Transcript chapter 10 notes

Foundations of Trigonometry
Pre-calculus chapter 10
A little history of trigonometry:
While the work of the Alexandrians Hipparchus (c. 150 BC), Menelaus (c. AD 100),
and Ptolemy (c. AD 150) in astronomy laid the foundations of trigonometry,
further progress was piecemeal and spasmodic. From about the time of
Aryabhata I (c. AD 500), the character of the subject changed, and it began to
resemble its modern form. Subsequently it was transmitted to the Middle
Easterners, who introduced further refinements. From the Middle East, the
knowledge spread to Europe, where a detailed account of existing trigonometric
knowledge first appeared under the title De triangulis omni modis, written in
1464 by Regiomontanus.
Pre-calculus chapter 10
A little history of trigonometry:
On account of their shapes, the arc of a circle, the arc ACB, was known as the
"bow" (capa) and its full chord, the line segment AMB, as the "bow string"
(samastajya). In their study of trigonometric functions, Indian mathematicians
more often used the half chord, the segment AM or MB. The half chord was
known as ardhajya orjyardha, later abbreviated tojya to become the Indian sine.
Three functions were developed, whose modern equivalents are defined here
with reference to the figure:
jya a = AM = r sin a,
kojya a = 0M = r cos a,
utkramajya a = MC = OC = r — r cos a
= r(l — cos a) = r versin a.
Pre-calculus chapter 10
A little history of trigonometry:
As with so many other areas of mathematics, the Middle Eastern scholars (the
author calls them Islamic, but I think that the name is misleading, for example we
don’t call European scholars Christian!) selected Hellenistic and Indian concepts of
trigonometry and combined them into a distinctive discipline that bore little
resemblance to its precursors. It then became an essential component of modern
mathematics.
1. The introduction of six basic trigonometric
functions, namely sine and cosine, tangent and
cotangent, secant, and cosecant.
2. The derivation of the sine rule and establishment
of other trigonometric identities.
3. The construction of highly detailed trigonometric
tables with the aid of various interpolation
procedures.
Pre-calculus chapter 10
A little history of trigonometry:
There were two types of trigonometry: one based on the geometry of chords and
best exemplified in Ptolemy's Almagest, and the other based on the geometry of
semi-chords, which was an Indian invention.
Pre-calculus chapter 10
A little history of trigonometry:
From the tenth century onward, starting with the work of Abu Nasr Mansur
(c. 960 —1036), Islamic mathematicians brought the sine function closer to its
modern form with a few defining it for the first time in terms of a circle of unit
radius, although it remained defined for an arc of a circle rather than the angle
subtended at the center. The etymology of the word "sine" is instructive, for it
shows what can happen as a result of imperfect linguistic and cultural filtering.
The Sanskrit term for sine in an astronomical context wasjya-ardha (half chord),
which was later abbreviated tojya. From this came the phonetically derived
Arabic wordjiba, which, following the usual practice of omitting vowels in
Semitic languages, was written asjyb. Early Latin translators, coming across this
word, mistook it for another word, jaib, which had among its meanings the
opening of a woman's garment at the neck, or bosom; jaib was translated as
sinus, which in Latin had a number of meanings, including a cavity in facial
bones (whence sinusitis), bay, bosom, and, indeed, curve. And hence the
Pre-calculus chapter 10
present word "sine."
A little history of trigonometry:
During the ninth century the Islamic astronomer Habash al-Hasib examined the
length of the shadow of a rod of unit length horizontally mounted on a wall
when the sun was at a given angle to the horizontal. It is easily shown, see the
figure, that the length s of the shadow on the wall can be calculated as
sin 
s
 tan 
cos 
where alpha is the angle of elevation
of the sun above the horizon. The
length t of the shadow cast by a
vertical rod, see the figure, is
cos 
t
 cot 
sin 
Pre-calculus chapter 10
A little history of trigonometry:
During the ninth century the Islamic astronomer Habash al-Hasib examined the
length of the shadow of a rod of unit length horizontally mounted on a wall
when the sun was at a given angle to the horizontal. It is easily shown, see the
figure, that the length s of the shadow on the wall can be calculated as
sin 
s
 tan 
cos 
where alpha is the angle of elevation
of the sun above the horizon. The
length t of the shadow cast by a
vertical rod, see the figure, is
cos 
t
 cot 
sin 
Pre-calculus chapter 10
Angles and their Measure
A ray is usually described as a ‘half-line’
and can be thought of as a line segment
in which one of the two endpoints is
pushed off infinitely distant from the other,
as pictured below. The point from which
the ray originates is called the initial point
of the ray.
When two rays share a common initial point they form an angle and the common
initial point is called the vertex of the angle.
Pre-calculus chapter 10
Angles and their Measure
The measure of an angle is a number which indicates the amount of rotation that
separates the rays of the angle. There is one immediate problem with this, as
pictured below.
One commonly used system to measure angles is degree measure. Quantities
measured in degrees are denoted by the familiar ‘◦’ symbol. One complete
revolution as shown below is 360◦, and parts of a revolution are measured
proportionately.
Pre-calculus chapter 10
Angles and their Measure
The measure of an angle is a number which indicates the amount of rotation that
separates the rays of the angle. There is one immediate problem with this, as
pictured below.
One commonly used system to measure angles is degree measure. Quantities
measured in degrees are denoted by the familiar ‘◦’ symbol. One complete
revolution as shown below is 360◦, and parts of a revolution are measured
proportionately.
Pre-calculus chapter 10
Angles and their Measure
Using our denition of degree measure, we have that 1 represents the measure of an angle which
1
constitutes
of a revolution. Even though it may be hard to draw, it is nonetheless not dificult
360
to imagine an angle with measure smaller than 1 . The choice of `360' is most often attributed to
the Babylonians.
Pre-calculus chapter 10
Angles and their Measure
Using our denition of degree measure, we have that 1 represents the measure of an angle which
1
constitutes
of a revolution. Even though it may be hard to draw, it is nonetheless not dificult
360
to imagine an angle with measure smaller than 1 . The choice of `360' is most often attributed to
the Babylonians.
There are two ways to subdivide degrees. The first, and most familiar, is decimal degrees.
The second way to divide degrees is the Degree - Minute - Second (DMS) system.
In this system, one degree is divided equally into sixty minutes, and in turn, each minute
is divided equally into sixty seconds.
Note :
15
45
117 

60 3600
Pre-calculus chapter 10
Angles and their Measure
Pre-calculus chapter 10
Angles and their Measure
Pre-calculus chapter 10
Angles and their Measure
We need to extend our notion of `angle' from merely measuring an extent of
rotation to quantities which can be associated with real numbers.
To that end, we introduce the concept of an oriented angle. As its name
suggests, in an oriented angle, the direction of the rotation is important. We
imagine the angle being swept out starting from an initial side and ending at a
terminal side, as shown below.
When the rotation is counter-clockwise from initial side to terminal side, we say
that the angle is positive; when the rotation is clockwise, we say that the angle
is negative.
Pre-calculus chapter 10
Angles and their Measure
To further connect angles with the Algebra which has come before, we
shall often overlay an angle diagram on the coordinate plane. An angle
is said to be in standard position if its vertex is the origin and its initial
side coincides with the positive x-axis. Angles in standard position are
classified according to where their terminal side lies. For instance, an
angle in standard position whose terminal side lies in Quadrant I is
called a `Quadrant I angle'. If the terminal side of an angle lies on one
of the coordinate axes, it is called a quadrantile angle.
Pre-calculus chapter 10
Angles and their Measure
Two angles in standard position are called conterminal if they share the same
terminal side.
Pre-calculus chapter 10
Angles and their Measure
Graph each of the (oriented) angles below in standard position and classify
them according to where their terminal side lies. Find three conterminal angles,
at least one of which is positive and one of which is negative.
Pre-calculus chapter 10
Radian Measure
Let  be the central angle subtended
by this arc, that is, an angle whose
vertex is the center of the circle and
whose determining rays pass through
the endpoints of the arc. Using
proportionality arguments, it stands
s
to reason that the ratio should also
r
be a constant among all circles, and
it is this ratio which defines the radian
measure of an angle.
Pre-calculus chapter 10
Radian Measure
Pre-calculus chapter 10
Radian Measure
Graph each of the (oriented) angles below in standard position and classify them
according to where their terminal side lies. Find three conterminal angles, at
least one of which is positive and one of which is negative.
Pre-calculus chapter 10
Converting degree to radian
Pre-calculus chapter 10
The Unit Circle
In order to identify real numbers with oriented angles, we make good use of
this fact by essentially `wrapping' the real number line around the Unit Circle
and associating to each real number t an oriented arc on the Unit Circle with
initial point (1, 0).
Pre-calculus chapter 10
The Unit Circle
Sketch the oriented arc on the Unit Circle corresponding to each of the following
real numbers.
Pre-calculus chapter 10
Applications of Radian Measure: Circular Motion
Suppose an object is moving as pictured below along a circular path of radius r
from the point P to the point Q in an amount of time t. Here s represents a
displacement so that s > 0 means the object is traveling in a counter-clockwise
direction and s < 0 indicates movement in a clockwise direction.
Average velocity
displacement s
v

time
t
Average angular velocity
angle measure in radian 


time
t
Velocity for Circular Motion: For an object moving on a circular path of
radius r with constant angular velocity  , the (linear) velocity of the object
is given by v  r.
Pre-calculus chapter 10
Applications of Radian Measure: Circular Motion
Assuming that the surface of the Earth is a sphere, any point on the Earth can
be thought of as an object traveling on a circle which completes one revolution
in (approximately) 24 hours. The path traced out by the point during this 24
hour period is the Latitude of that point.
San Jacinto Community College is at 29.6911° N latitude, and it can be shown
that the radius of rotation at this Latitude is approximately 3,439 miles. Find the
linear velocity, in miles per hour, of San Jacinto Community College as the world
turns.
2 radians

the angular velocity of earth 

24 hours
12 hours

miles
linear velocity  r  3, 439 miles
 900
12 hours
hour
Pre-calculus chapter 10
Applications of Radian Measure: Circular Motion
A
r
1 2
2

 2A  r  r  r  A  r 
2
r
2 r
2
Pre-calculus chapter 10
The Unit Circle: Cosine and Sine
If an object is moving on a circular path of radius r with a fixed angular velocity
(frequency)  , what is the position of the object at time t? The answer to this question
is the very heart of Trigonometry. Consider an angle  in standard position and let P
denote the point where the terminal side of  intersects the Unit Circle. By associating
the point P with the angle  , we are assigning a position on the Unit Circle to the angle  .
The x-coordinate of P is called the cosine of  , written cos( ), while the y-coordinate of P is
called the sine of  , written sin( ).
Pre-calculus chapter 10
The Unit Circle: Cosine and Sine
Find the cosine and sine of the following angles.
Pre-calculus chapter 10
The Unit Circle: Cosine and Sine
Example) Using the given information about, find the indicated value.
3
9
4  in Q 2
4
2
1) sin( )   cos ( ) 
 1  cos( )    cos( )  
5
25
5
5
5
5
2
2
 in Q 3
2
2) cos( )  

 sin ( )  1  sin( )  
 sin( )  
5
25
5
5
3) sin( )  1  cos 2 ( )  1  1  cos( )  0
Pre-calculus chapter 10
The Unit Circle: Cosine and Sine
Reference Angle Theorem. Suppose  is the reference angle for  . Then
cos( ) =  cos( ) and sin( ) =  sin( ), where the choice of the ( ) depends
on the quadrant in which the terminal side of  lies.
Pre-calculus chapter 10
The Unit Circle: Cosine and Sine
Example) Find the cosine and sine of the following angles.
 is in Q 3
1)     180  225  180  45 
 cos( )  0 and sin( )  0
cos(225)   cos(45)  
2
2
and sin(225)   sin(45)  
2
2
11   is in Q 4
 
 cos( )  0 and sin( )  0
6
6
11

3
11

1
cos(
)   cos( ) 
and sin(
)   sin( )   .
6
6
2
6
6
2
5
  is in Q 2
3)      
   
 cos( )  0 and sin( )  0
4
4
5

2
5

2
cos( )   cos( )  
and sin( )  sin( ) 
.
4
4
2
4
4
2
7
  is in Q1
4)     2 
 2  
 cos( )  0 and sin( )  0
3
3
7

1
7

3
cos( )  cos( )  and sin( )  sin( ) 
.
3
3
2
3
3
2
2)   2    2 
Pre-calculus chapter 10
The Unit Circle: Cosine and Sine
The Reference Angle Theorem in conjunction with the table of cosine and sine values can be used to generate the
following figure, which you should memorized.
Pre-calculus chapter 10
The Unit Circle: Cosine and Sine
5
Suppose  is an acute angle with cos( ) =
.
13
1. Find sin( ) and use this to plot  in standard position.
2. Find the sine and cosine of the following angles:
(a)     
(b)   2   (c)   3   ( d )  

2

5
25
12
2
cos( ) =

+ sin ( ) = 1  sin( ) = 
13
169
13
12
 isin Q1

 sin( ) =
13
5 12
(cos( ),sin( ))  ( , ).
13 13
Pre-calculus chapter 10
The Unit Circle: Cosine and Sine
5
Suppose  is an acute angle with cos( ) =
.
13
1. Find sin( ) and use this to plot  in standard position.
2. Find the sine and cosine of the following angles:
(a)     
(b)   2   (c)   3   ( d )  

2

Pre-calculus chapter 10
The Unit Circle: Cosine and Sine
Example) Find all of the angles which satisfy the given equation.
One solution in Quadrant I is  =
coterminal with

3
, we find  =

3

3
, and since all other Quadrant I solutions must be
 2 k for integers k.
Similarly in Q4, we find the solution to cos( ) =
1
5
here is
2
3
5
so our answer in this Quadrant is  
 2 k for integers k.
3
Pre-calculus chapter 10
The Unit Circle: Cosine and Sine
Example) Find all of the angles which satisfy the given equation.
7
, and since all other Quadrant III solutions must be
6
7
7
coterminal with
, we find  =
 2 k for integers k.
6
6
11
Similarly in Q4, one solution is  
6
11
so our answer in this Quadrant is  
 2 k for integers k.
6
In Quadrant III one solution is  =
Pre-calculus chapter 10
The Unit Circle: Cosine and Sine
Example) Find all of the angles which satisfy the given equation.
One solution is  =
coterminal with

2

2
, and since all other solutions must be
, we find  =

2
Similarly for the second angle  
coterminal to
 2 k for integers k.
3
, so our answer for all other angles
2
3
3

is  
 2 k for integers k. From the figure, both are shortened     k .
2
2
2
Pre-calculus chapter 10
Beyond the Unit Circle
If Q( x, y ) is the point on the terminal side of an angle  , plotted in standard
position, which lies on the circle x 2  y 2  r then x  r cos( ) and y  r sin( ).
Moreover,
x
x
y
y
cos( )  
and
sin( )  
r
r
x2  y 2
x2  y 2
OPA OQB
x r
  x  rx  x  r cos( )
x 1
y r
  y  ry  y  r sin( )
y 1
Pre-calculus chapter 10
Beyond the Unit Circle
Suppose that the terminal side of an angle  , when plotted in standard position, contains the
point Q(4, 2). Find sin( ) and cos( ).
x  4 and y  2, we have r  42  (2) 2  20  2 5
cos( ) 
x
4
y
2
5

and sin( )  

.
r 2 5
r 2 5
5
Pre-calculus chapter 10
Beyond the Unit Circle
Assuming that the surface of the Earth is a sphere, any point on the Earth can
be thought of as an object traveling on a circle which completes one revolution
in (approximately) 24 hours. The path traced out by the point during this 24
hour period is the Latitude of that point.
San Jacinto Community College is at 29.6911° N latitude, and it can be shown
that the radius of rotation at this Latitude is approximately 3,439 miles. Find the
linear velocity, in miles per hour, of San Jacinto Community College as the world
turns.
In this problem, now we can
figure out why the radius of
rotation for San Jacinto is
Radius=3,959cos(29.6911°)
R=3,439.
2 radians

the angular velocity of earth 

24 hours
12 hours

miles
linear velocity  r  3, 439 miles
 900
12 hours
hour
Pre-calculus chapter 10
Beyond the Unit Circle
Suppose an object is traveling in a circular path of radius r centered at the origin with
constant angular velocity . If t  0 corresponds to the point (r , 0), then the x and y
coordinates of the object are functions of t and are given by x  r cos(t ) and y  r sin(t ).
Here,   0 indicates a counter-clockwise direction and   0 indicates a clockwise direction.
x  r cos(t )
y  r sin(t ).
  0 counter-clockwise
  0 clockwise
Pre-calculus chapter 10
Beyond the Unit Circle
Find the equations of the rotation of the San Jacinto San Jacinto Community
College is at 29.6911° N latitude.
In this problem, now we can figure out why the radius of rotation for San
Jacinto is Radius=3,959cos(29.6911°)
R=3,439.
2 radians

The angular velocity of earth 

24 hours
12 hours
r of rotation is 3439

x  r cos(t )  3439 cos( t )
12
y  r sin(t )  3439sin(

12
t)
Pre-calculus chapter 10
Beyond the Unit Circle
Suppose  is an acute angle residing in a right triangle. If the length of the side
adjacent to  is a, the length of the side opposite  is b, and the length of the
hypotenuse is c, then
a
b
cos( ) 
and
sin( )  .
c
c
Pre-calculus chapter 10
Beyond the Unit Circle
Find the measure of the missing angle and the lengths of the missing sides of:
180  30  90  60  cos(30 ) 
7
7
c
c
cos(30 )
3
7
c
14 3

.
3
2
b
14 3 1 7 3
 b  c sin(30 ) 
 
.
c
3 2
3
14 3 2
7 3
2
2
Or 7  b  (
) b
.
3
3
sin(30 ) 
Pre-calculus chapter 10
Beyond the Unit Circle
We close this section by noting that we can easily extend the functions cosine and sine to real
numbers by identifying a real number t with the angle   t radians.
Pre-calculus chapter 10