Chapter 9: The Unit Circle and the Functions of Trigonometry - Precalculus Study Notes

Chapter 9: The Unit Circle and the Functions of Trigonometry

9.1 Angles, Arc, and Their Measures

Angles and Arcs Basic Terminology

Understanding the basic geometric elements is essential for studying trigonometry. Two distinct points, A and B, can define a line, a segment, or a ray:

• Line AB: Extends infinitely in both directions through points A and B.

• Segment AB: The portion of the line between A and B, including both endpoints.

• Ray AB: Starts at A and passes through B, extending infinitely in one direction.

An angle is formed by rotating a ray (the initial side) around its endpoint (the vertex) to a terminal side.

Degree Measure

The degree is a unit for measuring angles, dividing the circle's circumference into 360 parts. Key angle types:

• Acute angle: Between 0° and 90°

• Right angle: Exactly 90°

• Obtuse angle: Greater than 90° but less than 180°

• Straight angle: Exactly 180°

Complementary and Supplementary Angles

• Complementary angles: Two positive angles whose sum is 90°.

• Supplementary angles: Two positive angles whose sum is 180°.

Finding Measures of Angles

Given figures, you may be asked to find the measures of angles. For example:

• Angles are 60° and 30°.

• Angles are 72° and 108°.

Calculating With Degrees, Minutes, and Seconds

Angles can be measured in degrees (°), minutes ('), and seconds (").

• 1 minute (1') = 1/60 of a degree

• 1 second (1") = 1/60 of a minute

Calculations often require converting between these units or adding/subtracting them.

Converting Between Decimal Degrees, Minutes, and Seconds

Conversion between decimal degrees and DMS (degrees, minutes, seconds) is a common task:

• To convert DMS to decimal degrees: $\text{Decimal Degrees} = \text{Degrees} + \frac{\text{Minutes}}{60} + \frac{\text{Seconds}}{3600}$

• To convert decimal degrees to DMS: Multiply the decimal part by 60 for minutes, and again by 60 for seconds.

Quadrantal Angles

Quadrantal angles are angles in standard position whose terminal sides lie along the x- or y-axis (e.g., 0°, 90°, 180°, 270°, 360°).

Coterminal Angles

Coterminal angles share the same initial and terminal sides but differ by multiples of 360°.

Radian Measure

The radian is another unit for measuring angles. One radian is the angle subtended by an arc equal in length to the radius of the circle.

• The circumference of a circle is $C = 2\pi r$.

• 360° corresponds to $2\pi$ radians.

Converting Between Degrees and Radians

• To convert degrees to radians: $\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$

• To convert radians to degrees: $\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$

Equivalent Angle Measures in Degrees and Radians

Common angles and their radian equivalents:

Degrees	Exact Radians	Approximate Radians
0°	0	0
30°	$\frac{\pi}{6}$	0.5236
45°	$\frac{\pi}{4}$	0.7854
60°	$\frac{\pi}{3}$	1.0472
90°	$\frac{\pi}{2}$	1.5708
180°	$\pi$	3.1416
270°	$\frac{3\pi}{2}$	4.7124
360°	$2\pi$	6.2832

Arc Length

The length s of an arc intercepted by a central angle $\theta$ (in radians) in a circle of radius r is:

• $s = r\theta$

Applications: Latitude and Distance

Latitude can be used to find distances between cities using the arc length formula. For example, the distance between Reno and Los Angeles is calculated using their latitudes and Earth's radius.

Area of a Sector

The area A of a sector of a circle of radius r and central angle $\theta$ (in radians) is:

• $A = \frac{1}{2} r^2 \theta$

Linear and Angular Speed

• Angular speed ($\omega$): Rate of change of angle, in radians per unit time.

• Linear speed ($v$): Rate of change of arc length, $v = r\omega$.

9.2 The Unit Circle and Its Functions

The Unit Circle

The unit circle is a circle with center at the origin and radius 1. It is fundamental for defining trigonometric functions for all real numbers.

Circular (Trigonometric) Functions

If (x, y) is a point on the unit circle corresponding to real number s:

• $x = \cos s$

• $y = \sin s$

Domains of Trigonometric Functions

• Sine and Cosine: Domain is all real numbers.

• Tangent and Secant: Domain excludes values where $\cos s = 0$.

• Cotangent and Cosecant: Domain excludes values where $\sin s = 0$.

Trigonometric Identities

• $\sin^2 s + \cos^2 s = 1$

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

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

Finding Function Values Using the Unit Circle

For specific values of s, the unit circle provides exact values for sine, cosine, and other functions. For example:

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

Function Values at Multiples of $2\pi$

Adding integer multiples of $2\pi$ to s does not change the sine or cosine values:

• $\cos(s + 2n\pi) = \cos s$

• $\sin(s + 2n\pi) = \sin s$

Signs of Trigonometric Functions in Quadrants

The sign of each trigonometric function depends on the quadrant:

Quadrant	Sign of sin	Sign of cos	Sign of tan
I	+	+	+
II	+	-	-
III	-	-	+
IV	-	+	-

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

These angles correspond to important points on the unit circle:

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

9.3 Graphs of the Sine and Cosine Functions

Periodic Functions

A periodic function repeats its values in regular intervals. Sine and cosine are classic examples:

• $f(x) = f(x + np)$ for all real x, integer n, and period p.

• The smallest positive p is the period.

Amplitude

The amplitude of a sinusoidal function is half the difference between its maximum and minimum values. For $y = a \sin x$ or $y = a \cos x$, amplitude is $|a|$.

Graphing y = a sin x and y = sin bx

Changing the coefficient a stretches or compresses the graph vertically. Changing b affects the period:

• Period of $y = \sin bx$ is $\frac{2\pi}{b}$

Graphing y = a sin bx

For negative a, the graph is reflected across the x-axis.

Horizontal and Vertical Translations

• Horizontal translation (phase shift): $y = f(x - d)$ shifts the graph d units right if d > 0.

• Vertical translation: $y = c + f(x)$ shifts the graph c units up if c > 0.

9.4 Graphs of the Other Circular Functions

Graphs of Cosecant and Secant Functions

Cosecant and secant are reciprocals of sine and cosine, respectively. Their graphs have vertical asymptotes where the guide function (sine or cosine) is zero.

Graphing y = a sec bx and y = a csc bx

To graph these functions:

• Graph the corresponding guide function (cosine or sine).

• Draw vertical asymptotes at the x-intercepts of the guide function.

• Sketch U-shaped branches between asymptotes.

Graphing y = a csc(x – d)

Graph the guide function, identify asymptotes, and sketch the branches.

Graphs of Tangent and Cotangent Functions

Tangent and cotangent functions have vertical asymptotes and periods determined by their coefficients:

• Period of $y = a \tan bx$ is $\frac{\pi}{b}$

• Period of $y = a \cot bx$ is $\frac{\pi}{b}$

Determining Equations for Graphs

Given a graph, you can determine its equation by identifying period, amplitude, phase shift, and vertical translation.

Additional info: These notes cover the essential concepts, definitions, and graphical representations of trigonometric functions, their properties, and applications, as required for a precalculus course.