# Precalculus

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Teacher: Sarah
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#### 2 Unit Circle 13:38

The Unit Circle is a concept that is used eminsely in trigonometry and into calculus. This video coveres the basics of the Unit Circle (degrees, radians, and coordinates) such that students understand where each of the values come from.

#### 3 Angle Measures in Radians and Degrees 04:52

Converting between radian and degree measurements.

#### 5 Sine and Cosine Ratios 18:46

Finding sine and cosine ratios in right triangles and using the unit circle.

#### 7 Inverse Sine and Cosine 09:16

Using the restricted parameters of the inverse functions to simplify/evaluate expressions.

#### 9 Graphing Trig Functions: Sine and Cosine 23:37

Use the information learned in the previous lesson and in the unit circle lesson to graph the sine and cosine functions.

#### 11 Modeling with Trigonometric Functions 10:02

Model different situations using trigonometric functions.

#### 13 Tangent and Cotangent Ratios 15:01

Finding tangent and cotangent ratios using the unit circle, how they are each related to sine and cosine, and finding their ratios given other trig ratios.

#### 15 Secant and Cosecant Ratios 16:44

Finding their relationships with sine and cosine, finding the ratios on the unit circle, calculating their ratios given other trig ratios.

#### 17 Tangent and Slope 05:29

Finding the slope of an angle give side lengths, and find the angle of a line given the slope.

Unit Circle Test

#### 20 Pythagorean Identity 22:53

We will discuss what the pythagorean theorem that relates sine to cosine is and how the other pythaogrean identities are related to it. Then, we will find trig ratios using the identities.

#### 22 Addition/Subtraction Formulas for Sine and Cosine 12:23

Finding exact values of trig functions for angles not on the unit circle. Simplifying expressions into one trig function.

#### 24 Half/Double Angle Formulas 17:02

Use the half and double angle formulas to find exact values for trig functions.

#### 26 Solving Trig Equations 16:07

Solving trig equations.

#### 28 TEST Analytical Trigonometry 00:00

Analytical Trig Test

#### 29 Right Triangle Trigonometry 13:00

Identify the trig ratios of a right triangle, solve for unknown side lengths, find unknown angle measures, and solve a right triangle.

#### 31 Law of Sines 23:24

Using the Law of Sines with non-right triangles to find side lengths and angle measures.

#### 33 Law of Cosines 10:13

Using the law of cosines with non-right triangles to find side lengths and angle measures.

#### 35 Area of a Triangle 05:47

Finding the area of a triangle if given two sides and the included angle or if given all three sides of the triangle.

#### 37 Applying Trigonometry 09:35

Applying trigonometry to real world situations.

#### 39 TEST Triangular Trigonometry 00:00

Triangular Trigonometry Test

#### 40 Polar Coordinates 14:07

Graphing polar coordinates, identifying polar coordinates with more than one way, and calculating the distance between polar coordinates.

#### 42 Converting Points 10:02

Convert between polar to rectangular and rectangular to polar coordinates.

#### 44 Basics of Vectors in Cartesian Coordinates 18:07

Define vectors between two points, find the magnitude of a vectors, find the directional angle of a vector, and find a unit vector

#### 46 Operations with Vectors in Cartesian Coordinates 11:04

Using addition, subtraction, scalar multiplication, and the dot product with vectors. Find the angle between two vectors.

#### 48 Vectors in Polar Coordinates 16:31

Find components of vectors given its magnitude and angle and add two vectors given their magnitude and angles.

#### 50 Basics of Complex Numbers 11:59

Simplifying square roots of negative numbers, simplify expression with i, finding complex conjugates, and graphing complex numbers

#### 52 More on Complex Numbers 11:00

Finding the magnitude of a complex number, finding the angle of a complex number, and writing a complex number in trigonometric form

#### 54 De'Moivre's Theorem 10:40

Multiply and divide complex numbers, find powers of complex number, and find roots of complex numbers

#### 56 TEST Polar Coordinates, Vectors, and Complex Numbers 00:00

Polar Coordinates, Vectors, and Complex Numbers Test

#### 57 Proofs by Mathematical Induction 12:57

Proving statements using mathematical induction.

#### 59 Fundamental Theorem of Algebra 16:20

The Fundamental Theorem of Algebra

#### 61 Intro to Linear Algebra and Matrices 04:52

Identify the order (dimensions of a matrix and identify the entries of a matrix.

#### 63 Row Operations with Matrices 06:51

Manipulate matrices using row operations

#### 65 Row Echelon and Reduced Row Echelon Form 19:48

Use row operations in order to put matrices into row echelon and reduced row echelon forms.

#### 67 Coefficient Matrices 07:59

Representing a system of equations with matrices

#### 69 Gaussian and Guass-Jordan Elimination 11:26

Using the Guassian and Guass-Jordan elimination methods in order to solve a system of equations.

#### 73 Test Proofs, Fund Theorem, Linear Algebra 00:00

Proofs, Fund Theorem, Linear Algebra Test.

Course Description:

This Precalculus video course puts together the many concepts taught in Algebra 1, Geometry, and Algebra 2 in order to prepare students for Calculus and other subsequent math courses. In general, this course covers unit circle trigonometry, analytical trigonometry, right triangle trigonometry, the polar coordinate system, vectors, complex numbers, proofs by mathematical induction, the fundamental theorem of algebra, and an introduction to linear algebra. In each of the lessons, students are exposed to the basics skills of the concepts and challenged to use and improve their critical thinking skills.

The course includes:

12+ hours of video

5 units

34 instructional videos

34 blank pdf notes for students to use and fill in while watching the instructional videos

34 practice worksheets for students to practice and master the concepts being taught

34 answers key and videos (I explain step by step how to arrive at the correct answer)

350+ practice problems and step by step resolutions

5 tests (1 at the end of each unit)

5 test answer keys (one for each unit test)

Course Goals: Upon completion of this course, students will be well prepared for Calculus and have an introduction to a Linear Algebra course. Students will have a solid foundation on trigonometry, vectors, and matrices. Students will not only be able to use these skills to solve problems, but they will also be able to apply them to different situations.

Target Audience: This course is for students who have mastered the concepts taught in a traditional Algebra 2 course and are planning to take Calculus, or for those currently enrolled in a Precalculus class but need additional practice with the concepts being taught.

Technology: Although the course is taught such that a calculator is not necessary, a student may be more comfortable using one. If a student chooses to use a calculator make sure it has trigonometric functionality (a basic scientific calculator would not suffice).

Course Concepts:

Unit 1 – Unit Circle Trigonometry

The Unit Circle

Angle Measures in both radians and degrees

Sine and Cosine Ratios

Inverse Sine and Cosine

Graphing Sine and Cosine

Modeling real world situations with trigonometric functions

Tangent and Cotangent Ratios

Secant and Cosecant Ratios

Tangent and Slope

Unit 2 – Analytical Trigonometry

Pythagorean Identities

Half and Double Angle Formulas

Solving trig equations

Unit 3 – Triangular Trigonometry

Right Triangular Trig

Law of Sines

Law of Cosines

Area of a Triangle

Applying trig to real world situations

Unit 4 – Polar Coordinates, Vectors, and Complex Numbers

Polar Coordinates

Converting between polar and cartesian coordinates

Basics of vectors in cartesian coordiantes

Operations with vectors in cartesian coordiantes

Vectors in polar coordinates

Basics of Complex Numbers

More on Complex Numbers

De'Moivre's Theorem

Unit 5 – Proofs by Mathematical Induction, The Fundamental Theorem of Algebra, and Intro to Linear Algebra

Proofs by Mathematical Induction

Fundamental Theorem of Algebra

Intro to Linear Algebra and Matrices

Row Operations with Matrices

Row Echelon and Reduced Row Echelon Form

Coefficient Matrices

Guassian and Guass-Jordan Elimination

Basic Operations on Matrices

• Teacher: Sarah
• Areas of expertise: Secondary Mathematics
• Education: M.A. in Mathematics @ The University of Texas at Austin B.S. in Mathematics @ The University of Texas at Austin
• Interests: Outdoors, Sports, Arts & Crafts, Cooking, Mathematics
• Skills:
• Associations: National Council of Teachers of Mathematics

#### Unit Circle and Table of Values

This document includes and completed Unit Circle (with degrees, radians, and coordinates). Also, there is a tabular version that includes all six trig functions.

#### Formula Sheet

This document contains many formulas used with trigonometric functions. It will be referenced many times throughout the course.