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Angles and their

Measurement Algebra and TrigonometryMs. Sharon P. LubagCOS100D

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The exact origins of trigonometry are lostin prehistory.Early man became interested inastronomy for several reasons, for itsrelation to religion (and astronomy), topredict the seasons and planting time,and as an aid to navigation andgeography.

The mathematics that people developedto describe their observations inastronomy formed the beginnings of

trigonometry.

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Hipparchus - founder of trigonometry forproducing the very first known table of chords in Greece in about 140 BC (12 books).

In Greek, the word trigonometry came from the

two words

t r i gonon

meaning triangle and metron meaning measure. Thus,trigonometry means measurement of triangles.

The term trigonometry first appeared as thetitle of a book Tr igonomet r i a by B. Pitiacus published in 1595.

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Definition:

An angle AOB consists of two rays R 1 (initial side) and R 2(terminal side) with

a common vertex O.

Terminal side

Initial side

B

AO

O

Terminal

side

Initial side

A

B

R1

R2

Positive Angle Negative Angle

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Definition

The measure of an angle is the amountof rotation about the vertex required to

move R 1 onto R 2.

Note: Measurements for angles Degree Radian

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Note

An angle measure of 1 degree is formedby rotating the initial side of a

complete revolution.

Radian measure is the amount an angleopens measured along the arc of acircle of radius 1 with its center at the

vertex of the angle.

3601

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Definition

If a circle of radius 1 is drawn with thevertex of an angle at its center, then the

measure of this angle in radians (abbreviated as rad ) is the length of thearc that subtends the angle.

1

RadianMeasure of

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Relationshipo betweenDegrees and Radians

radradrad180

1180

1180 0

00

To convert radians to degrees,multiply by /180 0

To convert degrees to radian,multiply by 180 0/ Note: 1 rad 57.296 0

10 0.01745 rad

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Example

Express the following angles in degreesto radians (v.v)

1. 60 0

2. /3

Note: When no unit is given, the angle measureis assumed to be measured in radians.

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Definition

An angle is in standard position if it isdrawn in the xy-plane with its vertex at

the origin and its initial side on thepositive x-axis.

An angle is called quadrantal angle ifthe angle is in standard position and itsterminal side lies on an axis.

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Definition

Two angles in standard position arecoterminal if their terminal sides coincide.

Example1: Find angles that arecoterminal with (a) 30 0 and (b) /3 in

standard position.Example2: Find smallest positive anglethat is coterminal with (a) -3 /4 and (b)

19 /6.

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Length of a Circular Arc

An angle whose radian measure is issubtended by an arc that is the fraction /2 of the circumference of a circle. Thus, in acircle of radius r, the length S of an arc thatsubtends the angle is

)circleaof ncecircumfere(S 2

)r(S 22

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Example

Find the length of an arc of a circle withradius 10m that subtends a central angle

of 300

.

A central angle in a circle of radius 4mis subtended by an arc of length 6m.Find the measure of in radians.

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Area of a Circular Sector

In a circle of radius r, the area A of a sectorwith central angle of radians is

2

2

1 r A

Find the area of a sector of a circle with

central angle 600

if the measure of the cirlceis 3m.Find (a) arc length and (b) area of the sectorof the circle having (1) r=9in and =2 /3 and

(2) r=6cm and =135 0.

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Application

Two points A and B on the surface of theearth are on the same arc which is a

meridian having center at C, where C isthe center of the earth. If A has latitude10 0N and B has latitude 4.6 0S, what is

the distance between A and B?If the minute hand of a clock has lengthof 6in, how far does its tip travel in 18

minutes?