# Section 1.2 The Trig Functions on the Unit Circle

Section Objectives

1. Define and evaluate the trigonometric functions on the unit circle.
2. Use periodicity to evaluate the trigonometric functions.

### Sine and Cosine

The unit circle is described by the equation $x^2+y^2=1$. Any point $(x,y)$ on the unit circle satisfies this equation.

Let $t$ be any real number. Imagine starting at the point $(1,0)$ on the unit circle and traveling along the circle through an arc length of $t$ units (counterclockwise if $t$ is positive, clockwise if $t$ is negative). Notice that $t$ is the radian measure of the angle through which you've turned!

Now let $(x,y)$ be the point at which you arrived. The sine and cosine of the real number $t$ are defined as follows:

$\sin t = y$

$\cos t = x$

Not to get ahead of ourselves, but notice that it must be true that $(\cos t)^2 + (\sin t)^2 = 1$. Do you see why?

The other trigonometric functions are the tangent, secant, cosecant, and cotangent:

$\displaystyle \tan t = \frac{y}{x} = \frac{\sin t}{\cos t}, \quad x \ne 0$

$\displaystyle \sec t = \frac{1}{x} = \frac{1}{\cos t}, \quad x \ne 0$

$\displaystyle \csc t = \frac{1}{y} = \frac{1}{\sin t}, \quad y \ne 0$

$\displaystyle \cot t = \frac{x}{y} = \frac{1}{\tan t} = \frac{\cos t}{\sin t}, \quad y \ne 0$

The exact values of the trigonometric functions at certain special angles should be memorized! For now, we can read them off our unit circles.

### Domains of the Sine and Cosine Functions

Notice that the sine and cosine functions are defined for all real numbers---there was no restriction on the real number $t$.

On the other hand, the outputs of the sine and cosine functions are the coordinates of points on the unit circle. Therefore,

$-1 \le \cos t \le 1$

$-1 \le \sin t \le 1$

for all real numbers $t$.

The sine and cosine functions take, as inputs, any real numbers, and they return, as outputs, real numbers between -1 and 1.

### Periods of the Sine and Cosine Functions

Let's go back to the unit circle definitions of the sine and cosine. Imagine that the number $t$ is so large ($\gt 2\pi$) that the entire unit circle is traced more than one time. The $(x,y)$ points on the unit circle will be retraced, and the values of the sine and cosine function will be repeated. In fact, the values will be repeated every $2\pi$ units.

The sine and cosines functions are periodic with period $2\pi$.

$\sin(t+2\pi n) = \sin t$

$\cos(t+2\pi n) = \cos t$

for any integer $n$.

The periodicity of the sine and cosine carries over to the other trig functions, but we'll come back to that later.

### Even/Odd

In addition to periodicity, the symmetry of the unit circle, invokes a kind of symmetry in the trig functions.

The cosine and the secant are even functions:

$\cos(-t) = \cos t \qquad \mbox{and} \qquad \sec(-t) = \sec t$

The sine, cosecant, tangent, and cotangent are odd functions:

$\sin(-t) = -\sin t \qquad \mbox{and} \qquad \csc(-t)=-\csc t$

$\tan(-t) = -\tan t \qquad \mbox{and} \qquad \cot(-t)=-\cot t$