Angles, Arc, and Their Measures

Angles and Arcs: Basic Terminology

Understanding the foundational terminology of angles and arcs is essential for studying trigonometry and its applications.

• Line AB: Determined by two distinct points A and B.

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

• Ray AB: Starts at A and continues through B indefinitely.

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

Degree Measure

Angles are commonly measured in degrees, which divide the circumference of a circle into 360 equal parts.

• There are 360° in one full rotation.

• 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

Angles can be classified based on their sums:

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

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

Example:

• If , then ; angles are 60° and 30°.

• If , then ; angles are 72° and 108°.

Degrees, Minutes, and Seconds

Angles can be measured more precisely using minutes and seconds.

• 1 minute (1') = of a degree

• 1 second (1") = of a minute = of a degree

Example Calculations:

Converting Between Decimal Degrees, Minutes, and Seconds

Conversion between decimal degrees and degrees-minutes-seconds is often required in trigonometry.

• To convert to decimal degrees:

• To convert to degrees, minutes, and seconds:

Quadrantal Angles

Quadrantal angles are angles in standard position (vertex at the origin, initial side along the positive x-axis) whose terminal sides lie along the x- or y-axis.

• Examples: 0°, 90°, 180°, 270°, 360°

Coterminal Angles

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

• To find a coterminal angle between 0° and 360°, add or subtract 360° as needed.

• Example:

Radian Measure

Radians are another unit for measuring angles, based on the radius of a circle.

• An angle intercepting an arc equal to the radius has a measure of 1 radian.

• The circumference of a circle is .

• radians, radians

Converting Between Degrees and Radians

• To convert degrees to radians: Multiply by

• To convert radians to degrees: Multiply by

Example:

• radians

• radians

• radians

Equivalent Angle Measures in Degrees and Radians

Common angles and their radian equivalents:

Arc Length

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

Example:

• For cm and radians: cm

• For radians: cm

Applications: Using Latitudes to Find Distance

Latitude can be used to calculate the north-south distance between two cities on Earth.

• Central angle between Reno (40°N) and Los Angeles (34°N):

• Convert to radians: radians

• Distance: km

Area of a Sector

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

Example:

• For a field shaped as a sector with m and radians: m2

Linear and Angular Speed

Angular and linear speed describe rotational motion.

• Angular speed (): (radians per unit time)

• Linear speed ():

Example:

• Point P on a circle of radius 10 cm, angular speed rad/s, time = 6 s:

• Angle generated: radians

• Distance traveled: cm

• Linear speed: cm/s

Example: Pulley and Belt

• 80 revolutions per minute, radius = 6 cm

• Angular speed: radians/sec

• Linear speed: cm/sec

Additional info: These notes cover the essential concepts and calculations for angles, arc length, sector area, and rotational motion, foundational for further study in trigonometry and its applications in Precalculus.