4.2 | Trigonometric Functions: The Unit Circle


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 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,

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

E 4.2 Exercises