©Fry Texas A&M University
Math 150
Chapter 8H
Spring 2016
Chapter 8H - Solving Trigonometric Equations
For what sin 2 x + cos 2 x = 1 ?
sin 2 x + cos 2 x = 1
is true for
_________________
because sin 2 x + cos 2 x = 1 is a trigonometric
1. For what values of x does sin x =
3
2
______________________.
? (i.e., Solve sin x =
3
.)
2
!206
©Fry Texas A&M University
2. Solve sin 2 ( x ) =
3
4
Math 150
Chapter 8H
Spring 2016
!207
©Fry Texas A&M University
3. a) Solve
Math 150
cos ( 3x ) = −1
b) Solve cos ( 3x ) = −1 , x ∈[ 0, 2π ]
Chapter 8H
Spring 2016
!208
©Fry Texas A&M University
Math 150
4. a) Solve ( sin x ) ( tan x ) = sin x
b) Which of these solutions is on x ∈[ −π , π ]
Chapter 8H
Spring 2016
!209
©Fry Texas A&M University
Math 150
5. Solve cos x = sin 2x , x ∈[ −π , π ] .
Chapter 8H
Spring 2016
!210
©Fry Texas A&M University
2
Math 150
Chapter 8H
⎛ ⎛ x⎞⎞
⎛ x⎞
6. Solve 2 ⎜ sin ⎜ ⎟ ⎟ + 3sin ⎜ ⎟ + 1 = 0 , x ∈[−4π , 4π ] .
⎝ 2⎠
⎝ ⎝ 2⎠⎠
Spring 2016
!211
©Fry Texas A&M University
Math 150
Chapter 8H
Spring 2016
⎛ 1⎞
7. Consider the identity sin 2x = 2sin x cos x . It could be rewritten as ⎜ ⎟ sin 2x = sin x cos x .
⎝ 2⎠
⎛ 1⎞
a) What is the amplitude of the function f (x) = ⎜ ⎟ sin 2x ? ______________________
⎝ 2⎠
b) What is the period? _________________! !
2
⎛ 1⎞
c) Plot f (x) = ⎜ ⎟ sin 2x
⎝ 2⎠
⎛⎛ 1⎞
⎞
2
This is the graph of g(x) = ⎜ ⎜ ⎟ sin 2x ⎟ = ( sin x cos x ) = ( sin 2 x ) ( cos 2 x )
⎝
⎠
⎝ 2
⎠
d)
For what values of x does (sin 2 x)(cos 2 x) = 1 ?
!212
©Fry Texas A&M University
Math 150
Chapter 8H
e) For what values of x on [−π , π ] does 4(sin 2 x)(cos 2 x) = 1 ?
Spring 2016
!213
©Fry Texas A&M University
Math 150
Chapter 4A
Spring 2016
! 214
Chapter 4A - Introduction to Functions
Difference Quotient.
Assume that the function represented by
this graph is a position function. The
horizontal axis is time given in seconds.
The vertical axis is distance an object has
moved from the starting point given in feet.
It will be helpful to remember that
d = rt
so r =
What is the position of the object at t = 0?
_______________
What is the position of the object at t = 4? _______________
How far did the object move during those 4 seconds? _______________
What is the average speed of the object during those 4 seconds? _______________
What is the position of the object at t = 8? _______________
How far did the object move during those 8 seconds? _______________
What is the average speed of the object during those 8 seconds? _______________
How far did the object move between t = 4 and t = 8 _______________
What is the average speed of the object during those 4 seconds? _______________
Connect the points on the graph for t = 4 and t = 8.
What is the slope of that line
segment? _______________
The slope of the secant line for a position function is __________________________
____________________________________________________________________
©Fry Texas A&M University
Math 150
Chapter 4A
Spring 2016
! 215
How might you go about calculating an estimate for the instantaneous velocity at a point, say
at t=4?
You might calculate the average velocity between t=4 and t=5.
Or better, you might calculate the average velocity between t=4 and t=4.5.
Or better yet, you might calculate the average velocity between t=4 and t=4.1
The smaller the interval of time, the closer the average velocity would be to instantaneous
velocity.
Here we have the graph of some function
f (x)
Choose a general point (x, f (x)) on the
graph.
Let h > 0 be some teeny-tiny number.
x + h would be just to the right of x .
(Imagine that we’ve zoomed in on the
graph so it looks big, but it’s really small.)
Plot the point at (x + h, f (x + h)) .
Connect those two points with a line
segment.
Calculate the slope of that line segment:
This is what is called the ______________________________________________.
In calculus, you will study what happens as h gets smaller and smaller.
In the meantime, we will practice the sometimes-messy algebra that surrounds the Difference
Quotient.
©Fry Texas A&M University
Math 150
1. If f (x) = x 2 +1 , determine the difference quotient:
2. If f (x) = x −1 , determine the difference quotient:
Chapter 4A
Spring 2016
! 216
©Fry Texas A&M University
Math 150
Chapter 4A
Spring 2016
! 217
3. If f (x) = x , determine the difference quotient:
“Do a job” problems
I think of them this way:
Amount of the job completed = (rate the job is completed) * (time working)
⎛ 1⎞
Also remember that if it takes someone 3 hours to do a job, then the rate is ⎜ ⎟ job per hour.
⎝ 3⎠
4. Amy can clean the house in 6 hours. Brian can do the same job in 9 hours. In how many
hours can they do the job if they work together?
©Fry Texas A&M University
Math 150
Chapter 4A
Spring 2016
! 218
5. Amy and Brian can rake their entire yard in 2 hours when working together. If Brian
requires 6 hours to do the job alone, how many hours does Amy need to do the job alone?
6. One pipe can fill a tank 1.25 times faster than a second pipe. When valves to both pipes
are opened, they fill the tank in five hours. How long would it take to fill the tank if only the
second pipe is used?
©Fry Texas A&M University
Math 150
Chapter 4A
Spring 2016
! 219
7. Suppose there is a piece of sheet metal that is rectangular: 1 foot long and 2 feet wide.
Four congruent squares are cut from the corners, so that the resulting piece of metal can be
folded and welded into a box. If the length of the sides of the squares is s, write a function that
describes the volume of the box in terms of s.
8. Suppose there is a wire 40 inches long. Suppose the wire is cut into 2 pieces, not
necessarily equal in length. If each of the pieces of wire is bent to form a square, write a
function that describes the sum of the areas of the two squares.
Extra Problems: Text: 1, 2, 6, 7, 10 c-g, 11c-g, 12 c-g, 13, 14, 19-23
©Fry Texas A&M University
Math 150
Chapter 3A
Spring 2016
! 220
Chapter 3A Rectangular Coordinate System
A _______________________ is any set of ordered pairs of real numbers.
The _________________ of the relation is the set of all first elements of the ordered pairs.
The _________________ of the relation is the set of all second elements of the ordered pairs.
Calculating Distance in the rectangular coordinate system:
Find the distance d between the points A(1, 2) and B(-3, 4).
Find the distance d between the points C(x1 , y1 ) and D(x2 , y2 )
Consider a general point B (x, y) that is 5 units from the origin.
This equation is true for all points that are exactly 5 units from
the origin. Sketch the graph of this equation.
©Fry Texas A&M University
Math 150
Chapter 3A
Spring 2016
! 221
Place the point B at (1, 2).
List and graph 4 points that are 3 units from B.
_______________________________________________
Is the origin 3 units from (1, 2)?______________________
Write an equation whose solution is all of the points that are
3 units from (1, 2)
We could repeat this procedure starting at an arbitrary point (h, k) and find all the points that
are a given distance called r from (h, k). In this case, we would find that
This is the standard form of the equation of a ____________________
centered at ______________ with radius ___________.
Write the equation of a circle centered at (-3, 0) with radius 2.
What is the center of the circle (x + 4)2 + (y − 5)2 = 6 ? __________________________
What is the radius of this circle? __________
Expand this equation and simplify:
©Fry Texas A&M University
Math 150
Chapter 3A
Spring 2016
! 222
The General Form of an Equation of a Circle is written
Note that there is both an _________ term and a __________ term, but no _______ term.
The coefficients on the x 2 term and the y 2 are both ______________
Consider the equation 4x 2 + 4y 2 − 24x + 40y +135 = 0
If so what is its center? ______________
Is this a circle? ___________
What is its radius? ___________________
Consider the equation 16x 2 +16y 2 +16x − 56y + 5 = 0
Is this a circle? ___________
If so what is its center? ______________________
What is its radius? ________________
©Fry Texas A&M University
Math 150
Chapter 3A
Spring 2016
! 223
The Midpoint of a Line Segment.
Consider the points A (-3, 2) and B ( 4, -4). Determine the midpoint of the segment
connecting these 2 points.
Remember: The number half between the numbers a
and b is the average of a and b
Given a line segment with endpoints (x1 , y1 ) and (x2 , y2 ) , the midpoint M of that segment is
Extra Problems: Text: 4, 5, 6a, 11-13, 14a, 15a, 16, 17a, 18-21
©Fry Texas A&M University
Math 150
Chapter 3B
Spring 2016
! 224
Chapter 3B Graphs of Equations
Intercepts
The y-intercept is where the graph crosses _______________________________________
It occurs when ______________________________________________________________
The x-intercept is where the graph crosses________________________________________
It occurs when ______________________________________________________________
Find the intercepts of the graph for the following equation:
x 2 + xy + 5y 2 = 25 .
To find the x-intercept _________________________________________________________
To find the y-intercept ________________________________
Find the intercepts of the graph for the following equation: (x − 4)2 + (y + 2)2 = 9 .
©Fry Texas A&M University
Math 150
Chapter 3B
Spring 2016
! 225
Symmetry about the x-axis
On the figure below, label the vertices in the first and second quadrants. Then reflect the
image through the x-axis to create a polygon that is symmetric about the x-axis.
Notice that if (a, b) is on the
polygon, then so is ______
A graph is symmetric about the xaxis iff for each (x, y) on the graph,
____________
is also on the graph.
We can test an equation to see if its graph will be symmetric about the x-axis.
Substitute -y for y in the equation. Simplify the equation. If the resulting equation is
_________________ to the original equation, then the graph will be
___________________________________________________________________
1. Show that
x + y2 = 4
is symmetric about the x-axis.
©Fry Texas A&M University
Math 150
Chapter 3B
Spring 2016
! 226
Symmetry about the y-axis
On the figure below, label the vertices in the second and third quadrants. Then reflect the
image through the y-axis to create a polygon that is symmetric about the y-axis.
Notice that if (a, b) is on the polygon,
then so is ______
A graph is symmetric about the yaxis iff for each (x, y) on the graph,
_______________
We can test an equation to see if its graph will be symmetric about the y-axis.
Substitute -x for x in the equation. Simplify the equation. If the resulting equation is
______________ to the original equation, then the graph will be __________________
2. Show that
−x 4 + y = 9
is symmetric about the y-axis.
©Fry Texas A&M University
Math 150
Chapter 3B
Spring 2016
! 227
Symmetry about the origin
Informally, a graph is symmetric about the origin if when it is rotated 180° about the origin, the
graph looks the same.
By definition, a graph is symmetric about the origin iff for each (x, y) on the graph, _________
is also on the graph.
We can test an equation to see if its graph will be symmetric about the origin.
Substitute both -x for x and -y for y in the equation. Simplify the equation. If the resulting
equation is ____________________
to the original equation, then the graph will be
_____________________________________________
3. Show that x 2 = −y 2 + 9 is symmetric about the origin.
©Fry Texas A&M University
4. Determine the symmetry of
Math 150
Chapter 3B
Spring 2016
y 3 = 6x 4 − x 2 + 2
a) Test for symmetry about the x-axis by substituting ________ for __________
Equivalent or Not Equivalent?
Symmetric or Not Symmetric about x-axis?
b) Test for symmetry about the y-axis by substituting ________ for __________
Equivalent or Not Equivalent?
Symmetric or Not Symmetric about y-axis?
c) Test for symmetry about the origin by substituting both
!
________ for __________ and ________ for __________
Equivalent or Not Equivalent?
Suggested Problems:!!
Symmetric or Not Symmetric about origin?
Text: 8-14
! 228
©Fry Texas A&M University
Math 150
Chapter 9
Spring 2016
! 229
Chapter 7B Systems of Non-Linear Equations
1. Plot xy = 1 and
x + y = 2 on the same graph.
Then use algebra to determine the exact
point(s) of intersection, if there are any.
2.
Plot y = x 2 and
y = 8 − x 2 on the same graph.
exact point(s) of intersection, if there are any.
Then use algebra to determine the
©Fry Texas A&M University
3. Plot x 2 + y 2 = 9 and
Math 150
Chapter 9
x 2 − y = 9 on the same graph.
the exact point(s) of intersection, if there are any.
Spring 2016
Then use algebra to determine
! 230
©Fry Texas A&M University
4. Plot x 2 + y 2 = 17 and
Math 150
Chapter 9
x + y = −3 on the same graph.
the exact point(s) of intersection, if there are any.
Spring 2016
Then use algebra to determine
! 231
©Fry Texas A&M University
Math 150
Chapter 9
5. Plot y = 2x 2 + 3x + 4 and y = x 2 + 2x + 3 on the same graph.
determine the exact point(s) of intersection, if there are any.
Spring 2016
Then use algebra to
! 232
©Fry Texas A&M University
6. Plot x 2 + y 2 = 25 and
Math 150
Chapter 9
(x − 6)2 + y 2 = 25 on the same graph.
the exact point(s) of intersection, if there are any.
Suggested Problems: Text: 2 - 6
Spring 2016
! 233
Then use algebra to determine
©Fry Texas A&M University
Math 150
Chapter 9
Spring 2016
! 234
Chapter 9 -- Vectors
Remember that ! is the set of real numbers, often represented by the number line,
! 2 is the notation for the 2-dimensional plane. (Think about the Cartesian plane which is
drawn by intersecting 2 numbers lines. ! × ! = ! 2 )
! 3 is the notation for 3-dimensional space. (Imagine another number line passing through the
origin, perpendicular to the plane.)
Vectors are used to represent quantities such as force and velocity which have both
_______________________ and
_______________________.
The magnitude of a vector corresponds to its _______________.
Vectors are directed line segments. (Like the cross between a segment and a ray.)
Each of these directed line segments has an initial (starting) point and a terminal (ending)
point.
Naming Vectors:
Vectors in ! 2 are defined by their terminal point on the Cartesian plane, assuming their initial
point is at the origin.
For example, the vector 3, 4 has its initial point at the origin and its terminal point at ( 3, 4 ) .
Notice that distinction between the vector 3, 4 and its terminal point ( 3, 4 ) .
3, 4 is a vector. It is a directed line segment. It has magnitude and direction.
( 3, 4 )
is simply an ordered pair of real numbers.
©Fry Texas A&M University
Math 150
Chapter 9
Spring 2016
! 235
!
Definition: A vector v in the Cartesian plane is defined by an ordered pair of real numbers in
the form vx , vy . We write
of vector
!
v.
!
v = vx , vy and call vx and vy the _______________________
Specifically
vx is the ______________________ and vy is the ______________________ of the vector.
The graphical representation of 3, 4 is given in Figure 1.
Figure 2
Note: Although the vectors in Figure 2 have different initial points and different terminal points,
they are all equivalent to 3, 4 because they have the same magnitude and direction. Usually
!
what is important about the vector v is not where it is, but how long it is and which way it
points.
Two or more vectors with the same magnitude and direction are __________________
When a vector’s initial point is NOT at the origin, we must figure out how to name the vector so
that it has the same magnitude and direction as the equivalent vector with initial point at the
origin.
Note that you can think of the vector 3, 4 as set of instructions to get from the initial point to
the terminal point: “Go to the right 3 and up 4.”
©Fry Texas A&M University
Math 150
Chapter 9
Spring 2016
! 236
Consider a vector with initial point at (3, -4) and terminal point at (-2, 0).
To name this vector think about how you would
instruct someone to move from the initial point to the
terminal point. Those instructions would be to move
right
up
left _______________
down _______________
So the name of the vector with initial point at (3, -4)
and terminal point at (-2, 0) is __________
What is the name of the vector with initial point at (x0 , y0 ) and terminal point at (xT , yT ) ?
What is the length of 3, 4 ?
Length of a vector:
The magnitude or length of a vector vx , vy
!
!
is denoted
!
vx , v y
In ! 3 the length of a vector vx , vy , vz
!
!
vx , v y
is denoted
=
v x , v y , vz
!
v x , v y , vz
=
©Fry Texas A&M University
1, 2, − 3
Math 150
Chapter 9
Spring 2016
! 237
=
Adding vectors
If
!
u=
then
3, − 5 and
! !
u +v
!
v=
−4, 1 ,
=
!
v
Geometrically, imagine “picking up” vector
and
!
putting its initial point at the terminal point of u .
!
v
Then the new terminal point of
will be at the
! !
terminal point of u + v .
Notice that algebraically and geometrically,
! !
u+v
.=
! !
v +u
In other words, vector addition is ________________________________
Definition: If
If
!
v=
!
u=
1, 3 , find
u x , u y and
!
v = vx , vy , then
! !
u+v
=
! ! ! !
v +v+v+v
This suggests the reasonableness of the following definition:
________________________
©Fry Texas A&M University
Math 150
Chapter 9
Spring 2016
! 238
Scalar Multiplication: (When working with vectors, the term scalar refers to a real number.)
If
!
u=
u x , u y and
If
!
v=
1, 2 , then
!
2v =
!
3v =
c is a real number, then the scalar multiple
_____________________
_____________________
Notice how
! !
v , 2v
and
!
3v
all have the
same _____________________
So scalar multiplication is simply the multiplying a vector by
a number. The result is another vector that lies on
________________________________________
A scalar of particular interest is -1.
If
!
u=
u x , u y , then
!
( −1) u =
If
!
v=
!
−u
1, 2 , then
Notice how
= _____________________
!
−v =
!
!
v and − v
_____________________
lie
________________________________________
but point in _______________________________
!
cu=
©Fry Texas A&M University
Math 150
Chapter 9
Spring 2016
! 239
Direction of a vector:
We say that two nonzero vectors
!
!
u and v
have the
same direction if ∃ c ∈! _____________________________________________________
opposite directions if ∃ c ∈! __________________________________________________
Give two examples of vectors that are not equivalent to 3, − 5 , but have the same direction.
Give two examples of vectors that have the opposite direction of 3, − 5 .
Do 6, 10 and 9, 15 have the same direction? If 6, 10 and 9, 15 have the same
direction then
_________ These two vectors have the _____________ direction :_____________________
©Fry Texas A&M University
Math 150
Chapter 9
Spring 2016
! 240
Do 4, 9 and 2, 3 have the same direction? If 4, 9 and 2, 3 have the same direction
then
Unit Vectors
A unit vector is a vector with length ________.
Is
!
v = 1, 2
a unit vector? ______
What is the length of 1, 2 ?
1, 2
= ________
What if you wanted a unit vector that is in the same direction
as 1, 2 ?
You would want a scalar c , so that c 1, 2 = c, 2c
c, 2c
=1
=
Remember that since c 1, 2 is in the same direction as 1, 2 , c > 0
Check:
so c = ________
©Fry Texas A&M University
Math 150
Chapter 9
Spring 2016
! 241
What if you wanted a unit vector that is in the same direction as vx , vy ?
You would want a scalar c , so that c vx , vy
=1
If
!
v=
vx , vy , then _______________________ is a unit vector in the same direction as
If
!
v=
−1, 3 , what is the unit vector that is in the opposite direction as
Start by finding a unit vector that is in the same direction as
!
v.
!
v?
!
v.
So _____________________________________ is a unit vector in the same direction as
!
v.
Since we want a unit vector in the opposite direction, we will multiply this vector by c = ______
_____________________________________ is a unit vector in the opposite direction as
There are two really important unit vectors:
In !
3
!
i=
!
___________ j =
Sometimes these are named
In !
3
!"
e1 =
___________
!
i=
___________!
!"
e1 =
!"
!
e2 =
___________
________
and
!
k
and
!
j=
________
= ___________
!"
!
and e2 = ___________
___________ and
!"
e3 =
___________
!
v.
©Fry Texas A&M University
Notice that if
If
!
v=
!
v=
Math 150
Chapter 9
Spring 2016
! 242
3,−5 , then
vx , vy , then
In other words every vector is a _______________________________ of
Direction angles: If
!
v
makes an angle θ with the positive x-axis, then
!
i
and
!
j.
©Fry Texas A&M University
If
!
u
If
!
u=
= 5, 2
ux , uy
Math 150
and
!
v = 1, 5
and
!
v=
, find the angle
vx , vy
Chapter 9
!
θ , between u
, find the angle
θ,
Spring 2016
and
between
!
u
!
v.
and
!
v.
! 243
©Fry Texas A&M University
Math 150
Chapter 9
Spring 2016
! 244
Vector Dot Product
If
!
u=
ux , uy
and
!
v=
vx , vy
, then the dot product
! !
u i v = ___________________________
Notice that the dot product of two vectors is a __________________.
The vector dot product is sometimes called the _______________________________
It is important to notice the distinction between
the scalar product of two vectors
!
!
an operation on two vectors which yields a scalar
and
scalar multiplication of a vector
!
!
an operation between a scalar and a vector that yields a vector.
Notice also that
! !
u i v = _________________________________________________
Calculate the dot product
3, 4 i −2, 5 = _______________________________
Now determine the angle between
the vectors 3, 4 and −2, 5
©Fry Texas A&M University
Math 150
Calculate the dot product
Chapter 9
Spring 2016
! 245
! !
i i j =_______________________________
!
What is the angle between i
and
!
j
?_________________
cos 90! = __________
! !
Notice that since u i v = ____________________________________
! !
!
!
if u and v are ⊥
then u i v = ______________
In fact
! !
u iv =0
iff ______________________________________________
Synonyms for perpendicular include ______________________________________________
In ! 3
If
!
u=
u x , u y , uz
then the dot product
and
! !
uiv
!
v=
v x , v y , vz
,
= ____________________________________________
1, 2, − 3 • 4, 5, 6 = ____________________________________________
Find a vector perpendicular to 1, 1
If vx , vy is perpendicular to 1, 1 , then
©Fry Texas A&M University
Math 150
Chapter 9
Spring 2016
! 246
How many vectors exist that are perpendicular to 1, 1 ? _____________________________
How many unit vectors exist that are perpendicular to 1, 1 . ? _______________________
How do you find a unit vector that is in the same direction as the original vector?
You divide the vector by its ___________________
____________is perpendicular to 1, 1 , and the length of _____________ = ____________
So ________________________________________________________________________
is a unit vector perpendicular to 1, 1
Find a vector
If
!
v perpendicular to
!
v is perpendicular to
and _______________________ _____ is the other.
3, -4 .
3, -4 , then ______________________________ so
vx , vy • 3, -4 = ____________________________________________________
Consider choosing vx = ______________ because then clearly vy = ______________
and so the vector perpendicular to 3, -4
is ______________________.
Find all the unit vectors perpendicular to 3, -4
©Fry Texas A&M University
Math 150
Chapter 9
Spring 2016
! 247
An airplane is flying at 300 miles per hour, heading 30 degrees North of East. What are the
magnitudes of the North and East components of the velocity?
A wind from due North starts blowing at 40 miles per hour. What is the new velocity of the
plane?
A river flows at 3 mph and a rower rows at 6 mph. What heading should the rower take to go
straight across the river?
What if the river flowed at 6 mph and the rower rowed at 3 mph?