Precalculus 3: Trigonometry

The Algebra 2 topic of trigonometry, preparing for studies of Calculus; contains a crash course in Euclidean geometry

4.96 (131 reviews)

Udemy

platform

English

language

Math

What you will learn

How to solve problems in trigonometry (illustrated with 215 solved problems), in both geometrical and functional contexts, and why these methods work.

You get a crash course in Euclidean geometry: angles, triangles, polygons, similar triangles (proportions), inscribed and circumscribed circles, bisectors, etc.

Number pi: its definition as the ratio of the perimeter to the diameter of a disk, relation to the area of a disk, some geometrical approximations.

The geometric definitions (by ratios in right triangles) of three trigonometric functions (sin, cos, tan) and their reciprocals (secant, cosecant, cotangent).

Exact values of trigonometric functions for angles of 15, 18, 30, 36, 45, 60, 72, 75, and 22.5 degrees: geometric derivations, and with help of formulas.

Solving triangles (finding side lengths and measures of all angles, knowing some of them), both right and oblique, with help of trigonometry.

Degree vs radian: how to use proportions for recalculating degrees to radians and back; reference angles.

The functional definition of sine, cosine and tangent, with help of unit circle and circular movement; properties of these functions.

The definition of trigonometric (circular) functions (sin, cos, tan) for any real number using the unit circle in the coordinate system.

Reference Angles Theorem with proof (by geometrical illustration) and applications; supplementary identities and the complementary angle properties.

Periodic functions. Sinusoids: period, amplitude, phase shift, vertical shift. Transformations of graphs of trigonometric functions.

Pythagorean theorem and Pythagorean triples. Law of Cosines, Law of Sines: formulation, proofs, and applications in problem solving.

Various trigonometric identities with proofs, geometrical illustrations, and applications for problem solving.

The Pythagorean Identities; Reciprocal Identities; Quotient Identities; Even/odd identities.

Sum and Difference Identities for sine and cosine with proofs, geometrical illustrations, and applications.

Sum To Product and Product To Sum Formulas for sine and cosine, with derivations and applications.

Double (Half) Angles Identities with geometrical illustrations, proofs, and applications in problem solving.

Inverse functions to sine, cosine and tangent, their definitions, properties and graphs.

Compositions of trigonometric functions with inverse trigonometric functions; identities involving inverse trigonometric functions.

Complex numbers and their trigonometric (polar) form; consequences of the Sum Identities for sin and cos for multiplication of complex numbers in polar form.

De Moivre's formula (positive natural powers of complex numbers) and its application to quick recreation of formulas for sine and cosine of multiples of angles.

Trigonometric equations: various types and corresponding methods for solutions; depicting the solution sets on the graphs and on the unit circle.

You get a sneak peek into trigonometry in a future Calculus class (how some trigonometric formulas are used there).

You get a plethora of geometric illustrations, supporting your intuition and understanding of trigonometry.

Why take this course?

Based on the outline you've provided, here's a summary of what you will learn in each section of the Precalculus course:

Section S10: Coordinate Geometry (Coordinates and Graphs)

Slopes of Straight Lines: Understanding how to find the slope of a line given two points, which is essential for equations of lines.
The Law of Cosines: A generalization of the Pythagorean Theorem for triangles that are not right-angled, allowing for the solution of any triangle side by side.
Area of a Triangle Using Sine: A formula for finding the area when one angle and two sides are known.
The Law of Sines: A relationship between the sides and angles in a triangle that allows for solving oblique triangles.
Heron's Formula: A way to calculate the area of a triangle when all three sides are known.
Vectors in the Plane (or 3-Space): An introduction to vectors, including dot products and cross products, which represent vectors in two or three dimensions.
Rotations and Their Matrices: Understanding rotations around origins or points in two or three dimensions through their mathematical representation using matrices.

Section S15: Trigonometry in Calculus

Limit of (sin x)/x as x approaches zero: This limit is central to the concept of continuity and the definition of the derivative.
Slope of a Straight Line: The slope's significance in representing rates of change, which is fundamental in differential calculus.
Differentiability of Sine and Cosine: Knowing which derivatives to use for trigonometric functions.
Derivatives of Trigonometric Functions: Finding the derivative of sine, cosine, tangent, arcsine, and arctangent and their applications in finding extremums and determining intervals of monotonicity.
Classes of Functions: Understanding C^0, C^1, C^2 functions, etc., with examples involving trigonometric functions.
Fourier Series and Spirographs: An introduction to Fourier series and spirographs as applications of trigonometry.
Euler's Formula: Exploring the connection between trigonometric functions, complex exponentials, and periodic phenomena.
Trigonometric Functions in Differential Equations: Solving differential equations using trigonometric functions.
Polar Coordinates: Understanding an alternative system of coordinates that is particularly useful in calculus for describing motion in a plane.
Cylindrical and Spherical Coordinates: Extending the concept of polar coordinates into three dimensions.
Parametric Curves: A way to describe curves using parametric equations involving trigonometric functions.
Power Reduction Formulas: Simplifying integrals with trigonometric substitutions.

Section S16: Problem Solving: Varia

This section includes a variety of problems across different difficulty levels for practice and problem-solving skills, without being tied to specific sections of the course.

Section S17: Extras

List of All Videos and Problems: A comprehensive guide to all the content covered in the course.
Future Courses: An overview of other courses offered and future course plans, including approximate release dates.

Remember to check with your professor which parts of the course are most relevant for your final exam, as the focus can vary based on your educational institution's curriculum requirements. The resource file "001 List_of_all_Videos_and_Problems_Precalculus_3.pdf" contains a detailed description of all the videos and problems solved throughout the course. This information is also covered in video 1 of the course.

Screenshots

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Our review

🌟 Overview of the Course 🌟

The course on Trigonometry has garnered an impressive global rating of 4.96, with all recent reviews being positive. It is widely regarded as the best undergraduate Math teacher available on Udemy, according to one reviewer. The course is commended for its comprehensive coverage of trigonometry, delivered in a manner that is both easy to follow and richly detailed.

Pros of the Course:

In-Depth Coverage: The course provides an in-depth exploration of trigonometry, ensuring that students gain a deep understanding of the topic.
Practical Examples: It includes plenty of worked examples that are thoroughly explained, helping students to apply what they learn in practical scenarios.
Step-by-Step Approach: The content is presented in a step-by-step approach with an organized structure, making it easier for learners to follow and retain information.
Educational Value: The course is suitable for both math and physics students, offering more information than traditional college trigonometry courses.
Quality of Instruction: The instructor's explanations are clear and detailed, catering to a wide range of learners, from beginners to those looking for a refresher course.
Ease of Understanding: Hania, the instructor, makes complex topics in trigonometry easy to understand with her exceptional teaching style.
Engagement and Pacing: The course is organized incrementally, allowing students to progress without feeling overwhelmed, and the instructor includes reminders and references to previous lessons or other courses for continuity.
Accessibility: Videos can be watched at accelerated speeds (1.5x or 2x) without loss of clarity in the voiceover.
Highly Recommended: The course is recommended by students who wish they had learned trigonometry with this level of instruction years ago.
Diverse Problem Sets: The course provides a range of problems from simple to very complex, which are solved in real time, offering valuable insights into problem-solving techniques.
Responsive Instructor: Questions are addressed in a clear and timely manner, with additional points and clarifications provided when necessary.

Cons of the Course:

Tediousness: Some students may find the course's level of detail and thorough explanation to be tedious initially, but it is definitely worth the investment in time and attention.

In summary, this Trigonometry course on Udemy is highly recommended for its exceptional teaching quality, comprehensive content, and the instructor's ability to simplify complex topics while maintaining a high level of detail. The positive feedback from students across various levels of expertise indicates that this course is an invaluable resource for anyone looking to understand trigonometry effectively.