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This lesson has four objectives:
1. Rationale for radian measure
2. Definition of radian measure
3. Finding arc length and radius from the definition of radian measure
4. Converting between degrees and radians.

Solutions to Worksheet 1

This lesson has three objectives:
1. Angles in Standard Position
2. Definition of Sine and Cosine in the Coordinate Plane
3. The Unit Circle

Lesson 2 - Worksheet Review

This lesson has two objectives.
1. Deriving the Pythagorean Identity.
2. Using the Pythagorean Identity to find sine or cosine values.

Lesson 3 - Worksheet Review

This lesson has two objectives:
1. Establishing the even/odd and co-function identities.
2. Using the Pythagorean Identity to prove other trigonometric identities and to simplify trigonometric expressions.

Lesson 4 - Worksheet Review

This lesson has four objectives:
1. Applying the idea of a function to trigonometric functions
2. Revisiting the unit circle to sketch the parent sine and cosine curves.
3. Establishing features of trigonometric curves such as period and amplitude.
4. Applying transformations towards sketching graphs of sine and cosine curves.

Lesson 5 - Worksheet Review

This lesson has three objectives:
1. Defining the tangent and cotangent values in the coordinate plane.
2. Finding cotangent and tangent values.
3. Sketching tangent and cotangent curves.

Lesson 6 - Worksheet Review

This lesson has four objectives:
1. Definition of secant and cosecant functions.
2. Finding secant and cosecant values.
3. Sketching graphs of secant and cosecant functions.
4. Proving identities and simplifying expressions.

Lesson 7 - Worksheet Review

This lesson establishes and uses the relationship between the angle of inclination of a line and the slope of that line through the tangent function.

Lesson 8 - Worksheet Review

In this lesson, we are going to derive and use the addition formulas for sine and cosine.

Lesson 9 - Worksheet Review

In this lesson, we derive the half-angle and double angle formulas and use them to evaluate trigonometric values, simplify trigonometric expressions and prove trigonometric identities.

Lesson 10 - Worksheet Review

In this lesson, we find trigonometric values in a right triangle, and use given trigonometric ratios to solve a right triangle. We also look at a real-life application regarding angles of elevation and depression.

Lesson 11 - Worksheet Review

In this lesson, we derive and use the area formula with the sine ratio. We also look at Heron's formula for the area of a triangle.

Lesson 12 - Worksheet Review

In this lesson, we derive and use the law of cosines and the law of sines. We also cover the ambiguous case of the law of sines.

Lesson 13 - Worksheet Review

In this last lesson for the trigonometry unit, we will look at applications of trigonometry in real life. We will solve examples regarding the length of a diagonal in a parallelogram, angles of elevation and depression, distance and bearing.
Please put your trigonometry knowledge to the test by completing the unit test in the Resources section.

Lesson 14 - Worksheet Review

This test is timed at 70 minutes. Solutions are uploaded under Resources.

In this first lesson in our second unit, we look at polar form. First, we plot points in polar form and secondly, we look at graphs of equations in polar form. Would you like to see what a cardioid looks like? We get to graph one in this lesson.

Lesson 15 - Worksheet Review

In this lesson, we have two objectives.
Our first objective is to convert between polar and rectangular coordinates. Our second objective is to convert polar equations into Cartesian equations.

Lesson 16 - Worksheet Review

This lesson introduces imaginary and complex numbers. We work on identifying the real and imaginary parts of a complex number and the conjugate of a complex number. We also plot complex numbers in the complex plane and finish the lesson with an example where we calculate the square roots of a complex number.

Lesson 17 - Worksheet Review

In this lesson, we add, subtract, multiply and divide complex numbers. We also look at powers of i and solve an equation with complex coefficients.

Lesson 18 - Worksheet Review

In this lesson, we are going to accomplish four objectives. First, we will convert complex numbers into polar form. Then, we will multiply and divide complex numbers in polar form. Next, we will use de Moivre's theorem to find powers and roots of complex numbers.

Lesson 19 - Worksheet Review

In this lesson, we cover the basics of vectors in Cartesian and polar coordinates. We discuss operations with vectors, unit vectors, conversion between Cartesian and polar coordinates, scalar product and angles between vectors.

Lesson 20 - Worksheet Review

This test is timed at 40 minutes. Solutions are uploaded under Resources.

In this lesson, we are going to use the process of mathematical induction to prove statements regarding sum formulas, divisibility and inequality.

Lesson 21 - Worksheet Review

In this lesson, we study the fundamental theorem of algebra with multiplicity of roots and complex conjugate roots.

Lesson 22 - Worksheet Review

In this lesson, we look at operations on matrices and certain features of matrices. We cover matrix addition and subtraction, multiplication by a scalar and matrix multiplication. We also look at identity matrices, the dimensions of a matrix, square matrices and the transpose of a matrix.

Lesson 23 - Worksheet Review

In this lesson, we go over row echelon form and reduced row echelon form. We identify whether a given matrix is in row echelon form or reduced row echelon form. We also intuitively convert matrices into these two forms.

Worksheet Review

In this lesson, we learn to write the coefficient matrix and the augmented matrix for a system of equations. Then, we work with the augmented matrix and convert it to row echelon and reduced row echelon forms.

Lesson 25 - Worksheet Review

In this lesson, we solve systems of linear equations using Gaussian Elimination and Gauss-Jordan Elimination.

Lesson 26 - Worksheet Review

In this lesson, we revisit matrix addition and multiplication with a particular emphasis on vector addition and scalar product of vectors. We also compute the additive and multiplicative inverses of matrices.

Lesson 27 - Worksheet Review

This test is timed at 50 minutes. Solutions are uploaded under Resources.

In this lesson, we solve trigonometric solutions for roots in a certain interval.

Lesson 28 - Worksheet Review

The difference quotient is an important concept in the definition of the derivative. In this lesson, we define the difference quotient and we work on finding the difference quotient for a number of functions.

Lesson 29 - Worksheet Review

In this last lesson for our course, we start a discussion on limits. We look at limits of functions graphically; we specify what it means for the limit to be defined. We also look at limits around vertical asymptotes and consider the indeterminate form, 0/0.

Lesson 30 - Worksheet Review

**Course Description**

This course is a gateway to Calculus and it provides you with all the knowledge and skills you need in Calculus.

**The course includes:**

The course comprises 5 units and 30 lessons

Over 13 hours of video-lessons!

Each lesson includes a video lecture, a worksheet, a worksheet review video and an online quiz.

30 video lectures

30 worksheets

30 worksheet review videos with every question solved step-by step by the instructor

30 online quizzes

3 unit tests (on paper)

1 final exam (on paper)

an exclusive Getting Ready for Calculus unit where we introduce the difference quotient and limits!

**COURSE GOALS**

Upon course completion, you will learn how to:

- work with trigonometric functions
- sketch graphs of trigonometric functions
- solve trigonometric equations
- simplify trigonometric expressions and prove trigonometric identities
- use laws and formulas from trigonometry in solving triangles
- apply trigonometric to real life situations
- work with Cartesian and polar coordinates
- work with complex numbers
- use de Moivre's theorem to find powers and roots of complex numbers
- work with vectors in Cartesian and polar coordinates
- prove mathematical statements using mathematical induction
- apply the Fundamental Theorem of Algebra
- perform operations on matrices
- solve a system of linear equations using Gaussian elimination and Gauss-Jordan elimination
- find the difference quotient of a function
- calculate some elementary limits and read limits from graphs

**TARGET AUDIENCE **

This video-course is primarily intended for 12th grade students who would like to solidify their mathematical skills and knowledge before they take Calculus.

**COURSE REQUIREMENTS**

Students taking this course will need to have completed Algebra I, Math for 11th grade or an equivalent Math course prior to enrolling in this course.

UNIT 1 TRIGONOMETRY

Angles Measured in Radians or Degrees

Introducing Sine and Cosine

The Pythagorean Identity

Proving Other Trigonometric Identities

Graphing Trigonometric Functions

Tangent and Cotangent Functions

Secant and Cosecant Functions,

Tangent and Slope

Addition Formulas for Sine and Cosine

Half-Angle and Double Angle Formulas

Right Triangle Trigonometry

The Area of a Triangle

The Law of Sines and the Law of Cosines

Applications of Trigonometry

UNIT 2 COMPLEX NUMBERS AND POLAR FORM

Polar Form

Converting Points

Complex Numbers

The Arithmetic of Complex Numbers

De Moivre’s Theorem

Vectors in Cartesian and Polar Coordinates

UNIT 3 PROOF BY MATHEMATICAL INDUCTION AND THE FUNDAMENTAL THEOREM OF ALGEBRA

Proofs by Mathematical Induction

The Fundamental Theorem of Algebra

UNIT 4 LINEAR ALGEBRA

Introduction to Linear Algebra and Matrices

Row Echelon Form and Reduced Row Echelon Form

Coefficient Matrices

Gaussian Elimination and Gauss-Jordan Elimination

Matrix Addition and Matrix Multiplication

UNIT 5 GETTING READY FOR CALCULUS

Trigonometric Equations

The Difference Quotient

Introduction to Limits

- Teacher: Ebru
- Areas of expertise: Mathematics
- Education: Oberlin College, BA in Mathematics and Anthropology 1993 Texas A&M University MSc in Mathematics 2011
- Interests: Epistemology, Acting
- Skills: Learning Languages
- Associations: National Council of Teachers of Mathematics (NCTM)
- Issues I care about: Sustainability, Social Responsibility, Giving Back to the Community

I am happy to start working with Lernsys students as I reach out to you all beyond the physical boundaries of the classroom.

Lesson 1 online quiz answers

Lesson 2 online quiz answers

Lesson 3 online quiz answers

Lesson 4 online quiz answers

Lesson 5 quiz answers

Lesson 6 quiz answers

Lesson 7 quiz answers

Lesson 8 quiz answers

Lesson 9 quiz answers

Lesson 10 online quiz answers

Lesson 11 online quiz answers

Lesson 12 online quiz answers

Lesson 13 online quiz answers

Lesson 14 online quiz answers

Unit Test on Trigonometry Solutions

Lesson 15 online quiz answers

Lesson 16 online quiz answers

Lesson 17 online quiz answers

Lesson 18 online quiz answers

Lesson 19 online quiz answers

Lesson 20 online quiz answers

Unit Test on Complex Numbers and Polar Form Solutions

Lesson 21 online quiz answers

Lesson 22 online quiz answers

Lesson 23 online quiz answers

Lesson 24 online quiz answers

Lesson 25 online quiz answers

Lesson 26 online quiz answers

Lesson 27 online quiz answers

Unit Test on Linear Algebra Solutions

Lesson 28 online quiz answers

Lesson 29 online quiz answers

Lesson 30 online quiz answers

Final Exam Solutions