copyright © 2005 pearson education, inc. chapter 1 trigonometric functions

Copyright © 2005 Pearson Education, Inc.

Chapter 1

Trigonometric Functions


1.1

Angles

Copyright © 2005 Pearson Education, Inc. Slide 1-3

Basic Terms

Two distinct points determine a line called line AB.

Line segment AB—a portion of the line between A and B, including points A and B.

Ray AB—portion of line AB that starts at A and continues through B, and on past B.

A B

A B

A B


Basic Terms continued

Angle-formed by rotating a ray around its endpoint.

The ray in its initial position is called the initial side of the angle.

The ray in its location after the rotation is the terminal side of the angle.


Naming Angles Unless it is ambiguous as to the meaning, angles may

be named only by a single letter (English or Greek) displayed at vertex or in area of rotation between initial and terminal sides

Angles may also be named by three letters, one representing a point on the initial side, one representing the vertex and one representing a point on the terminal side (vertex letter in the middle, others first or last)

B

c

:Names AcceptableAangle

ABangleC ACangle B

angle


Basic Terms continued

Positive angle: The rotation of the terminal side of an angle counterclockwise.

Negative angle: The rotation of the terminal side is clockwise.


Angle Measures and Types of Angles

The most common unit for measuring angles is the degree. (One rotation = 360o)

¼ rotation = 90o, ½ rotation = 180o, Angle and measure of angle not the same, but it

is common to say that an angle = its measure Types of angles named on basis of measure:

oo 900 oo 18090 o90 o180

01 rotation 3601


Complementary and Supplementary Angles

Two positive angles are called complementary if the sum of their measures is 90o

The angle that is complementary to 43o = Two positive angles are called supplementary if

the sum of their measures is 180o

The angle that is supplementary to 68o =

o47

o112


Example: Complementary Angles

Find the measure of each angle. Since the two angles form a right

angle, they are complementary angles. Thus,

k 16

k +20

The two angles have measures of:

43 + 20 = 63 and 43 16 = 27

1620 kk 90

9042 k

862 k

43k


Example: Supplementary Angles

Find the measure of each angle. Since the two angles form a straight

angle, they are supplementary angles. Thus,

6x + 7 3x + 2

These angle measures are:

6(19) + 7 = 121 and 3(19) + 2 = 59

2376 xx 180

18099 x

1719 x

19x


Portions of Degree: Minutes, Seconds

One minute, 1’, is 1/60 of a degree.

One second, 1”, is 1/60 of a minute.

or 1

1' 60' 160

00

13600"or 1'60"or 3600

1

60

'1"1


Example: Calculations

Perform the calculation.

Since 86 = 60 + 26, the sum is written:

Perform the calculation.

Hint write:

27 34' 26 52'

72 15 18'

27 34'

26 52'

86'53

53

1 26'

54 26'

71 60

15 18'

56 42'

72 as 71 60'


Converting Between Degrees, Minutes and Seconds and Decimal Degrees

Convert Convert 34.62474 12' 18"

1818"

360074 74

74 .2 .00

1212'

6

5

7

0

4.205

34.624 34 .624

34 .624(60')

34 37.44'

34 37 ' .44'

34 37 ' .44(60")

34 37 ' 26.4"

34 37 ' 26.4"

:degree a of fractions as

seconds and minutes Write

:seconds tominutes

fractional and minutes to

degrees fractional Change


Standard Position

An angle is in standard position if its vertex is at the origin and its initial side is along the positive x-axis.


Quadrantal Angles

Angles in standard position having their terminal sides along the x-axis or y-axis, such as angles with measures 90, 180, 270, and so on, are called quadrantal angles.

:Measure 0360


Coterminal Angles

A complete rotation of a ray results in an angle measuring 360. Given angle A, and continuing the rotation by a multiple of 360 will result in a different angle, A + n360,with the same terminal side: coterminal angles.


Example: Coterminal Angles

Find the angles of smallest possible positive measure coterminal with each angle.

a) 1115 b) 187 Add or subtract 360 as may times as needed to

obtain an angle with measure greater than 0 but less than 360.

a) b) 7553601115 395360755 35360395

035

173360187

0173


Homework

1.1 Page 6 All: 6 – 9, 14 – 17, 24 – 29, 32 – 35, 38 – 41,

46 – 51, 55 – 58 , 75 – 79 MyMathLab Assignment 1 for practice

MyMathLab Homework Quiz 1 will be due for a grade on the date of our next class meeting!!!


1.2

Angle Relationships and Similar Triangles


Vertical Angles

When lines intersect, angles opposite each other are called vertical angles

Vertical angles in this picture:

How do measures of vertical angles compare?

Vertical Angles have equal measures.

M

QR

PN

:and NMP :and QMNRMQ RMP


Parallel Lines

Parallel lines are lines that lie in the same plane and do not intersect.

When a line q intersects two parallel lines, q, is called a transversal.

m

n

parallel lines

qTransversal


Angles and Relationships

m

n

q

Angle measures are equal.2 & 6, 1 & 5, 3 & 7, 4 & 8

Corresponding angles

Angle measures add to 180.4 and 6

3 and 5

Interior angles on the same side of the transversal

Angle measures are equal.1 and 8

2 and 7

Alternate exterior angles

Angles measures are equal.4 and 5

3 and 6

Alternate interior angles

RuleAnglesName

:ipsrelationsh and names following

with theangleseight forms lines

parallel ngintersecti salA transver

Interior

Exterior

Exterior


Example: Finding Angle Measures

Find the measure of each marked angle, given that lines m and n are parallel.

What is the relationship between these angles?

Alternate exterior with equal measures

Measure of each angle? One angle has measure

6x + 4 = 6(21) + 4 = 130 and the other has measure

10x 80 = 10(21) 80 = 130

m

n(10x 80)

(6x + 4)

6 4 10 80

84 4

21

x x

x

x

Equation?


Angle Sum of a Triangle

The instructor will ask specified students to draw three triangles of distinctly different shapes. All the angles will be cut off each triangle and placed side by side with vertices touching.

What do you notice when you sum the three angles?

The sum of the measures of the angles of any triangle is 180.

angle)(straight linestraight a isresult The


Example: Applying the Angle Sum

The measures of two of the angles of a triangle are 52 and 65. Find the measure of the third angle, x.

Solution?

The third angle of the triangle measures 63.

52

65

x

52 65

11

180

7 180

63

x

x

x


Types of Triangles: Named Based on Angles


Types of Triangles: Named Based on Sides


Similar and Congruent Triangles

Triangles that have exactly the same shape, but not necessarily the same size are similar triangles

Triangles that have exactly the same shape and the same size are called congruent triangles

A

B

H

G

C

K

D

L

E F

M N


Conditions for Similar Triangles

Corresponding angles must have the same measure.

Corresponding sides must be proportional. (That is, their ratios must be equal.)

A

B C

D

E F

FCEBDA , ,

DF

AC

EF

BC

DE

AB


Example: Finding Angle Measures on Similar Triangles

Triangles ABC and DEF are similar. Find the measures of angles D and E.

Since the triangles are similar, corresponding angles have the same measure.

Angle D corresponds to angle:

Measure of D: Angle E corresponds to

angle: Measure of E:

A

C B

F E

D

35

112 33

112

A

Bo33

o35


Example: Finding Side Lengths on Similar Triangles

Triangles ABC and DEF are similar. Find the lengths of the unknown sides in triangle DEF.

To find side DE:

To find side FE:A

C B

F E

D

35

112 33

112

32

48

64

16

64

32 1024

3

32

6

2

1

xx

x

48

32

3

768

2

6

2

1

4

xx

x

:unknown one with sides ingcorrespond involving proportion a Write

32

24


Example: Application of Similar Triangles

A lighthouse casts a shadow 64 m long. At the same time, the shadow cast by a mailbox 3 m high is 4 m long. Find the height of the lighthouse.

The two triangles are similar, so corresponding sides are in proportion, so:

The lighthouse is 48 m high.

3

4 644 192

48

x

x

x

64

4

3

x


Homework

1.2 Page 14 All: 3 – 7, 9 – 13, 16 – 19, 25 – 36, 41 – 44,

46 – 49, 51 – 54, 57 – 60, 65 – 66, 69 – 70 MyMathLab Assignment 2 for practice

MyMathLab Homework Quiz 2 will be due for a

grade on the date of our next class meeting!!!


1.3



Trigonometric Functions Compared with Algebraic Functions

Algebraic functions are sets of ordered pairs of real numbers such that every first member, “x”, is paired with exactly one second member, “y”

Trigonometric functions are sets of ordered pairs such that every first member, an angle, is paired with exactly one second member, a ratio of real numbers

Algebraic functions are given names like f, g or h and in function notation, the second member that is paired with “x” is shown as f(x), g(x) or h(x)

Trigonometric functions are given the names, sine, cosine, tangent, cotangent, secant, or cosecant, and in function notation, the second member that is paired with the angle “A” is shown as sin(A), cos(A), tan(A), cot(A), sec(A), or csc(A) – (sometimes parentheses are omitted)



Let (x, y) be a point other the origin on the terminal side of an angle in standard position.

The distance, r, from the point to the origin is:

The six trigonometric functions of are defined as:

2 2 .r x y

sin cos tan ( 0)y x y

xr r x

csc ( 0) sec ( 0) cot ( 0)r r x

y x yy x y

yx,

r


Values of Trig Functions Independent of Point Chosen

For the given angle, if point (x1,y1) is picked and r1 is calculated, trig functions of that angle will be ratios of the sides of the triangle shown in blue.

For the same angle, if point (x2,y2) is picked and r2 is calculated, trig functions of the angle will be ratios of the triangle shown in green

Since the triangles are similar, ratios and trig function values will be exactly the same

11, yx

1r2r

22 , yx

1x2x

2y1y


Example: Finding Function Values

The terminal side of angle in standard position passes through the point (12, 16). Find the values of the six trigonometric functions of angle .

(12, 16)

16

12

2 2 2 216

1 244 256 0

1

0

2

40

r x y

:sdefinition use then and r""

find given, arey and x :Note


Example: Finding Function Values continued x = 12 y = 16 r = 20

4sin

53

cos54

ta

2

16

1

1

6

12

0

n3

2

20

y

rx

ry

x

5csc

4

5sec

33

co

1

121

20

20

6

2

416t

r

y

r

xx

y

:is functions trig theofeach of value thes,definition Using


Trigonometric Functions of Coterminal Angles

Note: To calculate trigonometric functions of an angle in standard position it is only necessary to know one point on the terminal side of that angle, and its distance from the origin

In the previous example six trig functions of the given angle were calculated. All angles coterminal with that angle will have identical trig function values

ALL COTERMINAL ANGLES HAVE IDENTICAL TRIGONOMETRIC FUNCTION VALUES!!!!


Equations of Rays with Endpoint at Origin:

Recall from algebra that the equation of a line is:

If a line goes through the origin its equation is:

To get the equation of a ray with endpoint at the origin we write an equation of this form with the restriction that:

intercept-y is and slope is where bmbmxy

tscoefficien their andy andonly x involvingequation any :or mxy

0or 0either xxrayleft rayright


Example: Finding Function Values

Find the six trigonometric function values of the angle in standard position, if the terminal side of is defined byx + 2y = 0, x 0.

We can use any point on the terminal side of to find the trigonometric function values.

512 22 r

1y:y calculate 0, xChoose

:r"" Calculate

2x


Example: Finding Function Values continued From previous

calculations:

Use the definitions of the trig functions:

1 1 5 5

55 5 5

2 2 5 2 5

55 5 51

52

sin

cos

tan csc

sec cot5

22

y

r

x

r

y r

x y

r x

x y

5 ,1 ,2 ryx


Finding Trigonometric Functions of Quadrantal Angles

A point on the terminal side of a quadrantal angle always has either x = 0 or y = 0 (x = 0 when terminal side is on y axis, y = 0 when terminal side is on x axis)

Since any point on the terminal side can be picked, choose x = 0 or y = 0, as appropriate, and choose r = 1

The remaining x or y will then be 1 or -1

0 ,1

1 ,0

1,0

0 ,1

1r


Example: Function Values Quadrantal Angles Find the values of the six trigonometric functions for an angle

of 270. Which point should be used on the terminal side of a 270

angle? We choose (0, 1). Here x = 0, y = 1 and r = 1. Value of the six trig functions for this angle:

1 0sin 270 1 cos270 0

1 11 1

tan 270 undefined csc270 10 1

1 0sec270 undefined cot 270 0

0 1


Undefined Function Values

If the terminal side of a quadrantal angle lies along the y-axis, then, because x = 0, the tangent and secant functions are undefined:

If it lies along the x-axis, then, because y = 0, the cotangent and cosecant functions are undefined.

x

r

x

y sec and tan

y

r

y

x csc and cot


Commonly Used Function Values

undefined1undefined010360

1undefined0undefined01270


1undefined0undefined0190


csc sec cot tan cos sin

memorize tonecessary not - calculatedquickly becan These


Finding Trigonometric Functions of Specific Angles

Until discussing trigonometric functions of specific quadrantal angles such as 90o, 180o, etc., we have found trigonometric functions of angles by knowing or finding some point on the terminal side of the angle without knowing the measure of the angle

At the present time, we know how to find exact trigonometric values of specific angles only if they are quadrantal angles

In the next chapter we will learn to find exact trigonometric values of 30o, 45o, and 60o angles

In the meantime, we can find approximate trigonometric values of specific angles by using a scientific calculator set in degree mode


Finding Approximate Trigonometric Function Values of Sine, Cosine and Tangent

Make sure your calculator is set in degree mode Depending on your calculator,

Enter the angle measure first then press the appropriate sin, cos or tan key to get the value

Press the sin, cos, or tan key first, then enter the angle measure

Practice on these:

o

o

o

30cos

60tan

270sin

866025403.0

732050808.1

1

chapter.next in the calculator theusingabout morelearn willWe


Exponential Notation and Trigonometric Functions

A trigonometric function defines a real number ratio for a specific angle, for example “sin A” is the real number ratio assigned by the sine function to the angle “A”

Since “sin A” is a real number it can be raised to any rational number power, such as “2” in which case we would have “(sin A)2”

However, this value is more commonly written as “sin2 A”

sin2 A = (sin A)2

Using this reasoning then if “tan A = 3”, then:

tan4 A = 8134


Homework

1.3 Page 24 All: 5 – 8, 17 – 28, 33 – 40 MyMathLab Assignment 3 for practice



1.4

Using Definitions of the Trigonometric Functions


Identities

Recall from algebra that an identity is an equation that is true for all values of the variable for which the expression is defined

Examples:

6232 xx xof valuesallfor trueis and x of valuesallfor defined is Expression

62

31

2

xx xof esother valu allfor trueisbut 0,for x definednot is Expression


Relationships Between Trigonometric Functions

In reviewing the definitions of the six trigonometric functions what relationship do you observe between each function and the one directly beneath it?

They are reciprocals of each other


xr r x

csc ( 0) sec ( 0) cot ( 0)r r x

y x yy x y


Reciprocal Identities

This relationship can be summarized:

Each identity is true for angles except those that that make a denominator equal to zero

These reciprocal identities must be memorized

1 1 1sin cos tan

csc sec cot

1 1 1csc sec cot

sin cos tan


Example: Find each function value.

cos if sec =

Since cos is the reciprocal of sec :

sin if csc 15

3

153

1sin

3

15

3 15

15 15

3 15 15

15 5

2

3

3

2

231

sec

1cos


Signs of Trig Functions by Quadrant of Angle

Considering the following three functions and the sign of x, y and r in each quadrant, which functions are positive in each quadrant?


xr r x

r

y

x

r

y

x

r

y

x

r

y

x

all

costan

sin

r y, x,of Signs Functions Trig Positive


Signs of Other Trig Functions by Quadrant of Angle

Reciprocal functions will always have the same sign

All functions have positive values for angles in Quadrant I

Sine and Cosecant have positive values for angles in Quadrant II

Tangent and Cotangent have positive values for angles in Quadrant III

Cosine and Secant have positive values for angles in Quadrant IV


Memorizing Signs of Trig Functions by Quadrant

It will help to memorize by learning these words in Quadrants I - IV:

“All students take calculus”

And remembering reciprocal identities

Trig functions are negative in quadrants where they are not positive

all

(sec) cos(cot)tan

(csc)sin all

calculustake

students


Example: Identify Quadrant

Identify the quadrant (or quadrants) of any angle that satisfies tan > 0, sin < 0.

tan > 0 in quadrants:I and III

sin < 0 in quadrants:III and IV

so, the answer satisfying both is quadrant:III


Domain and Range of Sine Function

Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, sin A = y/r

Domain of sine function is the set of all A for which y/r is a real number. Since r can’t be zero, y/r is always a real number and domain is “any angle”

Range of sine function is the set of all y/r, but since y is less than or equal to r, this ratio will always be equal to 1 or will be a proper fraction, positive or negative:

1Asin1

xyr yx,


Domain and Range of Cosine Function

Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, cos A = x/r

Domain of cosine function is the set of all A for which x/r is a real number. Since r can’t be zero, x/r is always a real number and domain is “any angle”

Range of cosine function is the set of all x/r, but since x is less than or equal to r, this ratio will always be equal to 1, -1 or will be a proper fraction, positive or negative:

1Acos1

xyr yx,


Domain and Range of Sine & Cosine

What relationship do you notice between the domain and range of the sine and cosine functions?

They are exactly the same:

Domain:

Range:

AngleAny

1 ,1


Domain and Range of Tangent Function

Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, tan A = y/x

Domain of tangent function is the set of all A for which y/x is a real number. Tangent will be undefined when x = 0, therefore domain is all angles except for odd multiples of 90o

Range of tangent function is the set of all y/x, but since all of these are possible: x=y, x<y, x>y, this ratio can be any positive or negative real number: Atan

xyr yx,


Domain and Range of Cosecant Function

Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, csc A = r/y

Domain of cosecant function is the set of all A for which r/y is a real number. Cosecant will be undefined when y = 0, therefore domain is all angles except for integer multiples of 180o

Range of cosecant function is the reciprocal of the range of the sine function. Reciprocals of numbers between -1 and 1 are:

A csc1or 1Acsc

xyr yx,


Domain and Range of Secant Function

Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, sec A = r/x

Domain of secant function is the set of all A for which r/x is a real number. Secant will be undefined when x = 0, therefore domain is all angles except for odd multiples of 90o

Range of secant function is the reciprocal of the range of the cosine function. Reciprocals of numbers between -1 and 1 are:

A sec1or 1Asec

xyr yx,


Domain and Range of Cotangent Function

Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, cot A = x/y

Domain of cotangent function is the set of all A for which x/y is a real number. Cotangent will be undefined when y = 0, therefore domain is all angles except for integer multiples of 180o

Range of cotangent function is the reciprocal of the range of the tangent function. The reciprocal of the set of numbers between negative infinity and positive infinity is:

Acot

xyr yx,


Ranges of Trigonometric Functions

For any angle for which the indicated functions exist:

1 sin 1 and 1 cos 1 tan and cot can equal any real number; sec 1 or sec 1 csc 1 or csc 1.

(Notice that sec and csc are never between 1 and 1.)


Deciding Whether a Value is in the Range of a Trigonometric Function

Tell which of the following is in the range of the trig function:sin A = 1.332cos A = ¼ tan A = 1,998,214sec A = ½csc A = 0.2485cot A = 0sin A = - 0.3359cos A = -3tan A = -3 Yes

YesYesNo

YesYes

No

No

No


Development of Pythagorean Identities

For every point (x,y) on the terminal side of an angle A at a distance of r > 0 from the origin, we have the following relationship based on the Pythagorean Theorem:

Dividing both sides by r2 gives:

xyr yx,

12

2

2

2

r

y

r

x

222 ryx

:Equation Trig toEquivalent

A

1AsinAcos 22




Dividing both sides by x2 gives:

xyr yx,

2

2

2

2

1x

r

x

y

222 ryx


A

AsecAtan1 22




Dividing both sides by y2 gives:

xyr yx,

2

2

2

2

1y

r

y

x

222 ryx


A

Acsc1Acot 22


Pythagorean Identities

MUST MEMORIZE!!!

2 2

2 2

2 2

sin cos 1,

tan 1 sec ,

1 cot csc


Development of Quotient Identities

Based on x, y, r definitions of sine and cosine functions:

Acos

Asin x

r

r

y

rxry

x

yAtan

AtanAcos

Asin


Development of Quotient Identities

Based on x, y, r definitions of sine and cosine functions:

A sin

A cos y

r

r

x

ryrx

y

xAcot

AcotAsin

Acos


Quotient Identities

MUST MEMORIZE!!!

sintan

cos

coscot

sin


Using Identities to Find Missing Function Values

Given the quadrant of the angle and the value of one trig function, the other five trig function values can be found using various identities

Examples that follow will illustrate the approach


Example: Other Function Values

Find sin and cos given that tan = 4/3 and is in quadrant III.

Since is in quadrant III, sin and cos will both be negative.

sin and cos must be in the interval [1, 1].

3? cos and 4sin say that tocos

sintan

:say oidentity tquotient theuse t wecan'Why


Example: Other Function Values continued There is no identity that directly gives sin or cos from tan, but which

one will give a reciprocal of sin or cos from tan? 2 2tan 1 sec 2 2

22

2

2

tan 1 sec

1 s

3

ec

161 sec

925

sec95

sec3

cos

4

5

3

2 2

22

2

2

4

5

Since sin 1 cos ,

sin 1

9sin 1

2516

sin25

sin

3

5

cos? fromsin give illidentity w what Now,

negative?Why

IIIQuadrant negative?Why IIIQuadrant

.identities reciprocal with found becan aluesfunction v Trig 3Other :Note


Solving Trigonometric Equations

In algebra there are many types of equations that involve a variable that are either true or false depending on the value of the variable

This equation is true only if x = 10, so we say that 10 is the solution to the equation

In trig we likewise have many types of equations that involve a variable representing an unknown angle that are true or false depending on the value of the variable

In this course we will develop methods for solving various types of trigonometric equations

50csc

1102sin

73 x


Using Identities to Find a Value of an Angle that Solves a Trigonometric Equation

Given a trigonometric equation with an unknown angle, one solution (not all) can be found by using identities to convert both sides to the same trig function and then setting the unknown angles equal to each other as shown in the following example:


Find One Solution:

50csc

1102sin

:sideright on theidentity reciprocal a Use

50sin102sin

:right on the one theas same theisleft on the angle when the

is way one true,becan thisother ways are hereAlthough t

50102 40

equations tric trigonome tosolutions all findingfor methods develop will wecourse in theLater


Homework

1.4 Page 33 All: 3 – 6, 9 – 10, 15 – 18, 21 – 24, 27 – 40,

47 – 54, 56 – 61, 65 – 70

MyMathLab Assignment 4 for practice


copyright © 2005 pearson education, inc. chapter 1 trigonometric functions

Documents

b slide

pearson education

complementary angles

supplementary angles

measure of angle

angle measures

positive angles

trigonometric functions