PLANE AND SPHERICAL TRIGONOMETRY

Angle Measure, Arc Length, Linear and Angular Velocities

Engr. Mark Edison M. Victuelles10/6/2010 1

SPECIFIC OBJECTIVES:

At the end of the lesson, the student is expected to :

1. Define trigonometry.

2. Measure angles in rotations, in degrees and in

radians.

3. Find the measures of coterminal angles.

4. Change from degree measure to radian measure and

from radian measure to degree measure.

5. Find the length of an arc intercepted by a central

angle.

6. Solve word problems involving arc length

7. Solve word problems involving angular velocity and

linear velocityEngr. Mark Edison M. Victuelles10/6/2010 2

Trigonometry

Trigonometry is the branch of mathematics that deals

with the measurement of triangle

Engr. Mark Edison M. Victuelles

Angle

An angle is defined as the amount of rotation to move a

ray from one position to another.

The original position of the ray is called the initial side of

the angle, and the final position of the ray is called the

terminal side.

The point about which the rotation occurs and at which the

initial and terminal side of the angle intersect is called the

vertex.10/6/2010 3

Angle

Engr. Mark Edison M. Victuelles

Terminal Side

Initial Side

Vertex

Note:

When the vertex of an angle is the origin of the rectangular

coordinate system and its initial side coincides with the positive

x-axis, the angle is said to be in the standard position.

Positive Angle

Negative Angle

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Angle Measurements

1. Degree

1 revolution = 360 degrees

a. Minute = 1/60 of a degree

b. Second = 1/60 of a minute

2. Radian

1 revolution = 2π radians

3. Gradian / Gradient / Grade

1 revolution = 400grads

4. Mil

1 revolution = 6400mils

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Conversion (angle measurement)

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θ (degrees) θ (radians)

1 revolution = 360 degrees 1 revolution = 2π radians

The ratio of

Conversion (angle measurement)

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Degrees to Radians:

Radians to Degrees:

Degrees to Gradians:

Gradians to Degrees:

Radians to Grad:

Grad to Radians:

Example:

I. Convert the following angles measured in

degrees, minutes and seconds to angles

measured to the nearest hundredth of a

degree:

a. 64°24’ 38”

b. 228° 23’ 10”

c. 145° 11’ 56”

d. 356° 09’ 34”

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Example:

II. Convert the following angles measured in

degrees to angles measured to the nearest

minute:

a. 56.39°

b. 273.8°

c. 323.28°

d. 163.18°

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Example:

III. Express each angle measure in degrees:

a.

b.

c.

d.

e.

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3

4

4

5

3

18

12

7

6

11

Example:

IV. Express each angle measure in radians. Give

answer in terms of :

a. 120°

b. 335°

c. -310°

d. 1035°

e. 450°

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Coterminal Angles

Coterminal angles are angles in standard position whose

initial and terminal sides are the same.

To find angles coterminal to a given angle, add or subtract

multiples of 360° to it.

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Example:

I. Draw the following angles and find two angles

(one positive and one negative) coterminal

with each.

a. 55°

b. 70°

c. 153°

d. 219°

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Example:

II. For each of the following angles, find a

coterminal angle with measure such that

a. -100°

b. 524°

c. 900°

d. 1250°

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3600

Classification of Angles

Angles are classified according to the measurement of its

angle.

1. Zero Angle – an angle formed by two coinciding rays

without rotation between them

2. Acute angle (0r sharp) – an angle formed between 0

and 90.

3. Right Angle – is a 90 angle. Angle formed by two

perpendicular rays.

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Classification of Angles

4. Obtuse Angle (Blunt) – angle formed between 90 and

180.

5. Straight Angle – an angle whose measure is exactly 180 .It is formed by two rays extending in opposite directions.

6. Reflex (Bent-Back) – angle formed between 180 and

360.

7. Circular Angle – angle whose measure is exactly 360

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Length of a Circular Arc , s

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An arc length refers to the measure of a position of a circle or part of its circumference. The arc length s for a given central angle can be found as follows:

where: s = length of the arcr = radius of the circle = measure of the central angle in radians

rs

s

Example:

I. Find the length of the arc of a circle whose

radius and whose central angle are as follows

a. = 2.5 radians, r = 20 cm

b. = 225°, r = 30.1 mm

c. = , r = 15 ft.

II. If the minute hand of a clock is 8 cm long, how

far does the tip of the hand move after 25

minutes?

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4

7

50 cm

118.2 mm

82.5 ft

cm3

20

Area of a Sector

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where: A = area of the sectorr = radius of the circle = measure of the central angle in radians

2

rA

2

A

Angular Velocity

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The angular velocity ( ) of a point on a revolving ray is the angular displacement per unit time.

t

Where: - is the angular velocity

- is the angular displacement

t - time

Linear Velocity

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The linear velocity (V) of a point on a revolving ray is the linear distance traveled by the point per unit time.

Where: V - is the linear velocity

s - is the linear displacement

t - time

rVt

sV or