The Unit Circle and the Functions of Trigonometry

Chapter 9.2: The Unit Circle and the Functions of Trigonometry

9.2 The Unit Circle and Its Functions

The unit circle is a fundamental concept in trigonometry, defined as the circle with center at the origin (0,0) and radius one unit. The trigonometric functions, when extended to all real numbers, are called circular functions. To define these functions for any real number s, start at the point (1,0) and measure an arc of length s counterclockwise if s > 0 and clockwise if s < 0.

Angles and Arcs: Basic Terminology

On the unit circle, the coordinates of a point corresponding to an arc length s are given by (cos s, sin s). The angle θ in radians is equal to the arc length s for a unit circle.

Circular (Trigonometric) Functions

If (x, y) is the point on the unit circle that corresponds to the real number s, then:

• sin s = y

• cos s = x

• tan s = y / x (undefined when x = 0)

• csc s = 1 / y (undefined when y = 0)

• sec s = 1 / x (undefined when x = 0)

• cot s = x / y (undefined when y = 0)

Relationship Between Sine and Cosine

For any real number s, the coordinates (x, y) = (cos s, sin s) satisfy the equation of the unit circle:

$x^2 + y^2 = 1$

Therefore,

$\cos^2 s + \sin^2 s = 1$

This is known as the Pythagorean Identity.

Domains of the Circular Functions

The domains of the six circular functions are as follows (where n is any integer and s is a real number):

• Sine and Cosine: $(-\infty, \infty)$

• Tangent and Secant: $\{s \mid s \neq \frac{\pi}{2} + n\pi\}$

• Cotangent and Cosecant: $\{s \mid s \neq n\pi\}$

For example, tan s and sec s are undefined when x = 0, which occurs at $s = \frac{\pi}{2} + n\pi$.

Identities Involving Circular Functions

Several important identities hold for all values in the domain of the variable:

• If $\cos s \neq 0$, then $1 + \tan^2 s = \sec^2 s$

• If $\sin s \neq 0$, then $1 + \cot^2 s = \csc^2 s$

An identity is a statement that is true for all values in the domain of the variable.

Finding Function Values by Using the Unit Circle

To find the value of a trigonometric function for a given real number s, determine the corresponding point on the unit circle and use the x- or y-coordinate as appropriate.

• Example a) s = 0: The point is (1, 0). So, sin 0 = 0, cos 0 = 1, tan 0 = 0, sec 0 = 1, csc 0 is undefined, cot 0 is undefined.

• Example b) s = \pi: The point is (–1, 0). So, sin \pi = 0, cos \pi = –1, tan \pi = 0, sec \pi = –1, csc \pi is undefined, cot \pi is undefined.

• Example c) s = \frac{3\pi}{2}: The point is (0, –1). So, sin $\frac{3\pi}{2}$ = –1, cos $\frac{3\pi}{2}$ = 0, tan $\frac{3\pi}{2}$ is undefined, sec $\frac{3\pi}{2}$ is undefined, csc $\frac{3\pi}{2}$ = –1, cot $\frac{3\pi}{2}$ = 0.

• Example d) s = –\frac{\pi}{2}: The point is (0, –1), same as part c.

Table: Exact Function Values for $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$

s	sin s	cos s	tan s	cot s	sec s	csc s
0	0	1	0	Undefined	1	Undefined
$\frac{\pi}{2}$	1	0	Undefined	0	Undefined	1
$\pi$	0	–1	0	Undefined	–1	Undefined
$\frac{3\pi}{2}$	–1	0	Undefined	0	Undefined	–1
$2\pi$	0	1	0	Undefined	1	Undefined

Using a Calculator to Find Function Values

To find trigonometric function values for angles not on the standard unit circle, use a calculator in radian mode. For example, for $\pi$:

• cos($\pi$) = –1

• sin($\pi$) = 0

• tan($\pi$) = 0

• sec($\pi$) = –1

• csc($\pi$) is undefined

• cot($\pi$) is undefined

Function Values at Multiples of $2\pi$

Because the circumference of the unit circle is $2\pi$, any integer multiple of $2\pi$ added to s corresponds to the same point as s on the unit circle. Thus, for any integer n:

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

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

Signs of Values of Trigonometric Functions

The sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies:

• Quadrant I (x > 0, y > 0): All functions are positive.

• Quadrant II (x < 0, y > 0): Sine and cosecant are positive.

• Quadrant III (x < 0, y < 0): Tangent and cotangent are positive.

• Quadrant IV (x > 0, y < 0): Cosine and secant are positive.

Using Circular Functions to Find the Quadrant

Given a value of s, determine the quadrant by examining the signs of cos s (x-coordinate) and sin s (y-coordinate). For example, if cos s < 0 and sin s > 0, the point is in quadrant II.

Finding the Cosine, Sine, and Quadrant for Arbitrary Angles

For non-standard angles, use a calculator to approximate values. For example, for s = 4.5:

• cos 4.5 ≈ –0.2108

• sin 4.5 ≈ –0.9775

Since both values are negative, the point lies in quadrant III.

Exact Function Values for $\frac{\pi}{4}$, $\frac{\pi}{6}$, and $\frac{\pi}{3}$

Special angles such as $\frac{\pi}{4}$, $\frac{\pi}{6}$, and $\frac{\pi}{3}$ have exact trigonometric values. For example, the line y = x bisects the first quadrant, giving:

• $\sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$

For $\frac{\pi}{6}$ and $\frac{\pi}{3}$:

• $\sin \frac{\pi}{6} = \frac{1}{2}$

• $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$

• $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$

• $\cos \frac{\pi}{3} = \frac{1}{2}$

The Unit Circle with Key Coordinates

The unit circle can be labeled with the coordinates of points corresponding to common angles, which helps in quickly finding exact values for sine and cosine.

Table: Exact Function Values for $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$

s	sin s	cos s	tan s	cot s	sec s	csc s
$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$	$\sqrt{3}$	$\frac{2}{\sqrt{3}}$	2
$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	1	1	$\sqrt{2}$	$\sqrt{2}$
$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$	$\frac{1}{\sqrt{3}}$	2	$\frac{2}{\sqrt{3}}$

Additional info: The above tables and diagrams are essential for memorizing and understanding the behavior of trigonometric functions on the unit circle, which is foundational for all further study in trigonometry and calculus.