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Copyright 2007 Pearson Education, Inc. Slide 8-2

Trigonometric Functions And

Applications

.1 Angles and Their Measures

.2 Trigonometric Functions and FundamentalIdentities

.3 Evaluating Trigonometric Functions

.4 Applications of Right Triangles

.5 The Circular Functions

.6 Graphs of the Sine and Cosine Functions

.7 Graphs of the Other Circular Functions

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.1 Angles and Arcs

Basic Terminology

Two distinct pointsA andB determine the line AB.

The portion of the line including the pointsA andB is the

line segment AB.

The portion of the line that starts atA and continues

through B is called ray AB.

An angle is formed by rotating a ray, the initial side,

around its endpoint, the vertex, to a terminal side.

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Degree Measure

Developed by the Babylonians around 4000 yrs ago.

Divided the circumference of the circle into 360 parts.

One possible reason for this is because there are

approximately that number of days in a year.

There are 360 in one rotation.

An acute angle is an angle between 0 and 90.

A right angle is an angle that is exactly 90.

An obtuse angle is an angle that is greater than 90

but less than 180.

A straight angle is an angle that is exactly 180.

.1 Degree Measure

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.1 Finding Measures of Complementary

and Supplementary Angles

If the sum of two positive angles is 90, the angles are

called complementary.

If the sum of two positive angles is 180, the angles are

called supplementary.

Example Find the measure of each angle in the given figure.

(a) (b)

6 3 90

10

m m

m

!

!

o

o

4 6 180

18

k k

k

!

!

o

o

(Supplementary angles)(Complementary angles)

Angles are 60 and 30 degrees Angles are 72 and 108 degrees

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.1 Calculating With Degrees, Minutes,

and Seconds

One minute, written 1', is of a degree.

One second, written 1", is of a minute.

Example Perform the calculation

Solution

601

601

QQ106or1 60

1 !d!d

106or1 36001601 d!dd!!ddQ

'

.64329251 ddQQ

5783

6432

9251

d

d

d

Q

Q

Q

Since 75' = 1 + 15', the sum is written as 8415'.

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.1 Converting Between Decimal Degrees

and Degrees, Minutes, and Seconds

Example

(a) Convert 748d14t to decimal degrees.

(b) Convert 34.817 to degrees, minutes, and

seconds.

Analytic Solution

(a) Since

.137.740039.1333.74

360014

6087441874

QQQQ

QQ

QQ

!}

!ddd

,1and1 36001

601 QQ !dd!d

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.1 Converting Between Decimal Degrees

and Degrees, Minutes, and Seconds

(b)

Graphing Calculator Solution

2.194342.19434

)06(02.943420.9434

20.4934)06(817.34

817.34817.34

ddd!

ddd!

ddd!

dd!

d!

d!

!

Q

Q

Q

Q

Q

Q

QQQ

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.1 Coterminal Angles

Quadrantal Angles are

angles in standard

position (vertex at the

origin and initial side

along the positive x-

axis) with terminal sides

along the x ory axis,

i.e. 90, 180, 270, etc.

Coterminal Angles areangles that have the same

initial side and the same

terminal side.

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.1 Finding Measures of Coterminal

Angles

Example Find the angles of smallest possible positivemeasure coterminal with each angle.

(a) 908 (b) 75

Solution Add or subtract 360 as many times asneeded to get an angle between 0 and 360.

(a)

(b)

Let nbe an integer, we have an infinite number ofcoterminal angles: e.g. 60 + n 360.

QQQ 1883602908 !

QQQ

285)75(360 !

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.1 Radian Measure

The radian is a real number, where the degree is a unit of

measurement.

The circumference of a circle, given by C= 2Tr, where ris the radius of the circle, shows that an angle of 360 has

measure 2T radians.

An angle with its vertex at the

center of a circle that intercepts

an arc on the circle equal in

length to the radius of the circle

has a measure of1 radian.

radians180orradians2360 TT !!QQ

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.1 Converting Between Degrees and

Radians

Multiply a radian measure by 180/T and simplify

to convert to degrees. For example,

Multiply a degree measure by T /180 and simplify

to convert to radians. For example,

radians.4

radians180

4545 TT !

!Q

.405180

4

9

4

9 QQ

!

!

T

TT

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.1 Converting Between Degrees and

Radians With the Graphing Calculator

Example Convert 249.8 to radians.

Solution

Put the calculator in radian mode.

Example Convert 4.25 radians to degrees.

Solution

Put the calculator in degree mode.

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.1 Equivalent Angle Measures in

Degrees and Radians

Figure 18 pg 9

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.1 Arc Length

Example A circle has a radius of 25 inches. Find the

length of an arc intercepted by a central angle of 45.

Solution

The length s of the arc intercepted on a circle of radius rby a

central angle of measure U radians is given by the product of

the radius and the radian measure of the angle, or

s = rU, U in radians

inches25.64

25

radians4180

4545

TT

TT

!

!

!

!

s

Q

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.1 Linear and Angular Speed

Angular speed [(omega) measures the speed of

rotation and is defined by

Linear speed Y is defined by

Since the distance traveled along a circle is given

by the arc length s, we can rewrite Y as

.in timeradiansin, tt

UU

[ !

.in timedistancelinearis, tst

s!Y

.or, [YUU

Y rt

rt

r

t

s!

!!!

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.1 Finding Linear Speed and Distance

Traveled by a Satellite

Example A satellite traveling in a circular orbit 1600 km

above the surface of the Earth takes two hours to complete an

orbit. The radius of the Earth is 6400 km.

(a) Find the linear speed of the satellite.

(b) Find the distance traveled in 4.5 hours.

Figure 24 pg 12

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.1 Finding Linear Speed and Distance

Traveled by a Satellite

Solution

(a) The distance from the Earths center is

r= 1600 + 6400 = 8000 km.

For one orbit, U = 2T, so s = rU = 8000(2T) km. Witht= 2 hours, we have

(b) s = Yt= 8000T(4.5) } 110,000 km

kph.000,258000

2

)2(8000}!!! T

TY

t

s

sect 1 trig

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