Precalculus Fundamentals provides a comprehensive review of essential precalculus topics, bridging the gap between algebra/geometry and calculus. Students will master trigonometric functions, solve equations, and analyze rational expressions, equipping them with the skills for success in advanced mathematics.

Trigonometric Functions and Ratios

• Review of basic trigonometric ratios (sine, cosine, tangent) using the SOH CAH TOA mnemonic — understanding opposite, adjacent, and hypotenuse relationships
• Extending to reciprocal trigonometric functions — cosecant, secant, and cotangent
• Solving for unknown side lengths and angles in right triangles using trigonometric ratios

Unit Circle Mastery

• Understanding the unit circle — relating angles in standard position to coordinates on the circle
• Determining trigonometric values (sine, cosine, tangent) for common angles — including radian measures
• Solving trigonometric equations using the unit circle

Solving Trigonometric Equations

• Isolating trigonometric functions in equations — algebraic manipulation to find solutions
• Finding all possible solutions — accounting for periodicity and multiple rotations around the unit circle
• Solving equations involving transformations of trigonometric functions — dealing with horizontal stretches/compressions

Law of Sines and Cosines

• Applying the Law of Sines — solving for unknown sides and angles in non-right triangles when given specific information (e.g., AAS, ASA, SSA)
• Using the Law of Cosines — solving for unknown sides and angles in non-right triangles given different information (e.g., SSS, SAS)
• Understanding the ambiguous case (SSA) — recognizing when multiple solutions are possible

Rational Expressions: Simplification and Operations

• Simplifying rational expressions — factoring and canceling common factors
• Performing arithmetic operations (addition, subtraction, multiplication, division) on rational expressions
• Identifying restrictions on variables — preventing division by zero

Asymptotes and Graphing

• Finding horizontal asymptotes — analyzing the degrees of the numerator and denominator
• Locating vertical asymptotes — determining values of x that make the denominator equal to zero
• Determining slant asymptotes — using long division to find the linear equation when the degree of the numerator is one greater than the denominator
• Graphing rational functions — combining information about asymptotes, intercepts, and key points to sketch the function

Applications

• Solving real-world problems using trigonometric functions and the laws of sines and cosines—navigation, surveying, and physics
• Modeling and analyzing situations using rational functions — optimizing processes and understanding relationships between variables