig functions of acute angles using right triangles
- Solve right triangles
- Solve applied problems
- Solve SAA or ASA triangles
- Solve SSA triangles with and without the ambiguous case
- Solve applied problems
- Solve SAS triangles
- Solve SSS triangles
- Solve applied problems
- Find the area of SAS triangles
- Find the area of SSS triangles

### Unit 4: Polar Functions

In this unit, students learn how rectangular coordinates can be represented by polar coordinates, r and Θ. Students will translate coordinates from rectangular to polar and back again, using the properties of polar coordinates. Students will also sketch graphs of polar functions using symmetry properties and graphing utilities. Students will also use DeMoivre’s Theorem to find roots and powers of complex numbers. This unit includes the following topics:

- Plot points using polar coordinates
- Convert polar coordinates to rectangular (and the inverse)
- Transform equations between rectangular and polar form
- Identify and graph polar equations by converting to rectangular form
- Test polar equations for symmetry
- Graph polar equations
- Plot points in the complex plane
- Convert a complex number between polar and rectangular form
- Find products and quotients of complex numbers in polar form
- Use De Moivre’s Theorem
- Find complex roots

### Unit 5: Vectors

This unit is the introductory unit for vectors, their dot products and their applications. It covers properties of vectors as well as many introductory applications that are seen again in physics courses and vector calculus courses. This unit covers the following topics:

- Graph vectors
- Find position vectors
- Add and subtract vectors algebraically
- Find scalar multiples and magnitudes of vectors
- Find unit vector
- Find vectors from direction and magnitude
- Model with vectors
- Find the dot product or two vectors
- Find the angle between two vectors
- Determine whether two vectors are parallel, orthogonal, or neither
- Decompose a vector into two orthogonal vectors
- Compute work
- Find position vectors in space
- Perform operations on vectors
- Find the dot product
- Find the and between two vectors
- Find direction angles of a vector

### Unit 6: Parametrics, Partial Fractions and Systems of Inequalities

This unit covers topics that students matriculating into Calculus will need in order to be successful. This unit includes the following topics:

- Graph parametric equations
- Find a rectangular equation for a curve defined parametrically
- Use time as a parameter in parametric equations
- Find parametric equations for curves defined by rectangular equations
- Decompose rational functions of the form P/Q
- Solve systems of nonlinear equations using substitution and elimination
- Graph inequalities and systems of inequalities

### Unit 7: Sequences, Series and Induction

This unit focuses mainly on Arithmetic and Geometric sequences and series, however, Induction is included in the unit because it is seen at all levels of higher mathematics. This unit includes the following topics:

- Write terms of a sequence
- Write the terms of a sequence defined by both a functions and recursive formulae
- Use summation notation
- Find the sum of a sequence
- Determine whether a sequence is arithmetic
- Find a formula for an arithmetic sequence
- Find the sum of an arithmetic sequence
- Determine whether a sequence is geometric
- Find the formula for a geometric sequence
- Find the sum of a geometric sequence
- Determine whether a geometric series converges or diverges
- Prove statements using mathematical induction

### Unit 8: Limits, Continuity and an Introduction to Derivatives

This is the actual unit referred to as “Precalculus”. The purpose of the unit is to provide students with the foundations for their Calculus course. This is the last section of this course and it will be the first section they see in Calculus. This unit includes the following topics:

- Find limits using tables and graphs
- Find limits of sums, differences, products, quotients polynomials, powers and roots
- Find the difference quotient Find one-sided limits of a function
- Determine whether a function is continuous