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Circular Functions (Trigonometry)Chapter 8.1:

SSMTH1: Precalculus

Science and Technology, Engineering and Mathematics (STEM) Strands

Mr. Migo M. Mendoza

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

Basic ConceptsLecture 8.1:

SSMTH1: Precalculus


Mr. Migo M. Mendoza

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).

Did you know?If two rays are drawn

with a common endpoint, then they will

form an angle.

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.

Initial Side and Terminal Side

Take Note:The ray can travel an

unlimited number of times around the circle and still end in the same terminal position.

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.

Standard Angle

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.

Quadrantal Angle

Something to think about…

How many quadrantal angles

do we have?

Did you know?There are infinitely many quandrantal angles and its

measure is an integral multiple of 90°.

Positive and Negative AnglesAngles formed by a counter

clockwise direction are considered positive angles; angles formed by a clockwise direction are considered

negative angles.

Positive and Negative Angles

Degree Measure and Radian MeasureLecture 8.2:

SSMTH1: Precalculus


Mr. Migo M. Mendoza

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.

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.

Units to Measure AnglesThe most commonly used unit to

measure angles are in terms of:

1. revolution;

2. degrees; and

3. radians.

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.

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 °.

Types of AnglesOne Revolution

Straight Angle

Right Angle

Acute Angle

Obtuse Angle

Reflex Angle

Straight Angle

A straight angle is an angle of 180°, or ½

revolution.

Right Angle

A right angle is an angle of 90°.

Acute Angle

An acute angle is an angle if its measure is between 0° and 90°.

Obtuse Angle

An obtuse angle is an angle if its measure is between 90° and 180°.

Reflex Angle

A reflex angle is an angle if its measure is

between 180° and360°.

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.

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.

Take Note:

Complementary and Supplementary angles are ALWAYS positive.

Complementary Angle

290

Supplementary Angle

180

Example 8.2.1:Find the complementary and

supplementary angles of:

6

Final Answer:The complement is

.3

Final Answer:The supplement is

.6

5



72

Final Answer:

The complement of 72°is

.18

Final Answer:

The supplement of 72°is

.108



3

2

Final Answer:

Since it has no complement.

,23

2

Final Answer:

The supplement is:

3

Did you know?In geometry you learned to measure angles in degrees. In trigonometry, angles are also

measured in radians.

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.

Radian Measure

Radian MeasureOne radian is equivalent to

the measure of a central angle θ that intercepts an arc

s = r of the circle.

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.

Understanding Radian Measure

Take Note:Radians have no units.

Therefore, when radians are being used it is customary that no units are indicated

for the angle.

Take Note:Note that θ is taken in degrees if it is indicated; otherwise θ is in radian

measure.

Did you know?There is a little more than 6

radius lengths that can be wrapped around one full

circle.

Radians

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

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.

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,

Formula for Finding the Area of a Sector

2

2

1rA

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

Final Answer:

The length of the intercepted arc is

.2 scentimeter

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.

Final Answer:The measure of the central angle intercepted by the arc

joining the two points on earth surface is 1.4 radians.

Example 8.2.6.Find the area of a sector of a

circle of radius 4 cm which subtends an angle of

.3

2radians

Final Answer:The area of a sector of a

circle is

.3

16 2cm

Something to think about…How do we find the

relationship between the degree and the radian measures of an angle?

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.

Conversions between Degrees and Radians:

To convert degrees to radians, multiply the given measure by:

180

Conversions between Degrees and Radians:

To convert radians to degrees, multiply the given measure by:

180

Example 8.2.7:

Convert from radians to degrees:

6

Final Answer:

The radians to degrees is:

30

Example 8.2.8:


9

4

Final Answer:


80

Example 8.2.9:


12

25

Final Answer:


375

Example 8.2.10:

Convert from degrees to radians:

60

Final Answer:The degrees to radians is:

3

Example 8.2.11:


210


6

7

Example 8.2.12:


480


3

8

To sum it up…If the measure of an angle is

given without units, it is understood to be given in

terms of radians.

Equivalence Between Degrees and Radian Measures of Some Special Angles

Classroom Task 8.1:

Please answer "Let's Practice (LP)"

Number 32.

Standard Angle and Coterminal AngleLecture 8.3:

SSMTH1: Precalculus


Mr. Migo M. Mendoza

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.

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.

Coterminal AnglesCoterminal angles are

angles that have the same initial side and the same

terminal side.

Coterminal Angles

Something to think about…From the previous figure,

how many angles are coterminal? Give each of

their measures.

Something to think about…How can we determine

an angle that is coterminal to a given

angle θ?

Answer:We can find an angle that is coterminal to a given angle θby adding or subtracting 2πor one revolution (or 360°).

Example 8.3.1:Find the coterminal angles and

sketch the graph:

3

7

Final Answer:The coterminal angle is:

3


sketch the graph:

4

5


4

3


sketch the graph:

6

7


6

5


sketch the graph:

85


275

Example 8.3.5:Find the value of the acute angle

which is coterminal with:

780

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.

Final Answer:The given angle is coterminal

with the angle that measure

60

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

Reference AngleReference angle of an angle is

the smallest positive angle formed between the terminal

side and the x-axis.

Something to think about…Which among the

following angles on the board are reference angle

of the given θ?

Reference Angle

Something to think about…

How can we find the reference angle to a

given angle θ?

Answer:We can find the reference

angle to a given angle θby adding or subtracting

π or 180°.

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

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.

Take Note:When the reference angle is

negative take the absolute value, since a reference angle

is always positive.

Example 8.3.6:Find the reference angle and

sketch the graph:

85

Final Answer:Since 85° is an acute angle, thus

the reference angle is:

85


sketch the graph:

1056

Final Answer:Since a reference angle is always positive. Thus, the

reference angle is:

24


sketch the graph:

3

14

Final Answer:The reference angle is:

3


sketch the graph:

405

Final Answer:Since a reference angle is always positive. Thus, the

reference angle is:

45


sketch the graph:

4

17

Final Answer:The reference angle is

radians4

Classroom Task 8.2:

Please answer "Let's Practice (LP)“

Number 33.

Documents

Chapter 1: Circles - migomendoza.weebly.com · Complementary Angles ... To find the relationship between the degree and the radian measures of an angle, ... Between the Measures of