Precalculus 3: Trigonometry

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.