Pre-Calculus Syllabus

Pre-Calculus is an engaging, rigorous course for students who have completed Algebra B (or Algebra 2) and Geometry, or an Integrated Math program covering both. Aligned with the Common Core Standards for Pre-Calculus, this course equips students with essential skills in trigonometric functions, polar and exponential coordinate systems, and advanced operations with vectors and matrices up to 3x3.

Students will dive into complex problem-solving exercises, logic puzzles, and competition-style challenges that stimulate critical thinking and prepare them for calculus. Ideal for students comfortable manipulating polynomials and complex numbers, solving for variables, and analyzing geometric shapes on the Cartesian plane, Pre-Calculus offers a solid foundation in higher-level math through an interactive and inspiring learning experience.
‍
Required Textbook: Art of Problem Solving: Pre-Calculus

Required for: Calculus

Concepts, skills, and learning tools students see in this course include, but are not limited to:

• Trigonometric Functions and Identities: Master trigonometric functions and apply key identities.
• Laws of Sines and Cosines: Solve complex problems using the Law of Sines and Law of Cosines.
• Coordinate Systems: Convert between parametric, rectangular, and polar coordinates.
• Vectors and Matrices: Work with 2D and 3D vectors and matrices, including complex numbers in trigonometric form.
• Problem-solving skills, logic puzzles, algebraic and geometric thinking, competition-style problems

Upon completing the course, students will be able to:

• Complex Numbers and Trigonometry: Perform operations with complex numbers, represent them on the complex plane, relate them to trigonometry, and solve complex functions and equations.
• Trigonometric Functions and Identities: Graph and solve trigonometric functions, solve triangles, recreate the Unit Circle, and simplify expressions using trigonometric identities.
• Vectors, Matrices, and Coordinate Systems: Convert between parametric, rectangular, and polar coordinates, operate on vectors and matrices up to 3x3, model shapes, and understand the determinant and inverse of square matrices.

Students registering for this course should be comfortable with the following Math:‍

• Geometry and Measurement: Understand triangle and circle properties, apply the Pythagorean Theorem, and calculate volumes of 3D shapes with straight edges.
• Algebra and Equations: Solve equations and systems with up to 3 unknowns, work with logarithms, exponents, and polynomials (factoring, roots, and graph behavior).
• Graphing and Complex Numbers: Graph functions on the Cartesian Plane and perform arithmetic and simplification with complex numbers.
• Completion of Algebra B (or Algebra 2) and Geometry or equivalent.

• Functions and Trigonometry (8 weeks): Develop a strong foundation in functions, graphing, composition, inverse functions, and trigonometric functions. Explore the Unit Circle, radians, and transformations of trigonometric graphs.
• Advanced Trigonometric Identities and Applications (7 weeks): Work with trigonometric identities, angle sums, double/half angles, and apply trigonometric laws to solve right and non-right triangles.
• Coordinate Systems and Complex Numbers (8 weeks): Learn parameterization, polar coordinates, and complex numbers, including operations, graphing, and roots of unity in both polar and exponential forms.
• Vectors and Matrices (8 weeks): Study 2D and 3D vectors, dot products, cross products, and matrix operations, including transformations, determinants, and inverses, with applications in geometry.
• 3D Geometry and Analytic Applications (7 weeks): Analyze lines, planes, and vector geometry in three dimensions, apply geometric determinants, and solve complex problems using vector and matrix techniques.