TRIGONOMETRY

ANGLE MEASURE
MODULE 3 – LESSON 1
PREPARED BY: JONDEL S. IHALAS
1.1 ANGLES AND THEIR MEASURE
• An angle is formed when a ray is rotated around its endpoint.
The common endpoint is called the vertex.
1.1 ANGLES AND THEIR MEASURE
• The ray in its original position is called the initial ray or the initial side of
an angle. In the Cartesian plane, we assume the initial side of an angle is
the positive x‐axis. The ray after it is rotated is called the terminal ray or
the terminal side of an angle. Rotation in a counterclockwise direction
corresponds to a positive angle, whereas rotation in a clockwise direction
corresponds to a negative angle.
1.2 DEGREE MEASURE
DEFINITION: Degree Measure of Angles
• An angle formed by one complete counterclockwise
rotation has measure 360 degrees, denoted 360°.
1.2 DEGREE MEASURE

WORDS MATH
360° represents 1 complete counterclockwise 360°
rotation. °
=1
360
1
180° represents a counterclockwise rotation. 180° 1
2
°
=
360 2
1
90° represents complete counterclockwise 90° 1
4
°
=
rotation. 360 4
1
1° represents a counterclockwise rotation. 1° 1
360
°
=
360 360
1.2 DEGREE MEASURE

• The Greek letter θ (theta) is the most common name for an

angle in mathematics. Other common names of angles
are α (alpha), β (beta), and γ (gamma).
1.2 DEGREE MEASURE
The Different Types of Angles

An angle measuring exactly 90° is

called a right angle.
A right angle is often represented by
the adjacent sides of a rectangle,
indicating that the two rays
are perpendicular.
1.2 DEGREE MEASURE

• An angle measuring exactly 180° is called

a straight angle.
1.2 DEGREE MEASURE

• An angle measuring greater than 0°, but less

than 90°, is called an acute angle.
1.2 DEGREE MEASURE

• An angle measuring greater than 90°, but less than

180°, is called an obtuse angle.
1.2 DEGREE MEASURE

• If the sum of the measures of two positive angles is

90°, the angles are called complementary. We say
that α is the complement of β (and vice versa).
1.2 DEGREE MEASURE

• If the sum of the measures of two positive angles is

180°, the angles are called supplementary. We say
that α is the supplement of β (and vice versa).
1.2 DEGREE MEASURE
EXAMPLE 1: Read and follow the instructions.
a. Draw an angle of 60°. Label the initial side and the
terminal side; show the direction of the angle.
1.2 DEGREE MEASURE
EXAMPLE 1: Read and follow the instructions.
b. Draw an angle of -120°. Label the initial side and the
terminal side; show the direction of the angle.
1.2 DEGREE MEASURE
EXAMPLE 1: Read and follow the instructions.
c. Find the complement of 67°.

The sum of complementary angles is 90°. 𝜃 + 67° = 90°

Solve for θ. θ = 90° − 67°
𝜃 = 23°
1.2 DEGREE MEASURE
EXAMPLE 1: Read and follow the instructions.
d. Find the supplement of 123°.
The sum of supplementary angles is 180°. 𝜃 + 123° = 180°
Solve for θ. θ = 180° − 123°
𝜃 = 57°
1.2 DEGREE MEASURE
EXAMPLE 1: Read and follow the instructions.
e. Find two supplementary angles such that the first
angle is twice as large as the second angle.

The sum of supplementary angles is 180°. 𝛼 + 𝛽 = 180°

Suppose 𝛽 = 2𝛼 . 𝛼 + 2𝛼 = 180°
Solve for α. 3𝛼 = 180°
𝛼 = 60°
Substitute 𝛼 = 60° into 2𝛼.
1.3 RADIAN MEASURE
A central angle is an angle that has its vertex at the center of a circle.
When the intercepted arc's length is equal to the radius, the measure
of the central angle is 1 radian.
1.3 RADIAN MEASURE
DEFINITION: Radian Measure
If a central angle θ in a circle with radius r intercepts an arc on
the circle of length s (arc length), then the measure of θ,
in radians, is given by
𝒔
𝜽(in radians) =
𝒓

2𝝅𝒓
𝜽𝒇𝒖𝒍𝒍 𝒓𝒐𝒕𝒂𝒕𝒊𝒐𝒏 = = 2𝝅
𝒓
1.3 RADIAN MEASURE

WORDS MATH

The measure of 𝜃 is 12 degrees. 𝜃 = 12°

The measure of 𝜃 is 12 radian. 𝜃 = 12 rad

Note: Units for arc length and radius must be the same to use. And radians are unitless, the word
radians (or rad) is often omitted. If an angle measure is given simply as a real number, then radians
are implied.
1.3 RADIAN MEASURE
EXAMPLE 2: Finding the radian measure of an Angle.
What is the measure (in radians) of a central angle θ that intercepts an arc
of length 8 centimeters on a circle with radius 2 meters?
Solution:
𝒔
Write the formula relating radian measure to arc length 𝜽(in radians) =
𝒓
and radius.

8 𝑐𝑚
Substitute s = 8cm and r = 2 m into the radian expression 𝜃=
2𝑚

8 𝑐𝑚
Convert the radius (2) meters to centimeters: 2 𝑚 = 200 𝑐𝑚. 𝜃=
200 𝑐𝑚

The units, cm, cancel and the result is a unitless real number. 𝜃 = 0.04
1.3 RADIAN MEASURE
Converting between Degrees and Radian

An angle corresponding to one full rotation is said to have

measure 360° or 2𝜋 radians. Therefore, 180° = 𝜋 rad.
▪ To convert degrees to radians, multiply the degree measure
𝜋
by .
180°
▪ To convert radians to degrees, multiply the radian measure
180°
by .
𝜋
1.3 RADIAN MEASURE
EXAMPLE 3: Convert:
a.-30° to radians.
𝜋 𝜋
Multiply −30° by . −30°
180° 180°
−30°𝜋
180°
1 𝜋
Simplify, cancel the degree − 𝜋 𝑜𝑟 − 𝑜𝑟 − 0.523 𝑟𝑎𝑑
6 6

𝜋
Note: − is the exact value. A calculator can be used to approximate this expression.
6
Scientific and graphing calculators have a 𝜋 button (on most scientific calculators, it requires
using a shift or second command). The decimal approximation rounded to three decimal
places is 0.524.
1.3 RADIAN MEASURE
EXAMPLE 3: Convert:
b. 495° to radians.

𝜋 𝜋
Multiply 495° by . 495°
180° 180°
495°𝜋
180°
11𝜋
Simplify, cancel the degree 𝑜𝑟 8.64 𝑟𝑎𝑑
4
1.3 RADIAN MEASURE
EXAMPLE 3: Convert:
3𝜋
c. to degrees.
4

3𝜋 180° 3𝜋 180°
Multiply by .
4 𝜋 4 𝜋
540°𝜋
4𝜋
Simplify, cancel 𝜋 135°
1.4 ANGLES IN STANDARD POSITION
DEFINITION: Standard Position
An angle is said to be in standard position if its initial side is along
the positive x‐axis and its vertex is at the origin.
1.4 ANGLES IN STANDARD POSITION
We say that an angle lies in the quadrant in which its terminal side lies. Angles
in standard position with terminal sides along the x‐axis or y‐axis (90°, 180°,
270°, 360°, etc.) are called quadrantal angles.
1.4 ANGLES IN STANDARD POSITION
EXAMPLE 4: Sketch the following angles in standard position.
a. 78° b. 219° c. -406°
1.5 COTERMINAL ANGLES
DEFINITION: Coterminal Angles
Two angles in standard position with the same terminal side are
called coterminal angles.

In the figure above, both 𝛼 and 𝛽 are coterminal angles.

1.5 COTERMINAL ANGLES
To find the measure of the smallest nonnegative coterminal angle
of a given angle measured in degrees, follow this procedure:
▪ If the given angle is positive, subtract 360° (repeatedly until the
result is a positive angle less than or equal to 360°).
▪ If the given angle is negative, add 360° (repeatedly until the
result is a positive angle less than or equal to 360°).

Similarly, if the angle is measured in radians, subtract or add

equivalently 2𝜋 until your result is a positive angle less than or
equal to 2𝜋.
1.5 COTERMINAL ANGLES
EXAMPLE 5: Determine the angle with the smallest
possible positive measure that is coterminal with each of
the following angles:
a.527°
Since 527° is positive, subtract 360°
527°– 360° = 167°
1.5 COTERMINAL ANGLES
EXAMPLE 5: Determine the angle with the smallest
possible positive measure that is coterminal with each of
the following angles:
b. -995°
1.5 COTERMINAL ANGLES
EXAMPLE 5: Determine the angle with the smallest
possible positive measure that is coterminal with each of
the following angles:
b. -995°

Since -995° is negative, add 360° −995° + 360° = −635°

Add 360° again −635° + 360 = −275°
Add 360° again until it becomes positive −275° + 360 = 85°
1.5 COTERMINAL ANGLES
EXAMPLE 5: Determine the angle with the smallest
possible positive measure that is coterminal with each of
the following angles:
13𝜋
c. 13𝜋
4 Since is positive, subtract 2𝜋
4
13𝜋 5𝜋
− 2𝜋 = 𝑜𝑟3.935 𝑟𝑎𝑑
4 4
1.6 APPLICATIONS OF RADIAN MEASURE

We now look at applications of radian measure that

involve calculating arc lengths, areas of circular
sectors, and angular and linear speeds. All of these
applications are related to the definition of radian
measure.

Remember to use the relationship 𝒔 = 𝒓𝜽 where the

angle θ must be in radians.
1.6.1 Arc Length

DEFINITION: Arc Length

If a central angle 𝜽 in a circle with radius 𝒓 intercepts an
arc on the circle of length s, then the arc length 𝒔 is
given by
𝒔 = 𝒓𝜽
where the angle θ must be in radians.
1.6.1 Arc Length
EXAMPLE 6:
The International Space Station (ISS) is in
an approximately circular orbit 500
kilometers above the surface of the Earth. If
the ground station tracks the space station
when it is within a 60° central angle of this
circular orbit above the tracking antenna,
how many kilometers does the ISS cover
while it is being tracked by the ground
station? Assume that the radius of the Earth
is 6400 kilometers. Round to the nearest
kilometer.
1.6.1 Arc Length
EXAMPLE 6:
Solution:

𝜋
Write the formula for arc length when the angle has 𝑠 = 𝑟𝜃
180°
degree measure

𝜋
Substitute 𝑟 = 6400 + 500 = 6900 𝑘𝑚 and 𝜃 = 60°. 𝑠 = 6900 60°
180°

Evaluate with a calculator. 𝑠 = 7225.663

Round to the nearest kilometer. 𝑠 = 7226 𝑘𝑚

The ISS travels approximately 7226 kilometers during the ground station tracking.
1.6.2 Area of a Circular Sector
DEFINITION: Area of a Circular Sector
The area of a sector of a circle with radius r and central angle θ is given by
𝟏 𝟐
𝑨= 𝒓 𝜽
𝟐
where the angle θ must be in radians.
1.6.2 Area of a Circular Sector
DERIVATION OF THE FORMULA

WORDS MATH

Write the ratio of the area of the sector to the area of the entire 𝐴
circle. 𝜋𝑟 2
Write the ratio of the central angle θ to the measure of one full 𝜃
rotation. 2𝜋
The ratios must be equal (proportionality of sector to circle). 𝐴 𝜃
=
𝜋𝑟 2 2𝜋
Multiply both sides of the equation by πr 2 2
𝐴 𝜃
𝜋𝑟 ∙ 2 = ∙ 𝜋𝑟 2
𝜋𝑟 2𝜋
Simplify 1 2
𝐴= 𝑟 𝜃
2
1.6.2 Area of a Circular Sector
EXAMPLE 7:
A sprinkler on a golf course fairway is set to spray water over a distance of
70 feet and rotates through an angle of 45°. Find the area of the fairway
watered by the sprinkler.
1.6.2 Area of a Circular Sector
Solution:

1 2 𝜋
Write the formula for circular sector area in degrees. 𝐴 = 𝑟 𝜃
2 180°

1 2 𝜋
Substitute 𝑟 = 70 and 𝜃 = 45°. 𝐴= 70 45°
2 180°

Evaluate with a calculator. 𝐴 = 1924.226

. 𝐴 = 1924.23 𝑓𝑡 2

The area of the fairway watered by the sprinkler is 𝟏𝟗𝟐𝟒. 𝟐𝟑𝒇𝒕𝟐 .

1.6.3 Linear Speed
Recall the relationship between distance, rate (assumed to be constant),
and time: d = rt. Rate is speed, and in words this formula can be rewritten
as
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑠𝑝𝑒𝑒𝑑 ∙ 𝑡𝑖𝑚𝑒 𝑜𝑟 𝑠𝑝𝑒𝑒𝑑 =
𝑡𝑖𝑚𝑒

It is important to note that we assume speed is constant. If we think of a car

driving around a circular track, the distance it travels is the arc length s; and
if we let v represent speed and t represent time, we have the formula for
speed along a circular path (linear speed):
1.6.3 Linear Speed

DEFINITION: Linear Speed

If a point P moves along the circumference of a circle at
a constant speed, then the linear speed 𝑣 is given by

𝒔
𝒗=
𝒕

where 𝑠 is the arc length and 𝑡 is the time.

1.6.3 Linear Speed
EXAMPLE 8:
A car travels at a constant speed around a
circular track with circumference equal to 3
miles. If the car records a time of 16 minutes for
9 laps, what is the linear speed of the car in
miles per hour?
1.6.3 Linear Speed
Solution:

3 𝑚𝑖𝑙𝑒𝑠
Calculate the distance traveled around the circular track. 𝑠 = 9 𝑙𝑎𝑝𝑠 = 27𝑚𝑖
𝑙𝑎𝑝

𝑠 27𝑚𝑖
Substitute 𝑡 = 16𝑚𝑖𝑛 and 𝑠 = 27𝑚𝑖 into 𝑣 = 𝑣=
𝑡 16 𝑚𝑖𝑛

27𝑚𝑖 60 𝑚𝑖𝑛
Convert the linear speed from miles per minute 𝑣=
16 𝑚𝑖𝑛 ℎ𝑟
to miles per hour.

Simplify 𝑣 = 101.25𝑚𝑝ℎ

The linear speed of the car in miles per hour is 𝟏𝟎𝟏. 𝟐𝟓.
Note: To calculate linear speed, we find how fast a position along the circumference of a
circle is changing.
1.6.4 Angular Speed

DEFINITION: Angular Speed

If a point P moves along the circumference of a circle at a
constant speed, then the central angle 𝜃 that is formed with the
terminal side passing through point P also changes over some
time 𝑡 at a constant speed. The angular speed 𝜔 (omega) is
given by

𝜽
𝝎=
𝒕
where 𝜃 is given in radians.
1.6.4 Angular Speed
EXAMPLE 9:
A lighthouse in the middle of a channel rotates its light in a circular
motion with constant speed. If the beacon of light completes 1 rotation
every 15 seconds, what is the angular speed of the beacon in radians
per minute?
1.6.4 Angular Speed
Solution:

Calculate the angle measure in radians associated 𝜃 = 2𝜋

with 1 rotation

𝜃 2𝜋 𝑟𝑎𝑑
Substitute 𝑡 = 15𝑠𝑒𝑐 and 𝜃 = 2𝜋 into 𝜔 = 𝜔=
𝑡 15𝑠𝑒𝑐
2𝜋 𝑟𝑎𝑑 60 𝑠𝑒𝑐
Convert the angular speed from radians per second to 𝜔= ∙
15𝑠𝑒𝑐 1 𝑚𝑖𝑛
radians per minute.
Simplify 𝜔 = 8𝜋 𝑟𝑎𝑑/𝑚𝑖𝑛

The angular speed of the beacon in radians per minute is 𝟖𝝅.

Note: To calculate angular speed, we find how fast the central angle is changing.
LESSON SUMMARY
• Angle measures can be converted between degrees and radians in the following way:
𝜋
➢ To convert degrees to radians, multiply the degree measure by
180°
180°
➢ To convert radians to degrees, multiply the radian measure by
𝜋

• (Remember that 𝜋 = 180°. )

• Coterminal angles in standard position have terminal sides that coincide.
• The length of a circular arc is given by 𝑠 = 𝑟𝜃, where 𝜃 is the central angle given in
radians and 𝑟 is the radius of the circle.
1
• The area of a circular sector is given by 𝐴 = 𝑟 2 𝜃, where 𝜃 is the central angle given in
2

radians and 𝑟 is the radius of the circle.

𝑠 𝜃
• Linear speed,𝑣 = , and angular speed,𝜔 = .
𝑡 𝑡