# 10: Foundations of Trigonometry

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• 10.1: Angles and their Measure
This section begins our study of Trigonometry and to get started, we recall some basic definitions from Geometry. A ray is usually described as a half-line' and can be thought of as a line segment in which one of the two endpoints is pushed off infinitely distant from the other, as pictured below. The point from which the ray originates is called the initial point of the ray.
• 10.2: The Unit Circle - Cosine and Sine
One of the goals of this section is describe the position of such an object. To that end, consider an angle θ in standard position and let P denote the point where the terminal side of θ intersects the Unit Circle. By associating the point P with the angle θ , we are assigning a position on the Unit Circle to the angle θ .
• 10.3: The Six Circular Functions and Fundamental Identities
We previously defined cos(θ) and sin(θ) for angles θ using the coordinate values of points on the Unit Circle. As such, these functions earn the moniker circular functions (we will start using the phrase trigonometric function' interchangeably with the term circular function'). It turns out that cosine and sine are just two of the six commonly used circular functions which we define in this Module.
• 10.4: Trigonometric Identities
Our first set of identities is the Even / Odd' identities.The properties of the circular functions when thought of as functions of angles in radian measure hold equally well if we view these functions as functions of real numbers. Not surprisingly, the Even / Odd properties of the circular functions are so named because they identify cosine and secant as even functions, while the remaining four circular functions are odd.
• 10.5: Graphs of the Trigonometric Functions
In this section, we return to our discussion of the circular (trigonometric) functions as functions of real numbers.. As usual, we begin our study with the functions f(t)=cos(t) and g(t)=sin(t).
• 10.6: The Inverse Trigonometric Functions
In this section we concern ourselves with finding inverses of the (circular) trigonometric functions. Our immediate problem is that, owing to their periodic nature, none of the six circular functions is one-to-one. To remedy this, we restrict the domains of the circular functions in the same way we restricted the domain of the quadratic function previously.
• 10.7: Trigonometric Equations and Inequalities
In Sections 10.2, 10.3 and most recently 10.6, we solved some basic equations involving the trigonometric functions. Below we summarize the techniques we’ve employed thus far. Note that we use the neutral letter ‘ 𝑢 u ’ as the argument1 of each circular function for generality.

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