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,

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