Angles, Arc, and Their Measures – Trigonometry Foundations

Chapter 9: Trigonometry Functions and Applications

9.1 Angles, Arc, and Their Measures

This section introduces the foundational concepts of angles, their measurement, and related applications in trigonometry. Understanding these basics is essential for further study in trigonometric functions and their real-world uses.

Angles and Arcs: Basic Terminology

• Line AB: Determined by two distinct points A and B; extends infinitely in both directions.

• Line Segment AB: The portion of the line between points A and B.

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

• Angle: Formed by rotating a ray (the initial side) around its endpoint (the vertex) to a terminal side.

Degree Measure

• The degree divides the circumference of a circle into 360 equal parts.

• There are 360° in one full rotation.

• Acute angle: $0° < \theta < 90°$

• Right angle: $\theta = 90°$

• Obtuse angle: $90° < \theta < 180°$

• Straight angle: $\theta = 180°$

Complementary and Supplementary Angles

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

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

Example: If $6m + 3m = 90°$, then $m = 10°$; angles are $60°$ and $30°$.

Degrees, Minutes, and Seconds

• 1 minute ($1'$) = $\frac{1}{60}$ of a degree

• 1 second ($1''$) = $\frac{1}{60}$ of a minute = $\frac{1}{3600}$ of a degree

Example: $51°29' + 32°46' = 84°15'$

Converting Between Decimal Degrees and DMS (Degrees, Minutes, Seconds)

• To convert $74°8'14''$ to decimal degrees: $74 + \frac{8}{60} + \frac{14}{3600} \approx 74.137°$

• To convert $34.817°$ to DMS: $34° + 0.817 \times 60' = 34°49'1.2''$

Quadrantal Angles

• Angles in standard position (vertex at the origin, initial side along the positive x-axis) with terminal sides on the x- or y-axis (e.g., $0°, 90°, 180°, 270°, 360°$).

Coterminal Angles

• Angles with the same initial and terminal sides but different rotations.

• Their measures differ by multiples of $360°$.

Example: $908°$ is coterminal with $188°$ ($908° - 2 \times 360°$).

Radian Measure

• 1 radian is the angle at the center of a circle that intercepts an arc equal in length to the radius.

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

• $360° = 2\pi$ radians, $180° = \pi$ radians

Converting Between Degrees and Radians

• Degrees to radians: Multiply by $\frac{\pi}{180}$

• Radians to degrees: Multiply by $\frac{180}{\pi}$

Example: $45° = 45 \left(\frac{\pi}{180}\right) = \frac{\pi}{4}$ radians

Equivalent Angle Measures in Degrees and Radians

Degrees	Radians (Exact)	Radians (Approximate)
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 on a circle of radius $r$ by a central angle $\theta$ (in radians) is $s = r\theta$.

Example: For $r = 18.20$ cm and $\theta = \frac{3\pi}{8}$ radians, $s = 18.20 \times \frac{3\pi}{8} \approx 21.44$ cm.

Applications: Using Latitudes to Find Distance

• Latitude difference gives the central angle between two locations on Earth.

• Distance $s = r\theta$, where $\theta$ is in radians and $r$ is Earth's radius.

Example: Reno (40°N) and Los Angeles (34°N), $r = 6400$ km, $\theta = 6° = \frac{\pi}{30}$ radians, $s = 6400 \times \frac{\pi}{30} \approx 670$ km.

Area of a Sector

• The area $A$ of a sector of a circle of radius $r$ and central angle $\theta$ (in radians): $A = \frac{1}{2} r^2 \theta$

Example: For $r = 321$ m, $\theta = \frac{\pi}{12}$, $A = \frac{1}{2} (321)^2 \frac{\pi}{12} \approx 13,500$ m2

Linear and Angular Speed

• Angular speed ($\omega$): $\omega = \frac{\theta}{t}$ (radians per unit time)

• Linear speed ($v$): $v = \frac{s}{t} = r\omega$

Example: For a point on a circle of radius 10 cm, rotating at $\frac{\pi}{18}$ radians/sec, in 6 seconds:

• $\theta = 6 \times \frac{\pi}{18} = \frac{\pi}{3}$ radians

• $s = 10 \times \frac{\pi}{3} \approx 10.47$ cm

• $v = \frac{10.47}{6} \approx 1.75$ cm/sec

Finding Angular Speed of a Pulley and Linear Speed of a Belt

• For a pulley of radius 6 cm at 80 revolutions/minute:

• Each revolution = $2\pi$ radians, so $80 \times 2\pi = 160\pi$ radians/minute

• Angular speed: $\omega = \frac{160\pi}{60} = \frac{8\pi}{3}$ radians/sec

• Linear speed: $v = r\omega = 6 \times \frac{8\pi}{3} = 16\pi \approx 50$ cm/sec

Additional info: These concepts form the basis for understanding trigonometric functions, their graphs, and applications in physics, engineering, and navigation.