Trigonometric functions are among the most fundamental tools in mathematics, science, and engineering. Originally developed to describe the relationships between angles and side lengths in triangles, they have grown into a far richer subject: a language for expressing any quantity that repeats in a regular cycle, from sound waves and light to planetary orbits and electrical currents. In calculus, trigonometric functions appear constantly, as integrands, as limits, and as solutions to differential equations, making a firm grasp of their definitions and properties essential before proceeding further.
This chapter builds the foundations of trigonometry systematically, beginning with angle measure and working up through identities, graphs, and inverse functions. We start by introducing the radian, the natural unit of angle measure that simplifies every formula in calculus. From there we define the six trigonometric functions first via right triangles and then more generally using the unit circle, which allows them to accept any real number as input. We then develop the major algebraic identities that relate these functions to one another, explore their periodic graphs, establish two geometric inequalities that underpin key limits in calculus, and finally define the inverse trigonometric functions needed to solve equations and evaluate integrals. The chapter covers the following sections:
- Angles: radian and degree measure, arc length, area of a circular sector, directed angles, and standard position
- Basic Trigonometric Functions: right-triangle and unit-circle definitions, values at special angles, the ASTC sign rule, and coterminal angles
- Trigonometric Identities: Pythagorean, even-odd, addition and subtraction, double-angle, half-angle, product-to-sum formulas, and the laws of sines and cosines
- Periodicity and Graphs of Trigonometric Functions: periods of sine, cosine, tangent, and cotangent; their graphs; and the effect of amplitude and phase transformations
- Two Important Inequalities: the geometric proof that $|\sin\theta| \leq |\theta|$ and $|1 - \cos\theta| \leq |\theta|$, with applications to fundamental limits in calculus
- Inverse Trigonometric Functions: restricted domains, arcsine, arccosine, arctangent, inverses of the secondary functions, and worked examples involving cancellation equations
