Lecture 25: Section 4.1

Angles and Their Measure

Angle - initial side, terminal side, vertex

Standard position of an angle

Positive and negative angles

Coterminal angle

Central angle

Radians

Complementary and suppplementary angles

Degree measure and radian measure

Arc length, s

Area of a sector

Linear speed

Angular speed

L25 - 1

An angle is formed by rotating a ray around its end-point. The starting position of the ray is the initial

side of the angle, and the position after rotation is theterminal side of the angle. The endpoint of the rayis called the vertex.

An angle θ is said to be in standard position if itsvertex is in the origin and its initial side coincides withthe positive x-axis.

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L25 - 2

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If the rotation is in the counterclockwise direction, theangle is positive; if the rotation is clockwise, the angleis negative.

Angles α and β are coterminal angles if they havethe same initial and terminal sides.

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A central angle is an angle whose vertex is at thecenter of a circle.

L25 - 3

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Radian Measure

Def. One radian is a measure of the centralangle that intercepts an arc whose length is equal tothe radius. Algebraically, this means that

θ =s

r

where θ is measured in radians.

NOTE: For the angle θ = 1 revolution:

The length of the arc (circumference) s = 2πr

Therefore, θ =s

r=

2πr

r= 2π.

We have,1 revolution = 2π radians

L25 - 4

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NOTE: If 0 ≤ θ ≤ 2π, the standard position of theangle θ in the Cartesian coordinate system is shownbelow:

NOTE: Given an angle θ, the coterminal angles to θare

θ + 2nπ

where n is an integer.

ex. Find the angle with the smallest positive measure

that is coterminal with θ = −21π

4.

Checkpoint: Lecture 25, problem 1

L25 - 5

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Def. Given two positive angles α and β,

1. α and β are complementary if

2. α and β are supplementary if

ex. Find the complement and supplement of the angle

θ =π

7.

Checkpoint: Lecture 25, problem 2

L25 - 6

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Degree Measure

Another way to measure angles is in terms ofdegrees, denoted by ◦.

1 counterclockwise revolution = 360◦

NOTE: 1 revolution = 360◦ = 2π rad

=⇒ 180◦ = π rad

Therefore, we have

1◦ =π

180rad and 1 rad =

180◦

π

L25 - 7

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Conversions between Radians and Degrees

Radian Degree

ex. Convert each angle in degrees to radians.

1) 60◦

2) 150◦

ex. Convert each angle in radians to degrees.

1)π

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2) −3π

4

Checkpoint: Lecture 25, problem 3

L25 - 8

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Arc Length

For a circle of radius r, a central angle θ intercepts anarc of length s given by

s = rθ

where θ is measured in radians.

ex. A circle has a radius of 6 inches. Find the lengthof the arc intercepted by a central angle of 120◦.

Checkpoint: Lecture 25, problem 4

L25 - 9

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Area of a Sector

For a circle of radius r, the area A of a sector withcentral angle θ is given by

A =1

2r2θ

where θ is measured in radians.

ex. A sprinkler sprays water over a distance of 30 feetwhile rotating through an angle of 150◦. What area oflawn receives water?

Checkpoint: Lecture 25, problem 5

L25 - 10

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