AP Pre-calculus Unit 3 Overview

Key Announcements

• Last video of required course content for the AP pre-calculus exam.
• Special message at the end about future plans.
• Live reviews will be conducted before the AP exam; dates and times will be announced on Instagram.

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• Channel Members get special content and a badge in comments.
• Incentive: Videos for Unit 4 will be made if two new members join.

Unit 3 Content Overview

Periodic Functions

• Definition: A graph is periodic if it repeats a cycle over equal intervals.
• Key Examples: Sine and cosine graphs are classic examples of periodic functions.
• Period Calculation: The horizontal distance between two max or min points. For sine, it's typically 2π.
• Importance of Pattern: Even if the graph appears irregular, as long as the pattern repeats, it remains periodic.

Trigonometry Essentials

• Circle Basics: Starting angles 0°, 90°, 180°, 270°, and 360°.
• Radians Conversion: Entire circle equals 2π radians; 180° = π, 90° = π/2.
• Unit Circle: Radius is 1; sine is y-coordinate, cosine is x-coordinate.
• Trigonometric Ratios:
  • Sine (sin): Vertical displacement (y/r).
  • Cosine (cos): Horizontal displacement (x/r).
  • Tangent (tan): Slope (rise over run or y/x).

Special Angles & Triangles

• Quadrantal Angles: Angles that are multiples of 90°.
• Finding Trig Values for Special Angles:
  • Use known triangles (30-60-90, 45-45-90) for calculations.
• Inverse Trigonometric Functions:
  • Certain inverse functions only exist in specific quadrants.
  • Inverse sine functions exist in quadrants 1 and 4 only.

Graphs of Sine, Cosine, and Tangent

• Sine Graph: Period of 2π, domain of all real numbers, range from -1 to 1.
• Cosine Graph: Similar to sine, can be viewed as a horizontally shifted sine graph.
• Tangent Graph: Unique due to vertical asymptotes, period of π.

Sinusoidal Functions

• Components:
  • Amplitude (A)
  • Period (determined by B, B = 2π/period)
  • Phase Shift (C)
  • Vertical Shift (D)
• Key Graph Features:
  • Middle line halfway between min and max.
  • Amplitude is vertical distance from midline to max.

Important Trig Identities

• Reciprocal Identities: Introduction of cosecant, secant, and cotangent.
• Pythagorean Identities: Used to simplify expressions.
• Sum and Difference Identities: Important for simplifying angle expressions.
• Double-angle Identities: Less critical but useful.

Polar Coordinates and Functions

• Polar Graphs: Use r, θ instead of x, y.
• Converting between Systems: Formulas provided for conversion.
• Graph Types: Circles, cardioids, limacon, roses.

Final Thoughts

• This video concludes the required content for the AP exam.
• Encouragement to follow on social media for updates and additional content.

Remember: Understanding periodic functions, trigonometry essentials, and graph characteristics are crucial for success in pre-calculus and the AP exam. The use of a calculator in radians mode is essential for accuracy in calculations.