math12 (week 1)
TRANSCRIPT
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