Download Chapter 1 Trigonometric Functions

Chapter 1
Trigonometric Functions
Copyright © 2005 Pearson Education, Inc.
1.1
Angles
Copyright © 2005 Pearson Education, Inc.
Basic Terms

Two distinct points determine a line called
line AB.
A

B
Line segment AB—a portion of the line
between A and B, including points A and B.
A

B
Ray AB—portion of line AB that starts at A and
continues through B, and on past B.
A
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B
Slide 1-3
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.
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Slide 1-4
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
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Acceptable Names :
angle A
angle 
angle CAB
angle BAC
Slide 1-5
Basic Terms continued

Positive angle: The
rotation of the terminal
side of an angle
counterclockwise.
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
Negative angle: The
rotation of the terminal
side is clockwise.
Slide 1-6
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, 1 360 rotation  10
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:
0o    90o
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  90 o
90o    180o
  180o
Slide 1-7
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 = 47 o
Two positive angles are called supplementary if
the sum of their measures is 180o
The angle that is supplementary to 68o = 112o
Copyright © 2005 Pearson Education, Inc.
Slide 1-8
Example: Complementary Angles


Find the measure of each angle.
Since the two angles form a right
angle, they are complementary
angles. Thus,
k  20  k 16  90
2k  4  90
2k  86
k  43
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k +20
k  16
The two angles have measures of:
43 + 20 = 63 and 43  16 = 27
Slide 1-9
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  180
9x  9  180
6x + 7
3x + 2
9x  171
x  19
These angle measures are:
6(19) + 7 = 121 and 3(19) + 2 = 59
Copyright © 2005 Pearson Education, Inc.
Slide 1-10
Portions of Degree: Minutes, Seconds

One minute, 1’, is 1/60 of a degree.
1
1' 
60

or
60'  1
One second, 1”, is 1/60 of a minute.
1'
10
1" 

or 60"  1' or 3600"  10
60 3600
Copyright © 2005 Pearson Education, Inc.
Slide 1-11
Example: Calculations

Perform the calculation.
27 34' 26 52'

Perform the calculation.
72  15 18'
27 34'

 26 52'
53 86'

Hint write: 72 as 71 60'
71 60
Since 86 = 60 + 26, the
sum is written: 53
15 18'
 1 26'
56 42'
54 26'
Copyright © 2005 Pearson Education, Inc.
Slide 1-12
Converting Between Degrees, Minutes and
Seconds and Decimal Degrees

Convert
74 12' 18"
Write minutes and seconds
as fractions of a degree :

Convert 34.624
Change fractional degrees
to minutes and fractional
minutes to seconds :
12
18

60 3600
 74  .2  .005
74 12' 18"  74 
 74.205
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"
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Slide 1-13
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.


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Slide 1-14
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 :  3600
Copyright © 2005 Pearson Education, Inc.
Slide 1-15
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.
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Slide 1-16
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) 1115  360  755
b) 187  360  173
755  360  395
0
173
395  360  35
35
Copyright © 2005 Pearson Education, Inc.
0
Slide 1-17
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!!!
Copyright © 2005 Pearson Education, Inc.
Slide 1-18
1.2
Angle Relationships and
Similar Triangles
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Vertical Angles

When lines intersect, angles opposite each other are
called vertical angles
Q
R
M
N

Vertical angles in this picture:
NMP and : RMQ

P
QMN and : RMP
How do measures of vertical angles compare?
Vertical Angles have equal measures.
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Slide 1-20
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.
Transversal
q
m
parallel lines
n
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Slide 1-21
Angles and Relationships
A transver sal intersecti ng parallel
lines forms eight angles with the
q
Exterior
m
Interior
following names and relationsh ips :
n
Exterior
Name
Angles
Rule
Alternate interior angles
4 and 5
3 and 6
Angles measures are equal.
Alternate exterior angles
1 and 8
2 and 7
Angle measures are equal.
Interior angles on the same
side of the transversal
4 and 6
3 and 5
Angle measures add to 180.
Corresponding angles
2 & 6, 1 & 5,
3 & 7, 4 & 8
Angle measures are equal.
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Slide 1-22
Example: Finding Angle Measures
Equation?

Find the measure of each
marked angle, given that lines
m and n are parallel.
(6x + 4)

21  x
m

(10x  80)

n
What is the relationship
between these angles?
Alternate exterior with equal
measures
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6 x  4  10 x  80
84  4 x


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
Slide 1-23
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 result is a straight line (straight angle)

The sum of the measures of the angles
of any triangle is 180.
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Slide 1-24
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?
52  65  x  180
117  x  180
x  63
65
x

The third angle of the
triangle measures 63.
52
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Slide 1-25
Types of Triangles: Named Based on
Angles
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Slide 1-26
Types of Triangles: Named Based on
Sides
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Slide 1-27
Similar and Congruent Triangles

Triangles that have exactly the same shape, but
not necessarily the same size are similar
triangles A
D

F
E
C
B
Triangles that have exactly the same shape and
the same size are called congruent triangles
G
L
H
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K
M
N
Slide 1-28
Conditions for Similar Triangles

Corresponding angles must have the same
measure.
A  D, B  E , C  F

Corresponding sides must be proportional.
(That is, their ratios must be equal.)
AB BC AC


DE EF DF
A
B
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D
C
E
F
Slide 1-29
Example: Finding Angle Measures on
Similar Triangles

Triangles ABC and DEF
are similar. Find the
measures of angles D
and E.


D
A
112
35
F
C
112
33
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B

E


Since the triangles are
similar, corresponding
angles have the same
measure.
Angle D corresponds to
angle: A
o
Measure of D: 35
Angle E corresponds to
angle: B
o
Measure of E: 33
Slide 1-30
Example: Finding Side Lengths on Similar
Triangles
Write a proportion involving correspond ing sides with one unknown :

Triangles ABC and DEF
are similar. Find the
lengths of the unknown
sides in triangle DEF.

32 64

16
x
32 x  1024
x  32
D
32
A

16
112
35
64
F
24
32
C
112
33
48
Copyright © 2005 Pearson Education, Inc.
B
To find side DE:
E
To find side FE:
32 48

16 x
32 x  768
x  24
Slide 1-31
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: 3
x

4 64
4 x  192
x  48
3
4
x

The lighthouse is 48 m
high.
64
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Slide 1-32
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!!!
Copyright © 2005 Pearson Education, Inc.
Slide 1-33
1.3
Trigonometric Functions
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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)
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Slide 1-35
Trigonometric Functions
x, y 
r


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:
r  x2  y 2 .
The six trigonometric functions of  are defined as:
y
sin  
r
x
cos 
r
r
csc  ( y  0)
y
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y
tan   (x  0)
x
r
sec  ( x  0)
x
x
cot  
(y  0)
y
Slide 1-36
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
x2 , y2 
y2
x1 , y1 
r2
y1
x2
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r1

x1
Slide 1-37
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 .
Note : x and y are given, find
" r" and then use definition s :
r  x 2  y 2  122  162
 144  256  400  20
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(12, 16)
16

12
Slide 1-38
Example: Finding Function Values
continued

x = 12
y = 16
r = 20
Using definition s, the value of each of the trig functions is :
y 16 4


r 20 5
x 12 3
cos  

r 20 5
y 16 4
tan   

x 12 3
sin  
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r 20 5


y 16 4
r 20 5
sec  

x 12 3
x 12 3
cot   

y 16 4
csc 
Slide 1-39
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!!!!
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Slide 1-40
Equations of Rays with Endpoint at Origin:
Recall from algebra that the equation of a line is:
y  mx  b where m is slope and b is y - intercept
 If a line goes through the origin its equation is:
y  mx
or :
any equation involving only x and y and their coefficien ts


To get the equation of a ray with endpoint at the
origin we write an equation of this form with the
restriction that:
either x  0 or x  0
left ray
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right ray
Slide 1-41
Example: Finding Function Values
Calculate " r":

Find the six trigonometric
function values of the
angle  in standard
position, if the terminal
side of  is defined by
x + 2y = 0, x  0.

We can use any point on
the terminal side of  to
find the trigonometric
function values.
r  2  1  5
2
2
Choose x  0, calculate y :
x2
y  1
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Slide 1-42
Example: Finding Function Values
continued

From previous
calculations:

Use the definitions of the
trig functions:
sin  
x  2, y  1, r  5
x
2
2
5 2 5




r
5
5
5 5
y
1
r
tan    
csc    5
x
2
y
cos 
sec 
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y 1 1 5
5




r
5
5
5 5
r
5

x
2
cot  
x
 2
y
Slide 1-43
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
1, 0
r 1
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0, 1
1, 0
0,1
Slide 1-44
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
sin 270 
 1
1
1
tan 270 
undefined
0
1
sec 270  undefined
0
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0
cos 270   0
1
1
csc 270 
 1
1
0
cot 270   0
1
Slide 1-45
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:
y
r
tan  
and sec 
x
x

If it lies along the x-axis, then, because y = 0,
the cotangent and cosecant functions are
undefined.
cot  
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x
r
and csc 
y
y
Slide 1-46
Commonly Used Function Values

sin 
cos 
tan 
cot 
sec 
csc 
0
0
1
0
undefined
1
undefined
90
1
0
undefined
0
undefined
1
180
0
1
0
undefined
1
undefined
270
1
0
undefined
0
undefined
1
360
0
1
0
undefined
1
undefined
These can be quickly calculated - not necessary to memorize
Copyright © 2005 Pearson Education, Inc.
Slide 1-47
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
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Slide 1-48
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:
sin 270o   1
tan 60o  1.732050808
cos 30o  0.866025403
We will learn more about using the calculator in the next chapter.
Copyright © 2005 Pearson Education, Inc.
Slide 1-49
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:
4
4
tan A = 3  81
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Slide 1-50
Homework




1.3 Page 24
All: 5 – 8, 17 – 28, 33 – 40
MyMathLab Assignment 3 for practice
MyMathLab Homework Quiz 3 will be due for a
grade on the date of our next class meeting!!!
Copyright © 2005 Pearson Education, Inc.
Slide 1-51
1.4
Using Definitions of the
Trigonometric Functions
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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:
2x  3  2x  6
Expression is defined for all values of x and is true for all values of x
1
 2
2  3    6
x
 x
Expression is not defined for x  0, but is true for all other valu es of x
Copyright © 2005 Pearson Education, Inc.
Slide 1-53
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?
y
sin  
r
csc 

r
( y  0)
y
x
cos 
r
sec 
y
tan   (x  0)
x
r
( x  0)
x
cot  
x
(y  0)
y
They are reciprocals of each other
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Slide 1-54
Reciprocal Identities



This relationship can be summarized:
1
sin  
csc
1
cos 
sec
1
tan  
cot 
1
csc 
sin 
1
sec 
cos
1
cot  
tan 
Each identity is true for angles except those that
that make a denominator equal to zero
These reciprocal identities must be memorized
Copyright © 2005 Pearson Education, Inc.
Slide 1-55
Example: Find each function value.


cos  if sec  =
3
2
Since cos  is the
reciprocal of sec  :
1
1 2
cos  
 
sec  3 3
2
Copyright © 2005 Pearson Education, Inc.

sin  if csc    15
3
sin  
1
 15
3
3

15
3
15


15 15
3 15  15


15
5
Slide 1-56
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?
sin  
x
y
r
x
y
r
y
r
x
y
y
r
r
Signs of x, y, r
x
Copyright © 2005 Pearson Education, Inc.
cos 
x
r
tan  
y
(x  0)
x
sin
all
tan
cos
Positive Trig Functions
Slide 1-57
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
Copyright © 2005 Pearson Education, Inc.
Slide 1-58
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
students
take

all
sin (csc) all
calculus
tan (cot) cos (sec)
Trig functions are negative in quadrants where they are
not positive
Copyright © 2005 Pearson Education, Inc.
Slide 1-59
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
Copyright © 2005 Pearson Education, Inc.
Slide 1-60
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:
r y x, y 
x
Copyright © 2005 Pearson Education, Inc.
1  sin A  1
Slide 1-61
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:
r y x, y 
x
Copyright © 2005 Pearson Education, Inc.
1  cos A  1
Slide 1-62
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:
Any Angle
Range:
1, 1
Copyright © 2005 Pearson Education, Inc.
Slide 1-63
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:
   tan A  
r y x, y 
x
Copyright © 2005 Pearson Education, Inc.
Slide 1-64
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:
   csc A  1 or 1  csc A  
r y x, y 
x
Copyright © 2005 Pearson Education, Inc.
Slide 1-65
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:
   sec A  1 or 1  sec A  
r y x, y 
x
Copyright © 2005 Pearson Education, Inc.
Slide 1-66
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:
   cot A  
r y x, y 
x
Copyright © 2005 Pearson Education, Inc.
Slide 1-67
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.)
Copyright © 2005 Pearson Education, Inc.
Slide 1-68
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 = 1.332
No
cos A = ¼
Yes
Yes
tan A = 1,998,214
No
sec A = ½
No
csc A = 0.2485
Yes
cot A = 0
Yes
sin A = - 0.3359
No
cos A = -3
Yes
tan A = -3
Copyright © 2005 Pearson Education, Inc.
Slide 1-69
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:
x, y 
r y
A x
x2  y 2  r 2

Dividing both sides by r2 gives:
2
2
x
y
 2 1
2
r
r
Copyright © 2005 Pearson Education, Inc.
Equivalent to Trig Equation :
cos 2 A  sin 2 A  1
Slide 1-70
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:
x, y 
r y
A x
x2  y 2  r 2

Dividing both sides by x2 gives:
y2 r 2
1 2  2
x
x
Copyright © 2005 Pearson Education, Inc.
Equivalent to Trig Equation :
1  tan 2 A  sec 2 A
Slide 1-71
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:
x, y 
r y
A x
x2  y 2  r 2

Dividing both sides by y2 gives:
x2
r2
1  2
2
y
y
Copyright © 2005 Pearson Education, Inc.
Equivalent to Trig Equation :
cot 2 A  1  csc 2 A
Slide 1-72
Pythagorean Identities
sin   cos   1,
2
2
tan 2   1  sec 2  ,
1  cot 2   csc2 
MUST MEMORIZE!!!
Copyright © 2005 Pearson Education, Inc.
Slide 1-73
Development of Quotient Identities

Based on x, y, r definitions of sine and cosine
functions:
y
y r
sin A
y
r
 

 tan A

x
r x
cos A
x
r
sin A
 tan A
cos A
Copyright © 2005 Pearson Education, Inc.
Slide 1-74
Development of Quotient Identities

Based on x, y, r definitions of sine and cosine
functions:
x
x r
x
cos A
r
 cot A

  
r y
y
y
sin A
r
cos A
 cot A
sin A
Copyright © 2005 Pearson Education, Inc.
Slide 1-75
Quotient Identities
sin 
 tan 
cos 
cos 
 cot 
sin 
MUST MEMORIZE!!!
Copyright © 2005 Pearson Education, Inc.
Slide 1-76
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
Copyright © 2005 Pearson Education, Inc.
Slide 1-77
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.
Why can' t we use the quotient identity t o say :
sin 
tan  
to say that sin   4 and cos   3?
cos 
 sin and cos must be in the interval [1, 1].
Copyright © 2005 Pearson Education, Inc.
Slide 1-78
Example: Other Function Values
continued

There is no identity that directly gives sin or cos from tan, but which
2
2
one will give a reciprocal of sin or cos from tan? tan   1  sec 
Now, what identity w ill give sin from cos?
tan 2   1  sec 2 
2
Since sin 2   1  cos 2  ,
4
2

1

s
ec

 
3
sin 2 
16
 1  sec 2 
9
sin 2 
25
 sec 2 
9
sin 2 
Why negative?  5  sec
3
Quadrant III
sin 
3
  cos 
5
Note : Other 3 Trig function v alues can be found
Copyright © 2005 Pearson Education, Inc.
2
 3
1  
 5
9
1
25
16

25
4
Why negative?

Quadrant III
5
with reciprocal identities .
Slide 1-79
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
x 3  7


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
1
sin 2  10 
csc  50

In this course we will develop methods for solving
various types of trigonometric equations
Copyright © 2005 Pearson Education, Inc.
Slide 1-80
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:
Copyright © 2005 Pearson Education, Inc.
Slide 1-81
Find One Solution:
1
sin 2  10 
csc  50
Use a reciprocal identity on the right side :
sin 2  10  sin   50
Although t here are other ways this can be true, one way is
when the angle on the left is the same as the one on the right :
2 10    50
  40
Later in the course we will develop methods for finding all solutions to trigonome tric equations
Copyright © 2005 Pearson Education, Inc.
Slide 1-82
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

MyMathLab Homework Quiz 4 will be due for a
grade on the date of our next class meeting!!!
Copyright © 2005 Pearson Education, Inc.
Slide 1-83