MATH 41
College Algebra II
and Trigonometry
Lecture Notes
MATH 41
1.1 Basic Equations
College Algebra II
This section is a review of basic equation solving techniques. Each of these
techniques will be used frequently throughout the semester. Remember that solving
an equation means isolating the variable on one side of an equation.
•
•
•
•
•
Linear
Equations
Solving linear equations ax + b = 0
Solving equations involving fractional expressions
Solving nth -degree equations xn = a
Solving equations with fractional exponents
Solving for one variable in terms of another
Solve each linear equation.
1. 4x − 6 = 14
2. 3t − 7 = t + 3
nth -degree
Equations
3.
1
t−1
+
t
3t−2
4.
1
3−t
+
4
3+t
=
+
1
3
16
9−t2
=0
Find all real solutions to the equation.
5. x3 + 8 = 0
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MATH 41
1.1 Basic Equations
6. x4 − 1 = 0
7. x2 − 12 = 0
8. 2(x + 1)2 − 4 = 0
9. x3/2 − 8 = 0
Solving for a
Specific
Variable
Solve for the indicated variable.
10. a2 + b2 = c2 , for a
11. e = mc2 , for c
12.
a+b
b
=
a−1
b
+
b+1
,
a
for a
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College Algebra II
MATH 41
1.2 Modeling with Equations
College Algebra II
To solve a story problem, first identify what the unknown quantity is, and then
assign it a variable. Then set up equations from the information given in the problem. Solve the equation, and check your answer. Don’t make the problems any
more difficult than they are!
•
•
•
•
Money
Problems
Money Problems
Mixture Problems
Geometric Problems
Distance, Rate, and Time Problems
1. Butch earns $8 an hour at his job, but if he works more than 40 hours in a week,
he is paid 1 12 times his regular salary for the overtime hours worked. One week, he
earns $392. How many overtime hours did Butch work that week?
2. Stu invested $2,000, part at 4% interest, and the rest at 10%. During that year,
Stu earned the same amount from interest as he would have if he had invested the
$2000 at 8.5% interest. How much did Stu invest at each interest rate?
3. My couch has $2.50 under the cushions in nickels, dimes, and quarters. If there
are three times as many nickels as quarters, and the same number of dimes and
quarters, how many coins of each type are there?
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MATH 41
1.2 Modeling with Equations
College Algebra II
Mixture
Problems
4. Hector’s favorite drink is Sprite ($2.70/gal) mixed with Eggnog ($4.00/gal). He
can’t recall the proper proportions to mix, but what he does remember is that the
total cost of the mixture is $3.00 per gallon. If Hector wants to mix up 5 gallons
for his big holiday party, how much of each drink should he use?
Geometric
Problems
5. A 6 ft tall man wants to estimate the height of a light post. He notes that his
shadow is 4 feet long, and light post’s shadow is 28 feet long. How tall is the light
post?
6. Find the length x in the figure, if the shaded area is 126 cm2 .
x
x
2x
x
3x
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MATH 41
Distance,
Rate, Time
Problems
1.2 Modeling with Equations
College Algebra II
7. Harry flew his broom 550 miles from Hogwarts to Paris and back in 8 hours. A
15 mi/hr tailwind assisted him on the way to Paris, but impeded him on his return
trip. What was Harry’s flight speed (without the wind)?
8. My wife (Jenny) can clean our cluttered living room in 30 minutes. My son
(Andrew) can clutter up the living room in 40 minutes. If Jenny starts cleaning the
cluttered living room while Andrew is busy cluttering it back up, how long will it
take for the room to get cleaned?
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MATH 41
1.3 Quadratic Equations
College Algebra II
A quadratic equation is an equation of the form
ax2 + bx + c = 0
a , 0.
Quadratic equations can be solved by factoring and using the property that AB = 0
if and only if A = 0 or B = 0. When a quadratic equation doesn’t factor, then it can
be solved by completing the square or using the quadratic formula.
•
•
•
•
Factoring
Solving quadratic equations by factoring
Completing the square
x2 + bx √+ −
2
Quadratic formula
x = −b± 2ab −4ac
Discriminant
b2 − 4ac
+c=0
Solve the equation by factoring.
1. y2 + 7y + 12 = 0
2. x2 + 8x + 12 = 0
3. 3x2 − 8x + 4 = 0
4. 2x(x + 1) = 7x − 2
Completing
the Square
Solve the equation by completing the square.
5. x2 − 4x − 12 = 0
6. w2 + 6w − 18 = 0
7. t2 + 5t − 3 = 0
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MATH 41
1.3 Quadratic Equations
College Algebra II
8. 4x2 − 8x + 8 = 0
9. −6y2 + 7y − 5 = 0
Quadratic
Formula
Solve the equation using the quadratic formula.
10. x2 − 4x − 5 = 0
11. 3θ2 + 4θ − 18 = 0
12. x2 − 2 = 0
Discriminant
Applications
Find the number of real solutions of the equation using the discriminant.
13. 5x2 + 7x + 2 = 0
14. A box with a square base and no top is to be made from a square piece of
cardboard by cutting out a 2 in × 2 in squares from the corner and folding up the
sides as shown. The box is to hold 72 in3 . How big a piece of cardboard is needed?
2
2
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MATH 41
1.4 Complex Numbers
College Algebra II
To solve equations of the form x2 = −1, we introduce a number i which satisfies
i2 = −1. Complex numbers are numbers of the form a + bi. We call a the real part
and b the imaginary part. To do calculations with complex numbers, simply treat i
like a variable, and always simplify i2 to be −1.
• Definition of complex numbers
• Calculations with complex numbers
• Complex numbers as solutions of equations
Complex
Numbers
Calculations
with
Complex
Numbers
Decide whether each statement is true or false.
1. T F
3 + 4i has imaginary part 4 and real part 3.
2. T
F
6 is a complex number.
3. T
F
i4 = −1.
4. T
F
√
−16 = 4i.
Perform the indicated operations
5. (3 + 4i) + (2 − 5i)
6. (3 + 4i) − (2 − 5i)
7. (3 + 4i)(2 − 5i)
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MATH 41
8.
3 + 4i
2 − 5i
9.
1
2
−
2 + i 1 + 2i
1.4 Complex Numbers
10. i17
11. i−17
√
√
√
√
12. ( 5 − −6)( 10 + −3)
Solving
Quadratics
Solve each equation.
13. x2 + 4 = 0
14. x2 − 3x + 3 = 0
15. 2x2 + x + 1 = 0
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College Algebra II
MATH 41
1.5 Other Types of Equations
College Algebra II
Polynomials are expressions such as x4 + 3x2 − 2x + 4. Polynomial equations
can often be solved by factoring. Equations involving radicals can be solved either
by substitution or by raising both sides of the equation to a power to remove the
radicals.
• Solving by factoring
• Equations involving radicals
• Equations of quadratic type
Solving by
Factoring
Find all real solutions of the equation.
1. x4 + 3x3 + 2x2 = 0
2. x4 + 3x3 − 2x2 − 6x = 0
3.
QuadraticType
Equations
1
x+3
+
x
x+4
=
x2 −4
x2 +7x+12
4. (x + 4)2 + 13(x + 4) + 36 = 0
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MATH 41
5.
1.5 Other Types of Equations
x 2
6x
−
+8=0
x−3
x−3
6. 3x4 + 2x7 − 6 = 0
7. x + x1/2 − 6 = 0
8. x8 + 15x4 − 16 = 0
9. x3/2 + 3x1/2 − 10x−1/2 = 0
10. x6 − 9x4 − 4x2 + 36
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College Algebra II
MATH 41
Radical
Equations
11. 2x +
12.
1.5 Other Types of Equations
√
x+1=8
√3
4x2 − 4x = x
r
13.
College Algebra II
1+
q
q
√
√
x + 2x + 1 = 5 + x
14. A students debt D (in dollars) can be modeled by the formula
√
D(t) = 10t + 36 t + 196,
where t is time (in weeks) since moving to State College. How many weeks will it
take for the student’s debt to reach $500?
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MATH 41
1.6 Inequalities
College Algebra II
Inequalities are preserved when quantities are added or subtracted from both
sides. Multiplying or dividing by a negative reverses the direction of the inequality. Polynomial or rational inequalities are solved by getting 0 on one side of the
inequality and then factoring the other side to break up the numberline into test
intervals.
• Linear inequalities
• Polynomial inequalities
• Rational inequalities
Linear
inequalities
Solve each inequality. Write the solution using interval notation.
1. 3x − 5 ≥ 7
2.
−3x + 4
≤2
2
3. 3 ≤ −2x + 7 ≤ 9
Polynomial
Inequalities
4. (x + 3)(x − 4) < 0
5. x2 ≤ −x + 2
6. x3 − 4x ≥ 0
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MATH 41
1.6 Inequalities
College Algebra II
7. x5 < x3
Rational
Inequalities
8.
x+4
≥0
x−2
9.
x+4
≥1
x−2
10.
Applications
3
4
− ≥1
x−1 x
11. A telephone company offers two long-distance plans.
Plan A: $25 per month and $.05 per minute
Plan B: $5 per month and $.12 per minute
For how many minutes of long-distance calls would plan B be financially advantageous?
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MATH 41
1.7 Absolute Value Equations
College Algebra II
The absolute value of a number is the distance from that number to 0. Thus,
|x| = c if and only if x = ±c,
|x| < c if and only if −c < x < c,
|x| > c if and only if x < −c or x > c.
• Absolute value equations
• Absolute value inequalities
Absolute
Value
Equations
Solve each equation.
1. |2x − 1| = 5
2. |3x + 1| = 6
3. 2|x + 3| − 4 = 6
4. |x + 2| = |2x − 4|
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MATH 41
Absolute
Value
Inequalities
1.7 Absolute Value Equations
College Algebra II
Write the solution to each inequality in interval form.
5. |3x + 4| ≤ 7
6. |2x + 1| > 7
7.
1
|x
3
+ 4| − 3 ≥ 1
8. 2 ≤ |x − 1| ≤ 5
9. Write an inequality that describes the set of all numbers that are at least 3 units
away from 5.
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MATH 41
2.1 The Coordinate Plane
College Algebra II
Algebraic relationships can be visualized as a graph of points in the plane. That
is, a graph represents the solution set to some equation or inequality. The vertical
axis is the y-axis, and the horizontal axis is the x-axis. Each point is specified as an
ordered pair (x0 , y0 ), where x0 is the x-coordinate, and y0 is the y-coordinate.
•
•
•
•
Areas and
Plots
Plotting points
Area of triangles and parallelograms
Distance formula
Midpoint formula
The area of a triangle is given by the formula 12 (base)(height). A parallelogram
has area given by (base)(height).
1. Draw the parallelogram with vertices (−1, 1), (−4, 1), (0, 5) and (3, 5), and find
its area.
2. Draw the triangle with vertices (−2, −3), (4, 2), and (−2, 4), and find its area.
Sketch the region given by the set.
3. {(x, y) | x ≥ −2}
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MATH 41
2.1 The Coordinate Plane
4. {(x, y) | 0 ≤ x ≤ 3 and y < 2}
5. {(x, y) | |x| ≥ 1 and |y| ≤ 2}
6. {(x, y) | xy > 0}
Distance
The distance between points A(x1 , y1 ) and B(x2 , y2 ) is
p
d(A, B) = (x2 − x1 )2 + (y2 − y1 )2 .
7. Which of the points A(5, 6) or B(7, 4) is closer to the origin?
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College Algebra II
MATH 41
2.1 The Coordinate Plane
College Algebra II
8. Show that the triangle A(−4, 2), B(0, −4), C(3, −2) is a right triangle by using
the Pythagorean theorem.
9. Show that the points A(−1, 0), B(4, 2), C(2, 7), and D(−3, 5) are the vertices of
a square.
Midpoints
x + x y + y 1
2
1
2
,
.
The midpoint between points (x1 , y1 ) and (x2 , y2 ) is
2
2
10. Find the midpoint of the points A(−2, 1) and B(4, −3).
11. If M(1, 4) is the midpoint of the line segment AB, and if A has coordinates
(−2, −2), find the coordinates of B.
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MATH 41
2.2 Graphs of Equations
College Algebra II
The graph of an equation in x and y is the set of all points (x, y) in the coordinate
plane that satisfy the equation. The intercepts of a graph are the points where the
graph meets the coordinate axis (i.e. either the x or y coordinate is 0).
•
•
•
•
Intercepts
Sketching graphs by plotting points
Intercepts
Symmetry in equations
Equation of a circle
1. Which of the points (2, 3), (3, 2), or (−1, −9) are on the graph of x3 y−y−x = 19?
Find the intercepts of each of the following equations.
2. x2 + y3 − x2 y = 64
3. y = x2 − 7x + 10
Symmetry
An equation is symmetric about the x-axis if it remains unchanged when x is
replaced by −x (e.g. y = x2 ). An equation is symmetric about the y-axis if it remains
unchanged when y is replaced by −y (e.g. x = y2 ). An equation is symmetric about
the origin if it remains unchanged when x and y are replaced by −x and −y (e.g.
x = y).
4. Discuss the symmetries of
• y = x3 + x
• y=
1
x2
+1
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MATH 41
2.2 Graphs of Equations
College Algebra II
Sketch the graph of each equation. Find the intercepts and test for symmetry.
5. y = 3x − 3
6. x + y2 = 4
7. y = 1 − |x|
8. y = |1 − x|
Circles
A circle of radius r with center (h, k) has a standard form equation
(x − h)2 + (y − k)2 = r2 .
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MATH 41
2.2 Graphs of Equations
College Algebra II
9. Find the equation of the circle with center at (2, −1) and radius 9.
10. Find the equation of the circle that has a diameter with endpoints (1, 2) and
(9, −2).
11. Find the center and radius of the circle x2 + 2x + y2 = 3.
12. Find the center and radius of the circle x2 + 8x + y2 − 6y = 0.
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MATH 41
2.2 Graphs of Equations
13. Sketch the region give by the set {(x, y) | 1 < x2 + y2 ≤ 9}.
Page 24 of 112
College Algebra II
MATH 41
2.4 Lines
College Algebra II
Lines are the backbone of geometry. Here we discuss the algebra behind lines
and the various forms of the equations of a line.
•
•
•
•
•
Slope
Slope
Point-slope form
Slope-intercept form
General equation of a line
Parallel and perpendicular
A nonvertical line through the points (x1 , y1 ) and (x2 , y2 ) has slope
m=
rise y2 − y1
=
.
run x2 − x1
1. Find the slope of the line through the points (1, 4) and (−3, 6).
Point-slope
Form
The equation of the line that passes through the point (x1 , y1 ) and has slope m is
y − y1 = m(x − x1 ).
Find the equation of the line that satisfies the given conditions.
2. Through (1, 2); slope −4
3. Through (−1, 3); slope 0
4. x-intercept 3; y-intercept 7
5. Through (−1, −3); parallel to the x-axis
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MATH 41
Slopeintercept
Form
2.4 Lines
College Algebra II
An equation of the line that has slope m and y-intercept b is
y = mx + b.
6. Find the equation of the line with slope −2 and y-intercept 4.
7. Find the y-intercept of the line 6x − 2y = 4.
General
Form
The general form of the equation of a line is
Ax + By + C = 0.
8. Sketch the graph of 5x + 2y − 10 = 0.
Perpendicular
Two lines with slope m1 and m2 are parallel if m1 = m2 . The lines are perpenand Parallel dicular if m1 m2 = −1.
Lines 9. Find the equation of a line through the point (3, 4) that is perpendicular to 2x +
3y = 17.
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MATH 41
Applications
of Lines
2.4 Lines
College Algebra II
10. Student A made me 12 cookies and earned an A (4.0) in the course. Student B
made me 4 cookies and earned a C (2.0) in this course. Write out a linear model
that relates the number of cookies given to the instructor (x-variable) and the grade
that a student can expect (y-variable). What grade can a student expect who gives
the instructor no cookies? Interpret the slope of the equation in the context of this
problem.
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MATH 41
3.1 What is a Function?
College Algebra II
A functions is a special relation between variable quantities. A correct understanding of the language and uses of functions is essential for success in this course
and all courses to come.
• Definition of a Function
• Domain and Range
Definition of
a Function
A function f is a rule that assigns to each element x in a set A exactly one
element, called f (x), in a set B. The set A is the domain of f (i.e. the input of
the function), and the set { f (x) | x ∈ A} is the range of f (i.e. the output of the
function).
1. Let f (x) = |x|. Evaluate f (−2) and f (1). Find the domain and range of f .
2. Let f (x) = (x − 1)2 . Evaluate f (−2), f (1), f (2a), and f (x3 ). Find the domain
and range of f .
3. Let g(x) =
x
. Evaluate g(−2), g(1), g(2a), and g(x2 ). Find the domain and
|x|
range of g.
√
4. Find the domain of f (x) =
x2 − 5x + 6
.
x
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MATH 41
Functions
Inside
Functions
3.1 What is a Function?
College Algebra II
5. Use the function f (x) = x2 − 1 to calculate the following:
• f (x2 )
• ( f (x))2
• f (x + 1)
• f (x) + 1
• f 2x
•
f (x)
2
6. Use the function f (x) = x2 to calculate the difference quotient:
f (a + h) − f (h)
f (x) =
h
Piecewise
Defined
Functions
7. Evaluate
f (−5), f (0), f (2), and f (5) for



3x
if x < 0




f (x) = 
x+1
if 0 ≤ x ≤ 2




(x − 2)2 if x > 2
8. Harvey earns 10/hour at his job. After 40 hours, he earns time-and-a-half. Write
a piecewise defined function that gives Harvey’s pay as a function of the number
of hours he works.
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MATH 41
3.2 Graphs of functions
College Algebra II
A graph is a tool that allows geometric intuition to aid in solving an algebraic
problem. The graph of a function f consists of points (x, f (x)).
•
•
•
•
Plots from
Tables
Graphs from tables
Domain and range
Vertical line test
Piecewise defined graphs
Plot the following functions after making a table of values.
1. f (x) = x3 + x2 − x − 1
2. f (x) = |x| − x
3. f (x) =
√
x+5
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MATH 41
PiecewiseDefined
Functions



x
4. f (x) = 

 x2
3.2 Graphs of functions
College Algebra II
if x < 0
if x ≥ 0
5. Find an equation of the function graphed below.
Vertical Line
Test
A graph represents a function if any vertical line intersects the graph at most
once (i.e. there is at most one y value for each x value).
6. Which of the following represents a function?
Determine whether the equation defines y as a function of x.
7. x2 y + y = x
8. 2x + |y| = 0
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MATH 41
3.2 Graphs of functions
College Algebra II
9. x2 = y2
10. Find the equation for the right half of the circle x2 + y2 = 16.
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MATH 41
3.3 Increasing and decreasing functions
College Algebra II
The concept of an increasing or decreasing function is a fundamental tool in calculus. Here we introduce the important concept of the rate of change of a function
on an interval.
• Increasing and decreasing functions
• Average rate of change
Increasing
and
Decreasing
Functions
A function, f , is increasing % on an interval I if f (x1 ) < f (x2 ) whenever
x1 < x2 in I. f is increasing & on an interval I if f (x1 ) > f (x2 ) whenever x1 < x2
in I.
1. On what intervals is the function pictured increasing?
f (x) = 18 x4 − x2
Average
Rate of
Change
The average rate of change of a function f on an interval [a, b] is the slope of
the line joining the points (a, f (a)) and (b, f (b)).
Average Rate of Change =
change in y
f (b) − f (a)
=
change in x
b−a
2. Find the average rate of change of the function pictured below between the
following x-values.
• x = 0 and x = 2
• x = −2 and x = 0
• x = 4 and x = 6
• x = −2 and x = 4
f (x) =
3x2
8
−
x3
16
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MATH 41
3.3 Increasing and decreasing functions
College Algebra II
Find the average rate of change of the function between the given values of the
variable.
3. f (x) = x2 − 2x; x = 1, x = 4
4. g(t) =
√
t − 1; t = 2, t = 5
5. The graph below shows the number of people who have told me that I am going
bald in each of the past few years.
(a) What is the average number of unwanted remarks about the status of my hair
per year from the year 2000 to the end of 2007?
(b) Between which two successive years did the number of hair criticisms increase most quickly?
(c) Interpret the data and make a prediction for what can be expected in the
coming years.
People Year
0
2000
1
2001
1
2002
8
2003
18
2004
22
2005
30
2006
26
2007
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MATH 41
3.4 Transformations of functions
College Algebra II
Here we cover some extremely useful techniques for transforming the graph of
a function.
•
•
•
•
•
Vertical
Shifts
Vertical and horizontal shifts
Reflections
Vertical stretching/shrinking
Horizontal stretching/shrinking
Even and odd functions
The graph of y = f (x) + c shifts the graph of y = f (x) upward by c units.
1. Plot the functions f (x) = x2 + 2 and g(x) = x2 − 4.
Horizontal
Shifts
The graph of y = f (x − c) shifts the graph of y = f (x) to the right by c units.
2. Plot the functions f (x) = (x − 3)2 and g(x) = (x + 1)2 .
Reflecting
Graphs
To graph y = − f (x) reflect the graph of y = f (x) across the x-axis. To graph
y = f (−x) reflect the graph of y = f (x) across the y-axis.
3. Plot the functions f (x) = −x2 and g(x) = (−x − 3)2 .
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MATH 41
3.4 Transformations of functions
College Algebra II
Vertical and
Horizontal
Stretching
and
Shrinking
To graph y = c f (x) stretch the graph of y = f (x) vertically by a factor of c. To
graph y = f (cx) stretch the graph of y = f (x) horizontally by a factor of 1/c.
2
4. Plot the functions f (x) = 2x2 , g(x) = 41 x2 , h(x) = (2x)2 and k(x) = 13 x .
Combining
Transformations
A function is given. Write the equation that gives the requested transformations.
5. f (x) = |x|; reflect across the x-axis, stretch vertically by a factor of 2, shift left
1 unit and up 3 units
6. f (x) = 1x ; reflect across the y-axis, shrink horizontally by a factor of 3, shift
right 2 units and down 1 unit
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MATH 41
3.4 Transformations of functions
College Algebra II
7. A function is pictured below. Plot each of the following functions.
y = f (x)
y = f (2x)
y = − f (x) + 1
y = f (−x) + 2
Even and
Odd
Functions
y = 14 f (x + 4) − 3
y = 2 f (4x + 4) − 2
A function f is even if f (−x) = f (x) for all x. f is odd if f (−x) = − f (x) for all
x.
8. Determine whether each function is even, odd, or neither.
• f (x) = x3 − x
• g(x) = x4 − x2
• h(x) =
• k(x) =
√3
x
1
|x|
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MATH 41
3.5 Quadratic Functions; Extrema
College Algebra II
Quadratic functions take the form f (x) = ax2 + bx + c. The graph of a quadratic
function is a parabola.
• Standard form of a quadratic function
• Extrema of a quadratic function
Standard
Form
The standard form of a quadratic function with vertex (h, k) is
f (x) = a(x − h)2 + k.
The parabola f opens upward if a > 0 and downward if a < 0.
Express each quadratic equation in standard form.
1. f (x) = 6x2 + 24x − 5
2. f (x) = x2 + 5x + 8
Maxima and
Minima
A quadratic equation f (x) = a(x − h)2 + k will always have either a maximum or
a minimum value of k when x = h (i.e. at the vertex of the graph). By completing
2
the square,
we can see that f (x) = ax + bx + c has a relative maximum or minimum
at f − 2ab .
A quadratic function is given. Express the function in standard form, sketch its
graph, and find its maximum or minimum value.
3. f (x) = x2 − 8x + 18
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MATH 41
3.5 Quadratic Functions; Extrema
College Algebra II
4. f (x) = 12 x2 + 2x − 1
5. g(x) = 3 − 2x − x2
Find the maximum or minimum value of each function.
6. f (x) = 2x2 + 8x − 9
7. g(x) = 100x2 − 1500x
8. Stu’s enjoyment of his week depends on how many dates d he goes on. On a
scale of 0 to 100 his enjoyment E is given by E(d) = d(48 − 6d) + 4. How many
dates should Stu go on each week to ensure the maximum enjoyment?
Page 39 of 112
MATH 41
3.6 Combining Functions
College Algebra II
Here we discuss different ways to make new functions by combining functions.
•
•
•
•
Adding,
Subtracting,
Multiplying,
and Dividing
Functions
Adding and subtracting functions
Multiplying and dividing functions
Composing functions
Domain of combined functions
Two functions f (x) and g(x) can be combined to form new functions in the
following elementary ways:
• ( f + g)(x) = f (x) + g(x)
• ( f − g)(x) = f (x) − g(x)
• ( f g)(x)
= f (x)g(x)
f (x)
f
• g (x) = g(x)
1. For f (x) = x2 and g(x) = x2 − 2 find f + g, f − g, f g, and f /g and evaluate each
of these new functions at x = 3
Composition
of Functions
Given two functions f and g, the composite functions f ◦ g is defined by
( f ◦ g)(x) = f (g(x)).
For each of the following pairs of functions, find f ◦ g, g ◦ f , f ◦ f , and g ◦ g,
and then evaluate each of the composite functions at x = 1.
2. f (x) = x2 − 1, g(x) = x + 1
Page 40 of 112
MATH 41
3.6 Combining Functions
College Algebra II
3. f (x) = 1x , g(x) = 2x
4. Find f ◦ g ◦ f where f (x) = x2 and g(x) = x − 1.
5. Find functions f and g such that F = f ◦ g, where F(x) =
1
.
x−2
6. A circular ripple in a pond is expanding outward at a rate of 6 in/sec.
(a) Find a function g that models the radius as a function of time.
(b) Find a function f that models the area of the circle as a function of the radius.
(c) Find f ◦ g. What does this function represent?
Page 41 of 112
MATH 41
Domain of
Combined
Functions
3.6 Combining Functions
College Algebra II
For the purposes of this course, the domain of a combined function will be the
set of all values on which the new function is defined.
7. Find the domain of each of the√following functions.
• f + g; f (x) = x + 2, g(x) = x
• f − g; f (x) = x2 , g(x) =
• f g; f (x) =
√
√
−x
√
1−x
x + 1, g(x) =
• f /g; f (x) = x, g(x) = x − 3
• f ◦ g, g ◦ f ; f (x) = x + 2, g(x) =
• f ◦ g, g ◦ f ; f (x) = x2 , g(x) =
√
√
4 − 2x
x
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MATH 41
3.7 1-1 Functions and Inverses
College Algebra II
The inverse of a function “undoes” or reverses what the function has done.
•
•
•
•
•
One-to-one
Functions
One-to-one functions
Horizontal line test
Inverse function definition
Finding an inverse function
Plotting an inverse function
A function with domain A is called a one-to-one function if no two elements of
A have the same image, that is
f (x1 ) , f (x2 )
whenever
x1 , x2 .
Decide whether each function is one-to-one.
1. f (x) = x4
2. g(x) = 2x − 1
3. h(x) =
Horizontal
Line Test
1
x
A function is one-to-one if and only if no horizontal line intersects its graph
more than once.
Decide whether each function is one-to-one using the horizontal line test.
4. f (x) = x4
5. g(x) = 2x − 1
Page 43 of 112
MATH 41
6. h(x) =
Inverse
Functions
3.7 1-1 Functions and Inverses
College Algebra II
1
x
Let f be a one-to-one function with domain A and range B. Then its inverse
function f −1 has domain B and range A defined by
f −1 (y) = x
⇔ f (x) = y
for any y in B. Alternatively, we can define the inverse function of f to be any
function f −1 that satisfies each of the following conditions.
f −1 ( f (x)) = x for every x in A
f ( f −1 (x)) = x for every x in B
Determine if the following pairs of functions are inverses.
7. f (x) = 3x + 1, g(x) = x−1
3
8. f (x) =
Finding
Inverse
Functions
√
x, g(x) = x2
To find the inverse of a function y = f (x), simply interchange x and y and then
solve for y in terms of x.
Find the inverse function.
9. f (x) = 1x
10. f (x) =
x−2
x+2
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MATH 41
3.7 1-1 Functions and Inverses
College Algebra II
11. f (x) = (3 + x7 )5
Plotting an
Inverse
Function
To plot f −1 (x), simply reflect the graph of f (x) across the line y = x (i.e. interchange x and y values of each point).
12. Plot f (x) = x3 and f −1 (x).
13. Going to the movie costs $8 for admission plus $3 per bag of popcorn. Thus
if you go to the movie and buy x bags of popcorn the total cost is given by the
function f (x) = 8 + 3x. Find and interpret the meaning of f −1 (x).
Page 45 of 112
MATH 41
Modeling with Functions
College Algebra II
The ability to set up an equation that models a relationship is an essential skill
that is required in calculus.
• Setting up models
• Optimization problems
The basic steps in modeling with functions are
1. Express the model in words.
2. Choose the variable(s).
3. Set up the model.
4. Use the model to answer the question.
Setting up
Models
1. A poster is 10 inches longer than it is wide. Find a function that models its area
A in terms of its width w.
2. The height of a cylinder is four times its radius. Find a function that models the
volume V of the cylinder in terms of its radius r.
3. Find a function that models the surface area S of a cube in terms of its volume
V.
4. The volume of a cone is 100 in3 . Find a function that models the height h of the
cone in terms of its radius r.
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MATH 41
Modeling with Functions
College Algebra II
5. The bases on a baseball diamond are 90 feet apart. A runner from 2nd is stealing
3rd . Find a function that models the distance from the runner to 1 st base when the
runner is x feet from 2nd base.
2nd
x
1 st
3rd
Home
Optimization
Problems
6. Find two positive numbers whose sum is 30 and the sum of whose squares is a
minimum.
7. Bessie the cow is building her dream home. She has 400 feet of fencing and
will make her pasture by dividing up a rectangular pen into four pens as pictured.
(a) Find a function that models the total area of the four pens.
(b) Find the largest possible total area of the four pens.
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MATH 41
4.1 Polynomials and Graphs
College Algebra II
A polynomial function of degree n is a function of the form
P(x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ,
where n is a nonnegative integer and an , 0.
•
•
•
•
•
Graphing polynomials
End behavior of a polynomial
Zeros and graphs of polynomials
Intermediate value theorem
Multiplicity of a zero
A polynomial function will always have a smooth graph without any corners or
holes.
Graphing
Polynomials
End
Behavior of
Polynomials
1. Plot f (x) = x5 , g(x) = (x − 2)5 , and h(x) = x5 − 2.
If f (x) has odd degree, then the graph of f (x) looks like -. . .
&
or
.
...%
depending on the sign of the lead coefficient. Think of f (x) = x3 .
If f (x) has even degree, then the graph of f (x) looks like . . . or -. . . %
. &
depending on the sign of the lead coefficient. Think of f (x) = x2 .
Page 48 of 112
MATH 41
4.1 Polynomials and Graphs
College Algebra II
2. Determine the end behavior of the following polynomials.
• f (x) = x4 + 2x2 + x − 7
• f (x) = −4x7 + x6
• f (x) = 3x3 − 2x2 + x − 78
Real Zeros of
Polynomials
If P(x) is a polynomials and c is a real number, then the following are equivalent.
1. c is a zero of P(x).
2. x = c is a solution of P(x) = 0.
3. (x − c) is a factor of P(x).
4. x = c is an x-intercept of the graph of P.
3. Sketch the graph of f (x) = (x + 4)(x − 2)(x − 5).
Intermediate
Value
Theorem
If a < b and f (a) and f (b) have opposite signs, then there is a zero of f (x)
between a and b.
4. Show that f (x) = x3 − x2 + 1 has a zero between x = −1 and x = 0.
Multiplicity
of Zeros
If c is a zero of a polynomial f (x) and the corresponding factor x − c occurs exactly m times in the factorization of f (x), then we say that c is a zero of multiplicity
m. Near the point x = c the graph of f (x) looks a lot like the graph of y = xm .
Page 49 of 112
MATH 41
4.1 Polynomials and Graphs
5. Graph f (x) = x3 (x + 4)(x − 1)2 (x − 3)2
6. Graph f (x) = x5 − 9x3
7. Graph f (x) = x4 − 2x3 + 8x − 16
Page 50 of 112
College Algebra II
MATH 41
4.1 Polynomials and Graphs
College Algebra II
8. Find a polynomial f such that f (0) = 0, f (1) = 1, f (2) = 2, f (3) = 3, f (4) = 4,
but f (5) , 5.
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MATH 41
4.2 Dividing Polynomials
College Algebra II
Polynomial division is a useful technique for simplifying problems and finding
roots of polynomials.
•
•
•
•
•
Long
Division of
Polynomials
Long division of polynomials
Division algorithm
Synthetic division
Remainder Theorem
Factor Theorem
To divide polynomials, just apply the same techniques as dividing real numbers.
The key is to only focus on the lead terms.
Perform the division in the following problems.
2780
1.
13
2.
Division
algorithm
4x3 + 3x2 + x − 1
2x + 1
If P(x) and D(x) are polynomials, with D(x) , 0, then there exist unique polynomials Q(x) and R(x), where R(x) is either 0 or of degree less than the degree of
D(x), such that
P(x)
R(x)
= Q(x) +
.
P(x) = D(x) · Q(x) + R(x)
or
D(x)
D(x)
We call Q(x) the quotient and R(x) the remainder.
For each P(x) and D(x), divide P(x) by Q(x) and express the result in both forms
of the division algorithm.
3. P(x) = x3 + 6x + 5, D(x) = x − 4
Page 52 of 112
MATH 41
4.2 Dividing Polynomials
College Algebra II
4. P(x) = 2x5 + 4x4 − 4x3 − x − 3, D(x) = x2 − 2
Synthetic
division
Synthetic division is a quick method of dividing by a polynomial of the form
x − c. It is basically a streamlined version long division. Use synthetic division to
divide the polynomial below.
5. P(x) = x3 + 6x + 5, D(x) = x − 4
Remainder
Theorem
If the polynomial P(x) is divided by x − c, then the remainder is the value P(c).
6. Use synthetic division and the Remainder Theorem to evaluate P(11) for P(x) =
2x3 − 21x2 + 9x − 200.
7. Find P(7) for P(x) = (x1 4 + x8 − 98x)(x − 7) + 19.
Factor
Theorem
The value c is a zero of P(x) if and only if x − c is a factor of P(x).
8. Show that c = 4 are zeros of P(x) = x3 + 3x2 − 36x + 32, and find all other zeros
of P(x).
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MATH 41
4.2 Dividing Polynomials
9. Is x − 1 a factor of P(x) = 17x3 − 18x2 + 5x − 4?
10. Find a polynomial of degree 4 with zeros −1, 3, 5, 7.
Page 54 of 112
College Algebra II
MATH 41
4.5 Rational Functions
College Algebra II
A rational function is a fraction of polynomial functions.
• Vertical and horizontal asymptotes
• Slant asymptotes
Vertical and
Horizontal
Asymptotes
An asymptote of a function is a line that the graph of a function gets closer
and closer to as one travels along that line. Vertical asymptotes often occur where
division by 0 would take place. Horizontal asymptotes occur when the denominator
gets large at least as fast as the numerator as x → ±∞.
1. Plot f (x) = 1x
2. Use a transformation of f (x) =
1
x
to plot g(x) =
−2
x+1
If the numerator and denominator have the same degree, then the y-value of the
horizontal asymptote can be found by taking the ratio of the lead coefficients.
Page 55 of 112
MATH 41
3. Plot f (x) =
Slant
Asymptotes
4.5 Rational Functions
College Algebra II
4x2 +1
x2 −x−6
When the degree of the numerator exceeds the degree of the denominator, then
there is no horizontal asymptote. Instead, the behavior of the graph as x → ±∞ can
be found by using long division.
4. Plot f (x) =
x2 +2x
x−1
Page 56 of 112
MATH 41
4.5 Rational Functions
5. Plot f (x) =
x3 −x
x+4
6. Plot f (x) =
x2 +x
x+1
College Algebra II
7. After a certain drug is injected into a patient, the concentration c (in mg/L) of
the drug t minutes since the injection is given by c(t) = t30t
2 +2 . Draw a graph of the
drug concentration and describe what eventually happens to the drug concentration.
c
t
Page 57 of 112
MATH 41
5.1 Exponential Functions
College Algebra II
Exponential functions are an important class of functions that describe naturally
occuring phenomena.
•
•
•
•
Exponential
Functions
Exponential functions
Natural exponential functions
Compound interest
Exponential decay
The exponential function with base a (with a > 0) is defined for all real numbers
to be f (x) = a x .
1. f (x) = 2 x . Find f (2), f (1), f (1/2), f (0), and f (−1). Find the domain, range,
and any asymptotes.
2. Plot f (x) = 2 x , g(x) = 2−x , and h(x) = −2 x+1 + 6 on the same coordinate axis.
3. Plot f (x) = 3 x , g(x) = 4 x , and h(x) = 5 x on the same coordinate axis.
Page 58 of 112
MATH 41
Natural
Exponential
Function
5.1 Exponential Functions
College Algebra II
The number e ≈ 2.718281828 is an important constant that we will use repeatedly in this course and in courses to come.
4. Plot f (x) = e x .
5. Find the domain, range, and asymptote(s) of the following functions.
• f (x) = e x
• g(x) = 3e−x
• h(x) = −e
• k(x) =
Compound
Interest
√
x
√
1 − ex
The amount A in an account after t years that had a principal investment P that
is compounded n times per year at an interest rate of r is given by the formula
r nt
A(t) = P 1 +
.
n
If the interest is compounded continuously (i.e. n → ∞) the formula is
A(t) = Pert .
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MATH 41
5.1 Exponential Functions
College Algebra II
6. Suppose you invest $100 in the bank at a rate of 10% interest. How much money
will you have at end of the year if interest is compounded annually? biannually?
10 times per year? continuously?
Page 60 of 112
MATH 41
5.2 Logarithmic Functions
College Algebra II
Exponential functions are an important class of functions that describe naturally
occuring phenomena.
•
•
•
•
Log base a
Properties of logarithms
Plots of logarithms
Common log and natural log
For any positive number a , 1, the logarithmic function with base a is defined
Log base a
by
loga x = y
⇔
ay = x.
So, loga x is the exponent to which the base a must be raised to give x.
1. Convert the following from exponential form to logarithmic form or vice versa.
• 104 = 10, 000
• 41/2 = 2
• 20 = 1
• log4 16 = 2
• log3 81 = 4
2. Evaluate each of the following expressions.
• log2 16
• log16 4
• log10 .01
• log4 1/2
• log3 1
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MATH 41
Properties of
Logarithms
5.2 Logarithmic Functions
College Algebra II
3. Simplify each of the following expressions.
• loga 1
• loga a
• loga a x
• aloga x
Plotting
Logarithmic
Functions
The logarithmic function base a and the exponential function base a are inverses, so their graphs are obtained by reflecting across the line y = x. The domain
of loga is (0, ∞) and the range is (−∞, ∞).
4. Plot f (x) = log2 x.
5. Plot f (x) = log3 x and g(x) = log5 x.
Page 62 of 112
MATH 41
5.2 Logarithmic Functions
College Algebra II
6. Plot f (x) = − log2 (−x) and g(x) = 12 log2 (x + 4) + 1.
Common
and Natural
Log
When the base is 10, we often omit it from the notation log x = log10 x. We call
log base 10 common log. When the base is e, we write ln x = loge x and call this
natural log.
Page 63 of 112
MATH 41
5.3 Laws of Logarithms
College Algebra II
Logarithms allow us to do math with exponents, so the laws of exponents convert to laws of exponents
• Laws of logarithms
• Expanding and combining logarithmic expressions
• Change of base formula
Laws of
Logarithms
For A > 0, B > 0, and any C the following properties hold:
• loga (AB)
= loga A + loga B
• loga AB = loga A − loga B
• loga ((Ac )) = C loga A
1. Evaluate each expression.
• log2 160 − log2 5
• log 8 + log 125
• log
√
1000
• log3 100 − log3 18 − log3 50
• ln 7 + ln 17
4
• ln ln ee
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MATH 41
Combining
and
Expanding
Logarithms
5.3 Laws of Logarithms
College Algebra II
2. Expand each expression using the Laws of Logarithms.
• log2 (7/2)
√3
• log7 ab
• log4
4x2
y
r q
√
• ln x y z
• ln x+4
3e x
3. Combine each expression using the Laws of Logarithms.
• log(a + b) + log(a − b) − log(a2 − b2 )
• log 8 + log 50 − log 4
• 3 log2 x − 2 log2 x2
• ln(x2 − 5x) + ln 4 − ln x
• ln 7 + ln 17
•
Change of
Base
Formula
1
[ln a
2
− ln(a + b)
logb x =
loga x
loga b
The change of base formula is especially important when using a calculator for
computations.
Page 65 of 112
MATH 41
5.3 Laws of Logarithms
4. Simplify (log2 3)(log3 5)
5. Rewrite using natural log. log2 20
Page 66 of 112
College Algebra II
MATH 41
5.4 Exponential and Logarithmic Equations College Algebra II
Exponential and logarithmic equations will be important in calculus.
• Exponential equations
• Logarithmic equations
Exponential
equations
Exponential equations have variables in the exponent. To solve an exponential
equation do the following steps.
1. Isolate the exponential expression.
2. Take the logarithm of both sides.
3. Solve.
Solve each equation.
1. 2 x+4 = 7
2. 3 x+1 = 4 x
3. 3 + 52x = 95
4. e3x−2 = 2 x−4
5.
100
1+e− x
= 10
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MATH 41
5.4 Exponential and Logarithmic Equations College Algebra II
6. e2x − 4e x − 12 = 0
Exponential
equations
Logarithmic equations involve logarithms of variables. To solve an logarithmic
equation do the following steps.
1. Isolate the logarithmic expression.
2. Exponentiate both sides.
3. Solve.
Solve each equation.
7. ln(3 − x) = 6
8. log2 (x2 − 12) = 2
9. log5 x + log5 (x − 1) = log5 20
10. log(x − 2) = log x − log 2
11. $1000 is invested in an account for 4 years, and the interest was compounded
semiannually.
If the total after 4 years was $1400.00, find the interest rate. (Recall
r nt
A(t) = P 1 + n .)
Page 68 of 112
MATH 41
6.1 Angle Measure
Trigonometry
We introduce a new unit of measure for angles called a radian. It will have
various advantages over using degrees to measure an angle.
• Measure of θ = 1 rad
• Measure of θ ≈ 57.296◦
• Convert Radians → Degrees,
π
multiply by
180
• Convert Degrees → Radians,
180
multiply by
π
θ
1
•
•
•
•
•
Radians and
Degrees
Radians and degrees
Coterminal angles
Arc Length
Area of a circular sector
Linear speed and angular speed
An angle AOB consists of two rays R1 and R2 with a common vertex O. We
think of R1 as stationary and R2 rotating. If a circle has radius 1, then the measure
of an angle in radians is the length of the arc that subtends the angle.
1. Draw and label the following angles.
•
•
•
•
θ = π rad
α = 1 rad
β = π/2 rad
φ = 2 rad
To convert radians to degrees, multiply by
multiply by 180
.
π
π
.
180
Page 69 of 112
To convert degrees to radians,
MATH 41
6.1 Angle Measure
Trigonometry
2. Convert the following angle measures to radians.
• 60◦
• −135◦
• 90◦
• 750◦
3. Convert the following angle measures to degrees.
• 3π rad
•
7π
6
•
π
9
rad
rad
• 4 rad
Coterminal
Angles
Two angles are coterminal if their sides coincide when graphed (e.g. 360◦ =
0◦ ).
4. Determine whether the following angles are coterminal
• 30◦ , 330◦
• 125◦ , 845◦
•
π 7π
,
3 3
• 2, 2 + 6π
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MATH 41
6.1 Angle Measure
Trigonometry
In a circle of radius r, the length s of an arc that subtends a central angle of θ rad
Arc Length
is
s = rθ.
5. Find the length of an arc that subtends a central angle of 30◦ in a circle of radius
4 cm.
6. A central angle θ in a circle of radius 5 is subtended by an arc of length 6 m.
Find the measure of θ in radians.
Area of a
Circular
Sector
In a circle of radius r, the area A of a sector with central angle of θ rad is
1
A = r2 θ.
2
7. Find the area of a sector with central angle
Linear Speed
and Angular
Speed
2π
3
rad in a circle of radius 4 mi.
If a point moves along a circle of radius r with angular speed ω, then its linear
speed ν is given by
ν = rω.
8. The earth rotates about its axis once every ≈ 24 hours. The radius of the earth
is ≈ 4000 mi. Find the linear speed of a point on the equator in mi/hr.
Page 71 of 112
MATH 41
6.2 Trigonometry of Right Angles
Trigonometry
We introduce the 6 trigonometric ratios.
hypotenuse
opposite
θ
adjacent
opposite
hypotenuse
hypotenuse
csc θ =
opposite
sin θ =
adjacent
hypotenuse
hypotenuse
csc θ =
adjacent
cos θ =
• Trigonometric ratios
• Special Triangles
• Applications
Trigonometric
Find all six trigonometric ratios for each triangle.
Ratios 1.
5
θ
12
2.
10
8
θ
Page 72 of 112
opposite
adjacent
adjacent
cot θ =
opposite
tan θ =
MATH 41
6.2 Trigonometry of Right Angles
3. Calculate
• cos π2 + sin π2
• cos π3
• (cos π3 )2 + (sin π3 )2
Special
Triangles
4. 45◦ -45◦ -90◦ triangle
5. 30◦ -60◦ -90◦ triangle
Applications
Find all sides and angles of the triangle.
Page 73 of 112
Trigonometry
MATH 41
6.2 Trigonometry of Right Angles
6.
4
π
6
7. tan θ = 4
θ
9
Page 74 of 112
Trigonometry
MATH 41
6.3 Trigonometric Functions of Angles
Trigonometry
We introduce the 6 trigonometric ratios.
r
y
θ
x
•
•
•
•
Trigonometric
Functions
Reference
Angles
y
r
r
csc θ =
y
sin θ =
x
r
r
csc θ =
x
cos θ =
y
x
x
cot θ =
y
tan θ =
Trigonometric Functions
Reference angles
Trig identities
Area of a triangle
1. Describe the relationship between sin θ and csc θ. Do they share the same domain?
Let θ be an angle in standard position. The reference angle θ associated with θ
is the acute angle formed by the terminal side of θ and the x-axis.
2. Find the reference angle for each of the following angles.
• θ = 105◦
• θ = 225◦
• θ=
7π
6
• θ = − 17π
3
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MATH 41
6.3 Trigonometric Functions of Angles
3. Find the exact value of the trigonometric function.
• sin 135◦
• tan 45◦
• sec 300◦
• cos 7π
6
• cot − 17π
3
• csc 5π
4
4. In which quadrant does θ lie if sin θ > 0 and tan θ < 0?
5. If θ lies in quadrant II and cos θ = 45 , find tan θ.
6. If tan θ > 0 and csc θ = 3, find cos θ.
Page 76 of 112
Trigonometry
MATH 41
Trig
Identities
6.3 Trigonometric Functions of Angles
Trigonometry
Pythagorean Theorem tells us that
sin2 θ + cos2 θ = tan2 θ,
1 + tan2 θ = sec2 θ,
cot2 θ + 1 = csc2 θ.
7. Write tan θ in terms of cos θ.
Area of a
Triangle
The area A of a triangle with sides a and b with included angle θ is
A = 12 ab sin θ.
8. Find the area of the triangle with sides 3 and 12 with included angle θ = 30◦ .
9. An isosceles triangle has an area of 24 cm2 , and the angle between the sides is
5π/6. What is the length of the two sides?
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MATH 41
6.4 The Law of Sines
Trigonometry
A triangle with sides a, b, and c and angles A, B, and C can usually be determined if we know 3 of the 6 parts, as long as at least one of these three is a side.
C
b
A
a
sin A sin B sin C
=
=
a
b
c
B
c
• Law of Sines
• The ambiguous case
• Applications
Law of Sines
The Law of Sines
sin A sin B sin C
=
=
a
b
c
can be used to solve triangles in the cases of ASA, or SAA.
Find the indicated quantity in the following cases.
1. ∠A = π/6, ∠B = π/4, a = 4; Find b.
2. ∠A = π/6, ∠B = π/4, a = 4; Find c.
Page 78 of 112
MATH 41
6.4 The Law of Sines
Trigonometry
3. ∠A = 70◦ , ∠C = 25◦ , c = 10; Find a.
The
Ambiguous
Case
There will often be two solutions in the SSA case.
ind the indicated quantity in the following cases.
4. ∠A = π/6, c = 5, a = 4; Find b.
5. ∠B = 45◦ , b = 10, a = 9; Find c.
Applications
6. A tree on a hillside casts a shadow of length 100 feet down a hill that has slope
14◦ . If the angle of elevation of the sun (above horizontal) is 44◦ , find the height
of the tree.
Page 79 of 112
MATH 41
6.5 The Law of Cosines
Trigonometry
A triangle with sides a, b, and c and angles A, B, and C can usually be determined if we know 3 of the 6 parts, as long as at least one of these three is a side.
C
b
a2 = b2 + c2 − 2bc cos A
a
b2 = a2 + c2 − 2ac cos B
c2 = a2 + b2 − 2ab cos C
A
•
•
•
•
Law of
Cosines
c
B
Law of cosines
Solving triangles
Heron’s Formula
Applications
The Law of Cosines
a2 = b2 + c2 − 2bc cos A
b2 = a2 + c2 − 2ac cos B
c2 = a2 + b2 − 2ab cos C
can be used to solve triangles in the cases of SSS or SAS.
Find the indicated quantity in the following cases.
1. a = 2, b = 3, c = 4; Find A.
Page 80 of 112
MATH 41
6.5 The Law of Cosines
Trigonometry
2. ∠A = 5π/6, b = 18, c = 4; Find a.
3. ∠A = 70◦ , b = 8, c = 10; Find ∠B.
Solving
Triangles
Find the missing pieces using either the Law of Sines or Law of Cosines.
4. ∠C = π/3, c = 6, a = 5
Page 81 of 112
MATH 41
6.5 The Law of Cosines
Trigonometry
5. ∠B = 45◦ , c = 12, a = 8
Heron’s
Formula
The semiperimeter s of a triangle ABC is half the perimeter (i.e. s = 21 (a+b+c)).
Heron’s formula gives the area of this triangle:
p
Area(ABC) = s(s − a)(s − b)(s − c).
6. Butch wants to buy a triangular field for $100,000 per square kilometer. He
drives his car around the field and notes that the three sides measure 5 km, 6 km,
and 7 km respectively. How much will the field cost Butch?
Applications
7. two straight roads diverge an an angle of 30◦ . Two cars leave the intersection at
12:00 noon, one traveling 40 mi/h, the other at 60 mi/h. How far apart are the cars
at 1:30 p.m.?
Page 82 of 112
MATH 41
7.1 The Unit Circle
Trigonometry
All circles are just scale models of a circle with radius 1.
√
(−2 2,
(− 12 ,
√
3
)
2
2
)
2
√
(−
3 1
, )
2 2
(0, 1)
√
( 12 ,
π/2
π/3
2π/3
3
)
2 √
(
π/4
3π/4
√
√
2
2
,
)
2
2
√
( 23 , 12 )
π/6
5π/6
0 (1, 0)
2π
(−1, 0) π
7π/6
11π/6
√
(−
3
, − 21 )
5π/4
2
√
√
4π/3
( − 2 2 , − 22 )
√
(− 12 , − 23 )
√
3
, − 12 )
2
√
√
( 22 , − 22 )
√
( 12 , − 23 )
7π/4
5π/3
3π/2
(0, −1)
(
• The unit circle
• Terminal points
• Reference number
The Unit
Circle
The unit circle is the circle of radius 1 centered at the origin:
x2 + y2 = 1
1. Show that the point
√ 1 2 6
,
5
5
(1)
is on the unit circle.
2. The point P lies on the unit circle. The x-coordinate of P is
coordinate is negative. Find the y coordinate.
Page 83 of 112
3
5
and the y-
MATH 41
Terminal
Points
7.1 The Unit Circle
Trigonometry
A point P that is found by tracing the unit circle from (1, 0) in the counterclockwise direction for t units is called a terminal point for the angle t.
3. Find the terminal point P(x, y) on the unit circle determined by t.
• t = 2π
3
• t=
3π
2
• t = −π
Reference
Number
The reference number t is the shortest distance from the terminal point for t to
the x-axis along the unit circle.
4. Find the terminal point P(x, y) and the reference number determined by t.
• t = 7π
4
• t=
7π
6
• t = − 3π
4
• t=
31π
3
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MATH 41
7.2 Trig Functions of Real Numbers
Trigonometry
We compute exact values of trigonometric functions using the geometry of the
unit circle.
• Exact values of trig functions
• Signs and quadrants
• Trig functions in terms of trig functions
Exact Values
of Trig
Functions
1. Find the exact values of each of the trigonometric functions.
• cos π6
• tan 2π
3
• csc − π3
• cot 7π
6
• sec 11π
4
• sin 15π
2. Find
t, and tan t for the terminal point P(x, y) determined by t.
sin t, cos
• 135 , − 12
13
• − 35 , − 54
Signs and
Quadrants
3. Find the quadrant where each of the following is satisfied.
• cos t > 0 and sin t < 0
• tan t > 0 and sin t > 0
• csc t < 0 and cot t > 0
Page 85 of 112
MATH 41
7.2 Trig Functions of Real Numbers
4. Determine whether the function is even, odd, or neither.
• f (x) = cos x
• g(x) = sin x
• h(x) = x2 tan x
• k(x) = sin(cos x)
Trig
Functions in
Terms of Trig
Functions
5. If sin t < 0, write sin t in terms of cos t.
Page 86 of 112
Trigonometry
MATH 41
7.3 Trigonometric Graphs
y = cos x
•
•
•
•
Trigonometry
y = sin x
Periodic properties
Transformations
Amplitude and period
Mixing functions
Both y = sin x and y = cos x have period 2π. Thus their graphs repeat every 2π
units. Plot each of the following:
1. f (x) = sin x, g(x) = cos x
2. f (x) = 1 + 2 sin x
Page 87 of 112
MATH 41
7.3 Trigonometric Graphs
Trigonometry
3. f (x) = | cos x|
Amplitude
and Period
The functions y = a sin kx and y = a cos kx have amplitude |a| and period 2π/k.
Find the amplitude and period for each function and the plot the function.
4. f (x) = sin 2x
5. f (x) = − cos πx
6. f (x) = 3 cos 12 x
Page 88 of 112
MATH 41
7.3 Trigonometric Graphs
7. f (x) = 2 sin(x − π/2)
8. f (x) = cos(2x − π)
Mixing
Functions
9. f (x) = x sin x
Page 89 of 112
Trigonometry
MATH 41
7.4 More Trigonometric Graphs
Trigonometry
• Tangent and cotangent
• Secant and cosecant
Tangent and
Cotangent
Both y = tan x and y = cot x have period π. That is, tan(π + x) = tan x and
cot(π + x) + cot x. The period of y = a tan(kx) is 2π/k.
Plot each of the following:
1. f (x) = tan x
−3π
−2π
−1π
1π
2π
3π
Page 90 of 112
MATH 41
7.4 More Trigonometric Graphs
2. f (x) = cot x
−3π
−2π
−1π
1π
2π
3π
1π
2π
3π
3. f (x) = − cot(x + π/2)
−3π
−2π
−1π
Page 91 of 112
Trigonometry
MATH 41
7.4 More Trigonometric Graphs
Trigonometry
4. f (x) = 2 tan(πx)
Secant and
Cosecant
Both y = sec x and y = csc x have period 2π. That is, sec(2π + x) = sec x and
csc(2π + x) + csc x. The period of y = a sec(kx) is π/k.
5. f (x) = sec x and g(x) = cos x
−3π
−2π
−1π
1π
2π
3π
Page 92 of 112
MATH 41
7.4 More Trigonometric Graphs
6. f (x) = sec 2x
−3π
−2π
−1π
1π
2π
3π
1π
2π
3π
7. f (x) = 12 csc( 12 x)
−3π
−2π
−1π
Page 93 of 112
Trigonometry
MATH 41
7.4 More Trigonometric Graphs
8. f (x) = 12 csc(4x + 2π)
−3π
−2π
−1π
1π
2π
3π
Page 94 of 112
Trigonometry
MATH 41
8.1 Trigonometric Identities
Trigonometry
Trigonometric identities are formulas that are always true. They can be used to
simplify complicated expressions into forms that are equivalent but more friendly.
•
•
•
•
Reciprocal identities
Pythagorean identities
Even-Odd identities
Cofunction identities
Reciprocal
Identities
csc x =
1
sin x
sec x =
tan x =
Pythagorean
Identities
sin2 x + cos2 x = 1
Even-Odd
Identities
Cofunction
Identities
sin(−x) = − sin x
π
− x = cos x
2π
cos − x = sin x
2
sin
sin x
cos x
1
cos x
cot x =
tan2 x + 1 = sec2 x
cos(−x) = cos x
π
− x = cot x
2
π
cot − x = tan x
2
tan
cot x =
1
tan x
cos x
sin x
1 + cot2 x = csc2 x
tan(−x) = − tan x
π
− x = csc x
2
π
csc − x = sec x
2
sec
Write the trigonometric expression in terms of sine and cosine, then simplify.
1. cos2 θ(1 + cot2 θ)
2. tan θ csc θ
Page 95 of 112
MATH 41
8.1 Trigonometric Identities
Trigonometry
Simplify the trigonometric expression.
3. cos3 x + sin2 x cos x
4.
sin θ
+
csc θ
5.
cos x
cos x
+
1 − sin x 1 + sin x
cos θ
sec θ
Verify the identity. (Often things are easiest if you write everything in terms of
sines and cosines.)
cos θ
6.
= csc θ − sin θ
sec θ sin θ
Page 96 of 112
MATH 41
8.1 Trigonometric Identities
7.
sec t − cos t
= sin2 t
sec t
8.
sin x
tan x
=
sin x + cos x 1 + tan x
9. sec4 x − tan4 x = sec2 x + tan2 x
Page 97 of 112
Trigonometry
MATH 41
10.
8.1 Trigonometric Identities
Trigonometry
tan x + tan y
= tan x tan y
cot x + cot y
11. Show that the equation
1
= csc x + sec x is not an identity.
sin x + cos x
Page 98 of 112
MATH 41
8.2 Addition and Subtraction Formulas
(cos(s + t), sin(s + t))
Trigonometry
s
(cos t, sin t)
t
(1, 0)
−s
(cos s, − sin s)
• Formulas for sine
• Formulas for cosine
• Formulas for tangent
Formulas for
Sine
Formulas for
Cosine
Formulas for
Tangent
sin(s + t) = sin s cos t + cos s sin t
sin(s − t) = sin s cos t − cos s sin t
cos(s + t) = cos s cos t − sin s sin t
cos(s − t) = cos s cos t + sin s sin t
tan s + tan t
1 − tan s tan t
tan s − tan t
tan(s − t) =
1 + tan s tan t
The proof of the addition formula for cosine can be proved by finding the length
of the line segments pictured above using Pythagorean theorem. The other formulas
can be derived through a similar procedure or from the addition formula for cosine.
Use an addition or subtraction formula to find the exact value of each expression.
1. sin 15◦
tan(s + t) =
Page 99 of 112
MATH 41
8.2 Addition and Subtraction Formulas
2. cos 5π
12
3. Write the expression as a single value.
π
+ tan π9
tan 18
π
1 − tan 18
tan π9
Prove each identity.
4. sin π2 − x = sin π2 + x
Page 100 of 112
Trigonometry
MATH 41
5. tan x − tan y =
8.2 Addition and Subtraction Formulas
sin(x−y)
cos x cos y
6. cos(x + y) cos(x − y) = cos2 x − sin2 y
Skip problems 45-47,54,55.
Page 101 of 112
Trigonometry
MATH 41
8.3 Double-Angle/Half-Angle/Product-Sum Formulas Trigonometry
We provide some formulas that allows us to compute the values of trig functions
for nonstandard angles.
•
•
•
•
Double-angle formulas
Formulas for lowering powers
Half-angle formulas
Product-sum formulas
DoubleAngle
Formulas
sin 2x = 2 sin x cos x
cos 2x = cos2 x − sin2 x
= 1 − 2 sin2 x
= 2 cos2 x − 1
2 tan x
tan 2x =
1 − tan2 x
To prove these formulas, just use the addition formula for angles.
1. Find sin 2x, cos 2x, and tan 2x if sin x = 45 and x is in quadrant I.
2. Calculate sin 15◦ cos 15◦ .
Page 102 of 112
MATH 41
Formulas
For Lowering
Powers
Half-Angle
Formulas
8.3 Double-Angle/Half-Angle/Product-Sum Formulas Trigonometry
1 − cos 2x
2
1
+
cos
2x
cos2 x =
2
1
−
cos
2x
tan2 x =
1 + cos 2x
To prove these formulas, just take square roots in the appropriate double angle
formula for cos 2x
3. Write cos4 x in terms of the first power of cosine.
sin2 x =
r
1 − cos x
x
sin = ±
2
2
r
x
1 − cos x
cos = ±
2
2
sin x
x 1 − cos x
=
tan =
2
sin x
1 + cos x
To prove these formulas, just replace x with x/2 in the formulas for lowering powers.
4. Find the exact value of cos 75◦ .
Page 103 of 112
MATH 41
5. Simplify
Product-toSum
Formulas
Sum-toProduct
Formulas
8.3 Double-Angle/Half-Angle/Product-Sum Formulas Trigonometry
1−cos 4θ
sin 4θ
1
sin x cos y = [sin(x + y) + sin(x − y)]
2
1
cos x sin y = [sin(x + y) − sin(x − y)]
2
1
cos x cos y = [cos(x + y) + cos(x − y)]
2
1
sin x sin y = [cos(x − y) − cos(x + y)]
2
To prove these formulas, just combine the addition and subtraction formulas for the
sine function.
6. Write the product as a sum.
cos x sin 5x
x+y
x−y
cos
2
2
x+y
x−y
sin x − sin y = 2 cos
sin
2
2
x+y
x−y
cos x + cos y = 2 cos
cos
2
2
x+y
x−y
cos x − cos y = −2 sin
sin
2
2
To prove these formulas, just reverse engineer the product-to-sum formulas
(substituting x+y
for x and x−y
for y).
2
2
sin x + sin y = 2 sin
Page 104 of 112
MATH 41
8.3 Double-Angle/Half-Angle/Product-Sum Formulas Trigonometry
7. Write the sum as a product.
sin 75◦ + sin 15◦
8. Prove the identity
(sin x + cos x)2 = 1 + sin 2x
9. Show that sin 130◦ − sin 110◦ = − sin 10◦
Page 105 of 112
MATH 41
8.4 Inverse Trigonometric Functions
Trigonometry
We define the inverse trigonometric functions.
• Inverse sine and cosine
• Inverse tangent
• Other inverses
Inverse Sine
and Cosine
The inverse sine function is the function sin−1 with domain [−1, 1] and range
[−π/2, π/2] defined by
sin−1 y = θ ⇔ sin θ = y.
The inverse cosine function is the function cos−1 with domain [−1, 1] and range
[0, π] defined by
cos−1 x = θ ⇔ cos θ = x.
Evaluate each of the following.
1. sin−1 −
2. sin−1
√
3
2
1
2
Page 106 of 112
MATH 41
8.4 Inverse Trigonometric Functions
3. sin(sin−1 12 )
√
4. cos−1 −
2
2
5. cos−1 1
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Trigonometry
MATH 41
8.4 Inverse Trigonometric Functions
Trigonometry
)
6. cos−1 (cos 5π
3
Inverse
Tangent
The inverse tangent function is the function tan−1 with domain (−∞, ∞) and
range (−π/2, π/2) defined by
tan−1 x = θ ⇔ tan θ = x.
7. tan−1 1
8. tan−1
√
3
Page 108 of 112
MATH 41
8.4 Inverse Trigonometric Functions
Trigonometry
9. Plot f (x) = tan−1 x.
Other
Inverses
The inverses of sec, csc, and cot also exist and are defined analogously.
q
10. Show that sin(tan−1 x) = x21+1 . (Hint: Draw a triangle in the unit circle with
angle tan−1 x and set x = √ u 2 and then solve for u.)
1−u
Page 109 of 112
MATH 41
8.5 Trigonometric Equations
We solve trigonometric functions.
•
•
•
•
Intersection
Points
Intersection Points
Solving by Factoring
Using a trig identity
Extra solutions
1. Find the all points of intersection of y = sin x and y = tan x.
Solve each equation.
Solving by
Factoring
2. tan x sin x + sin x = 0.
Page 110 of 112
Trigonometry
MATH 41
8.5 Trigonometric Equations
3. sin2 x − cos2 x = 0.
4. 2 cos3 x + 2 cos2 x − 1 cos x − 1 = 0
Using a Trig
Identity
5. sin x + 1 = cos x (Hint: Square both sides.)
Page 111 of 112
Trigonometry
MATH 41
8.5 Trigonometric Equations
6. sin 2x + cos x = 0 (Hint: Use the double angle formula)
7. tan x − 3 cot x = 0
Extra
Solutions
8. 2 sin 4x cos x − cos x = 0
Page 112 of 112
Trigonometry