Chapter 5: Trigonometry

The orientation of the rotating line segment determines which of the four quadrants contains the angle, and this quadrant placement governs the sign of each trigonometric ratio according to a predictable pattern that students can remember through the ASTC mnemonic. The chapter establishes exact trigonometric values for commonly encountered angles such as thirty, forty-five, and sixty degrees, derived from properties of special triangles, providing reference points for calculations without calculator dependence. Graphical representations of sine, cosine, and tangent functions reveal their periodic nature, with sine and cosine completing a full cycle every three hundred sixty degrees while maintaining a maximum displacement of one unit from the horizontal axis, whereas tangent repeats its pattern every one hundred eighty degrees and exhibits undefined vertical asymptotes at specific points. The chapter then examines how trigonometric graphs transform under various operations, including vertical and horizontal stretches that alter amplitude and period respectively, along with translations that shift the entire curve without changing its fundamental shape. To address the many-to-one characteristic of trigonometric functions that prevents direct inversion, the chapter explains how restricting domains to specific ranges enables the definition of inverse trigonometric functions with uniquely determined principal values. Finally, the chapter covers essential trigonometric identities that express relationships between different functions and provides systematic approaches to solving trigonometric equations by combining calculator results with an understanding of function symmetry and periodicity to identify all solutions within given intervals, often requiring algebraic manipulation and substitution techniques to reduce complex equations to solvable forms.