- 4.1 | Radian and Degree Measure
- 4.3 | Right Triangle Trigonometry
- 4.4 | Trigonometric Functions of Any Angle
- 4.5 | Graphs of Sine and Cosine Functions
- 4.6 | Graphs of Other Trigonometric Functions
- 4.7 | Inverse Trigonometric Functions
- 4.8 | Applications and Models

**The Six Basic Trigonometric Functions:**If \((x, y)\) is a point of the unit circle, and if the ray from the origin \((0, 0)\) to \((x, y)\) makes an angle \(t\) from the positive x-axis, then we define

**sine**and

**cosine**by

- \(\sin t = y\)
- \(\cos t = x\),

That is, sine of \(t\) is the \(y\)-coordinate and cosine of \(t\) is the \(x\)-coordinate. In applications, it is often convienent to consider ratios and reciprocals of these two functions and so we define **tangent**, **cosecant**, **secant**, and **cotangent** in exactly this way:

- \(\tan t = \frac{\sin t}{\cos t} = \frac{y}{x}\)
- \(\csc t = \frac{1}{\sin t} = \frac{1}{y}\)
- \(\sec t = \frac{1}{\cos t} = \frac{1}{x}\)
- \(\cot t = \frac{\cos t}{\sin t} = \frac{x}{y}\)

where these definitions hold provided the right-hand sides are defined.

When exponentiating the six basic trigonometric functions it is convenient to define, for \(n \not = -1\):

- \(\displaystyle \sin^n \theta = (\sin \theta)^n \)
- \(\displaystyle \csc^n \theta = (\csc \theta)^n \)
- \(\displaystyle \cos^n \theta = (\cos \theta)^n \)
- \(\displaystyle \sec^n \theta = (\sec \theta)^n \)
- \(\displaystyle \tan^n \theta = (\tan \theta)^n \)
- \(\displaystyle \cot^n \theta = (\cot \theta)^n \)

For example, \(\sin^2 \theta = (\sin \theta)^2 = \sin \theta \cdot \sin \theta \). However, \(\sin^{-1} \theta \not = \frac{1}{\sin \theta}\). The notation \(\sin^{-1} \theta\) is used to denoted the arcsine function.

An ordered pair \((x,y)\) on the unit circle satisfies the equation

$$

x^2 + y^2 = 1.

$$

But \(x = \cos \theta\) and \(y = \sin \theta\), and so

**Pythagorean Identity:**

$$

\cos^2 \theta + \sin^2 \theta = 1

$$

A function \(f\) is

$$

f(t+c) = f(t)

$$

for all \(t\) in the domain of \(f\). The smallest number \(c\) for which \(f\) is periodic is called the

**periodic**if there exists a positive real number \(c\) such that$$

f(t+c) = f(t)

$$

for all \(t\) in the domain of \(f\). The smallest number \(c\) for which \(f\) is periodic is called the

**period**of \(f\).A function \(f\) is

**even**provided that \(f(-t) = f(t)\) and**odd**provided \(f(-t) = -f(t)\)Cosine and secant are even while sine, tangent cosecant, and cotangent are odd. That is,

- \(\cos(-t) = \cos t\)
- \(\sec(-t) = \sec t\)
- \(\sin(-t) = – \sin t\)
- \(\tan(-t) = – \tan t\)
- \(\csc(-t) = – \csc t\)
- \(\cot(-t) = – \cot t\)