chapter 1: circles - migomendoza.weebly.com · complementary angles ... to find the relationship...
TRANSCRIPT
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Circular Functions (Trigonometry)Chapter 8.1:
SSMTH1: Precalculus
Science and Technology, Engineering and Mathematics (STEM) Strands
Mr. Migo M. Mendoza
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Chapter 8.1: Circular Functions Lecture 8.1: Basic Concepts
Lecture 8.2: Degree Measure and Degree Radian Measure
Lecture 8.3: Standard Angle and Coterminal Angle
Lecture 8.4: Circular Functions
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Basic ConceptsLecture 8.1:
SSMTH1: Precalculus
Science and Technology, Engineering and Mathematics (STEM) Strands
Mr. Migo M. Mendoza
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Ray and VertexA ray consists of point O on a
line and extends indefinitely in one direction. The point O
is called the end point (or vertex).
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Did you know?If two rays are drawn
with a common endpoint, then they will
form an angle.
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Initial Side and Terminal SideThe initial position of the ray is called the initial side of the angle, while the position of
the ray after rotation is called the terminal side.
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Initial Side and Terminal Side
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Take Note:The ray can travel an
unlimited number of times around the circle and still end in the same terminal position.
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Standard AngleAn angle is said to be in standard position if its vertex is at the origin of a rectangular coordinate system
and its initial side coincides with the x-axis.
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Standard Angle
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Quadrantal AngleWhen the terminal side of an angle in standard position coincides with
one of the rays of either the x-axis or the y-axis, the angle is said to be a
quadrantal angle.
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Quadrantal Angle
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Something to think about…
How many quadrantal angles
do we have?
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Did you know?There are infinitely many quandrantal angles and its
measure is an integral multiple of 90°.
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Positive and Negative AnglesAngles formed by a counter
clockwise direction are considered positive angles; angles formed by a clockwise direction are considered
negative angles.
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Positive and Negative Angles
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Degree Measure and Radian MeasureLecture 8.2:
SSMTH1: Precalculus
Science and Technology, Engineering and Mathematics (STEM) Strands
Mr. Migo M. Mendoza
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Our Goal:This section focuses on the degree measure and radian measure of an angle. This
section also includes the different angles and its measure.
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Measure of an AngleThe measure of an angle is
given by stating the amount of rotation to revolve from the
initial position of the ray to the terminal position.
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Units to Measure AnglesThe most commonly used unit to
measure angles are in terms of:
1. revolution;
2. degrees; and
3. radians.
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One RevolutionOne revolution is the amount of
rotation needed for one full turn of a ray about its endpoints in which the
initial and terminal sides of the angle coincides.
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One DegreeA degree is the measure of an angle formed by rotation a ray
1/360 of a complete revolution, the symbol for degree is °.
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Types of AnglesOne Revolution
Straight Angle
Right Angle
Acute Angle
Obtuse Angle
Reflex Angle
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Straight Angle
A straight angle is an angle of 180°, or ½
revolution.
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Right Angle
A right angle is an angle of 90°.
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Acute Angle
An acute angle is an angle if its measure is between 0° and 90°.
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Obtuse Angle
An obtuse angle is an angle if its measure is between 90° and 180°.
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Reflex Angle
A reflex angle is an angle if its measure is
between 180° and360°.
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Complementary AnglesIf the sum of the two measures of
two angles is 90° (or α + β = 90°) the angles are complementary
angles and one angle is the complement of the other.
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Supplementary AnglesIf the sum of the two measures of
two angles is 180° (or α + β = 180°) the angles are supplementary
angles and one angle is the supplement of the other.
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Take Note:
Complementary and Supplementary angles are ALWAYS positive.
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Complementary Angle
290
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Supplementary Angle
180
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Example 8.2.1:Find the complementary and
supplementary angles of:
6
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Final Answer:The complement is
.3
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Final Answer:The supplement is
.6
5
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Example 8.2.2:Find the complementary and
supplementary angles of:
72
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Final Answer:
The complement of 72°is
.18
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Final Answer:
The supplement of 72°is
.108
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Example 8.2.3:Find the complementary and
supplementary angles of:
3
2
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Final Answer:
Since it has no complement.
,23
2
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Final Answer:
The supplement is:
3
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Did you know?In geometry you learned to measure angles in degrees. In trigonometry, angles are also
measured in radians.
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Radian Measure To define this unit of measure, consider a
circle with center at the origin and radius equal to r, and let θ be an angle in standard
position. Let s be the length of the arc intercepted by the angle. If s = r, then the measure of angle is said to be equal to one
radian.
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Radian Measure
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Radian MeasureOne radian is equivalent to
the measure of a central angle θ that intercepts an arc
s = r of the circle.
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In Other Words…One radian is the measure of a central angle that intercepts
an arc S equal in length to the radius of the circle.
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Understanding Radian Measure
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Take Note:Radians have no units.
Therefore, when radians are being used it is customary that no units are indicated
for the angle.
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Take Note:Note that θ is taken in degrees if it is indicated; otherwise θ is in radian
measure.
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Did you know?There is a little more than 6
radius lengths that can be wrapped around one full
circle.
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Radians
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Did you know?If θ is the measure of an angle
in radians, then, s r and θ are related by the equation:
rsr
s
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SectorIf θ is an angle in standard position,
the region bounded by the initial side and the terminal side of θ,
together with the intercepted arc, is called a sector of a circle.
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Did you know? The area of the sector is proportional to the
area of the circle of the same radius. If A is the area of this sector, and θ is the measure
of the corresponding central angle in radians, then the ratio of the area of the
sector and the area of the circle is proportional to the ratio of θ to 2π radians,
that is,
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Formula for Finding the Area of a Sector
2
2
1rA
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Example 8.2.4. A central angle of measure
intercepts an arc of a circle whose radius is 8cm. Find
the length of the intercepted arc.
4
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Final Answer:
The length of the intercepted arc is
.2 scentimeter
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Example 8.2.5. Two points on the surface of the earth
are 5,530 miles apart. If the radius of the earth is approximately 3,950 miles,
find the measure of the central angle intercepted by the arc joining the two
points on the earth surface.
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Final Answer:The measure of the central angle intercepted by the arc
joining the two points on earth surface is 1.4 radians.
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Example 8.2.6.Find the area of a sector of a
circle of radius 4 cm which subtends an angle of
.3
2radians
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Final Answer:The area of a sector of a
circle is
.3
16 2cm
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Something to think about…How do we find the
relationship between the degree and the radian measures of an angle?
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The Relationship… To find the relationship between the degree and
the radian measures of an angle, observe that when the terminal ray makes one complete
revolution, the measure of the angle in degrees is 360°, while the length of the intercepted arc is
equal to the circumference of the circle, which has the value 2πr.
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Conversions between Degrees and Radians:
To convert degrees to radians, multiply the given measure by:
180
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Conversions between Degrees and Radians:
To convert radians to degrees, multiply the given measure by:
180
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Example 8.2.7:
Convert from radians to degrees:
6
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Final Answer:
The radians to degrees is:
30
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Example 8.2.8:
Convert from radians to degrees:
9
4
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Final Answer:
The radians to degrees is:
80
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Example 8.2.9:
Convert from radians to degrees:
12
25
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Final Answer:
The radians to degrees is:
375
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Example 8.2.10:
Convert from degrees to radians:
60
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Final Answer:The degrees to radians is:
3
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Example 8.2.11:
Convert from degrees to radians:
210
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Final Answer:The degrees to radians is:
6
7
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Example 8.2.12:
Convert from degrees to radians:
480
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Final Answer:The degrees to radians is:
3
8
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To sum it up…If the measure of an angle is
given without units, it is understood to be given in
terms of radians.
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Equivalence Between Degrees and Radian Measures of Some Special Angles
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Classroom Task 8.1:
Please answer "Let's Practice (LP)"
Number 32.
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Standard Angle and Coterminal AngleLecture 8.3:
SSMTH1: Precalculus
Science and Technology, Engineering and Mathematics (STEM) Strands
Mr. Migo M. Mendoza
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Recall:An angle is said to be in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with
the x-axis.
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Did you know? If the terminal side of an angle in standard
position is allowed to continue rotating after it has made one complete revolution, it
is possible to obtain angles with measures that are greater than 360° or 2π radians, as
well as angles with measures that are less than -360° or -2π radians.
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Coterminal AnglesCoterminal angles are
angles that have the same initial side and the same
terminal side.
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Coterminal Angles
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Something to think about…From the previous figure,
how many angles are coterminal? Give each of
their measures.
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Something to think about…How can we determine
an angle that is coterminal to a given
angle θ?
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Answer:We can find an angle that is coterminal to a given angle θby adding or subtracting 2πor one revolution (or 360°).
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Example 8.3.1:Find the coterminal angles and
sketch the graph:
3
7
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Final Answer:The coterminal angle is:
3
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Example 8.3.2:Find the coterminal angles and
sketch the graph:
4
5
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Final Answer:The coterminal angle is:
4
3
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Example 8.3.3:Find the coterminal angles and
sketch the graph:
6
7
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Final Answer:The coterminal angle is:
6
5
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Example 8.3.4:Find the coterminal angles and
sketch the graph:
85
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Final Answer:The coterminal angle is:
275
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Example 8.3.5:Find the value of the acute angle
which is coterminal with:
780
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Relationship Between the Measures of Two Coterminal Angles in Standard Position
In general, two angles in standard position are
coterminal if their measures differ by an integral multiple
of 360° or 2π radians.
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Final Answer:The given angle is coterminal
with the angle that measure
60
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Reference AngleLet θ be an angle in standard position,
and let be the terminal side of this angle. The measure of the acute angle which the terminal side makes
with the x-axis is called the reference angle of θ.
OP
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Reference AngleReference angle of an angle is
the smallest positive angle formed between the terminal
side and the x-axis.
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Something to think about…Which among the
following angles on the board are reference angle
of the given θ?
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Reference Angle
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Something to think about…
How can we find the reference angle to a
given angle θ?
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Answer:We can find the reference
angle to a given angle θby adding or subtracting
π or 180°.
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How to Compute the Measure of the Reference Angle of an Angle in Standard Position
Terminal Side In
Quadrant I
QuadrantII
QuadrantIII
QuadrantIV
Reference Angle (θ in
degrees)θ
Reference Angle (θ in
radians)θ
180
180 360
2
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Take Note: If this is not the case, we first subtract
the largest integral multiple of 2π or 360° from the measure of θ. If the
measure of θ is negative, replace it with a coterminal angle with a positive
measure.
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Take Note:When the reference angle is
negative take the absolute value, since a reference angle
is always positive.
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Example 8.3.6:Find the reference angle and
sketch the graph:
85
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Final Answer:Since 85° is an acute angle, thus
the reference angle is:
85
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Example 8.3.7:Find the reference angle and
sketch the graph:
1056
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Final Answer:Since a reference angle is always positive. Thus, the
reference angle is:
24
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Example 8.3.8:Find the reference angle and
sketch the graph:
3
14
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Final Answer:The reference angle is:
3
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Example 8.3.9:Find the reference angle and
sketch the graph:
405
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Final Answer:Since a reference angle is always positive. Thus, the
reference angle is:
45
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Example 8.3.10:Find the reference angle and
sketch the graph:
4
17
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Final Answer:The reference angle is
radians4
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Classroom Task 8.2:
Please answer "Let's Practice (LP)“
Number 33.