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Algebra I, Algebra II, and Transcendentals Review
© Department of Mathematical Sciences
George Mason University
Revised September 2017
Contents
Introduction ........................................................................................... 1
Courses Requiring the Placement Test ............................................... 2
Policies ................................................................................................... 2
Test Format ............................................................................................ 3
Learning Resources ............................................................................... 4
Basic Algebra.......................................................................................... 5
Fractions ................................................................................... 6
Decimal Notation ...................................................................... 7
Percents ..................................................................................... 8
Ratios…. .................................................................................... 10
Algebraic Expressions .............................................................. 11
Linear Equations in One Variable ............................................ 11
Linear Equations in Two Variables .......................................... 12
Graph/Slope of a line ................................................................ 13
Algebra I ................................................................................................. 14
Linear Inequalities in 1 Variable .............................................. 14
Absolute Value ......................................................................... 14
Systems of Linear Equations .................................................... 15
Linear Inequalities in 2 Variables ............................................. 16
Multiplying Algebraic Expressions .......................................... 16
Factoring Algebraic Expressions .............................................. 17
Rational Expressions ................................................................ 17
More on Exponents ................................................................... 18
Radicals .................................................................................... 19
Quadratic Equations ................................................................. 19
Functions Notation .................................................................... 20
Equations with Rational Expressions ....................................... 21
Algebra II……………………………………………………...……… 22
More on Functions .................................................................... 22
Graphs of Functions................................................................... 24
Transcendentals ..................................................................................... 25
Exponential Functions ............................................................... 25
Logarithmic Functions .............................................................. 26
Angle Measure .......................................................................... 26
Right Triangle and Circular Functions Definitions ................... 27
Graphs of Trigonometric Functions .......................................... 28
Inverse Trigonometric Functions .............................................. 29
Trigonometric Formulas and Identities ..................................... 30
Answers .................................................................................................. 31
1
Courses that Require the Placement Test
The Math Placement Test is given by the Department of Mathematical Sciences
to determine the readiness for the following courses:
Math 104 Trigonometry and Transcendental Functions and Math 105
Pre-Calculus
Math 108 Introductory Calculus with Business Applications
Math 113 Analytic Geometry and Calculus I
Math 123 Calculus Algebra/Trigonometry A
Math 125 Discrete Mathematics
CS 112 Computer Science I
Students should talk to an academic advisor to determine which Math course(s)
they are required to take. All students admitted to the university are advised to
take the Math Placement Test during the orientation process.
Policies
For current policies and required minimum scores please visit the
webpage:
http://math.gmu.edu/placement_test.php
2
Basic Algebra
Basic Operations
Example: John’s car insurance premium is $1515.00. He has the option of
making four equal payments but will be charged an additional $25 processing
fee. If he chooses the four-payment option, what is the amount of each payment?
Solution: The processing fee is added to the premium
00.1540$
00.25
00.1515$
This amount is then divided by 4 385
15404
Each payment will be $385.00
Exercise 1: A shipment of 1344 CD’s is to be packed into cartons containing 24
CD’s each. How many cartons are necessary to ship these CD’s?
Exercise 2: A loan of $10,020.00 is to be paid off in 60 equal monthly payments.
How much is each payment (excluding interest)?
Exercise 3: Susan’s Cell phone provider charges her $35.00 per month for the
first 300 minutes of calls and $0.70 for each minute over 300. If she uses 340
minutes of calls in a given month, how much will she be charged?
3
Fractions
A fraction or rational number is of the form q
p where p and q are integers, and
0q
p is called the numerator and q is called the denominator. The fraction is said to
be reduced to lowest terms if p and q have no factors (other than 1) in common.
The same number can be expressed as a fraction in many ways. These fractions
are called equivalent fractions: 4
1,
12
3,
20
5,
40
10,
48
12
Only the first fraction is in reduced form.
► To change from a mixed number to an improper fraction, multiply the whole
number by the denominator of the fraction and add the result to the numerator of
the fraction. This gives the numerator of the improper fraction.
Example: 3
5
3
21 ( 2315 )
8
43
8
35 ( 38543 )
► The product of two fractions d
c
b
a is
bd
ac. That is, multiply numerators to
get the numerator of the product and multiply denominators to get the
denominator of the product.
Examples: 99
8
9
4
11
2
5
12
5
2
1
6
5
26
► The quotient of two fraction d
c
b
a is equal to
c
d
b
a or
cb
ad
Example: 49 ounces of a solution is to be poured into test tubes with capacity
2
13 ounces. How many test tubes will be filled?
Solution: 147
249
2
749
2
1349 14 test tubes will be filled.
4
Exercise 4: Perform the indicated operation and give the result in reduced form.
a. 5
1
3
1 b. 6
8
3 c.
4
32
5
21 d.
8
5
3
26
► The sum of two fractions with a common denominator is c
ba
c
b
c
a
If fractions do not have the same denominator then they must be changed to an
equivalent form.
Example: Find the sum 8
5
3
11
Change each fraction to an equivalent form
24
32
8
8
3
4
3
11
24
15
3
3
8
5
8
5
Then add the equivalent fractions to get 124
47
24
15
24
32
8
5
3
1 or
24
231
Exercise 5: Perform the indicated operation and give the result in reduced
form
a. 8
7
4
16 b.
4
3
10
7 c.
6
1
8
5 d.
12
7
4
12
Decimal Notation
Rational numbers or fractions can also be expressed in decimal notation.
100
137 can be written as 7.13
.125 is equivalent to 1000
125 or
8
1
► When adding or subtracting numbers in decimal notation, make sure to align
the decimal point.
5
Example: Find the sum 78.31.23
Solution:
88.26
78.3
1.23
► When multiplying two numbers in decimal notation, first multiply the
numbers disregarding the decimal point(s). Next put the decimal point in the
product. The number of places to the right of the decimal point in the product
should be the SUM of the places to the right of the decimal point in each of the
factors.
Example: Find the product 1.235.6
Solution:
13335
12700
635
21
635
The product should have three places to the right of the decimal point.
The result will be 13.335
Example: Carlos is getting paid $10.80 per hour for the first 40 hours he is
working and one time and a half that wage for each extra hour. If he works 43.5
hours this week, how much will he earn?
Solution: Carlos is paid 20.16$5.180.10$ per hour of overtime
He will earn: 70.488)20.16$5.3()80.10$40(
Percents
The symbol % is used to indicate the fraction 100
1 or the ratio: 1001 to
23% is equivalent to 23.100
23or
100% is equivalent to 00.1100
100or
6
Exercise 6: Change the decimal to percent
a. .25 b. .085 c. 1.52
Exercise 7: Change the percent to a decimal
a. 2% b. %2
15 c. 0.3%
Example: A class contains 12 female and 8 male students. What percent of the
class is female?
Solution: There are a total of 20 students in the class.
12 out of 20 or 20
12 are female
%60100
60
20
12 of the class is female
Exercise 8: Find the following
a. 30% of 75 b. 4% of 196 c. 120% of 300
Example: Suppose 60% of a class is female. Of these females, 20% are
freshman. What percent of the class is female upper class students?
Solution: If 20% of the females are freshman, then 80% of females are upper
class students. 80% of 60%= 48.60.80.
so 48% of the class are female upper class students
Example: Suppose we know that 40% of a class or 16 students are male. How
many students are in the class?
Solution: We know 40% of (the number) is 16
That is 1640. numberthe
So 4040.
16numberthe students
Example: Before graduating, Joe worked in a convenience store and earned
$5.85/hr. He started a new job that is paying him $18.95/hr. What is the
percent increase in his hourly rate?
Solution: original
increaseactualincrease % so
85.5
85.595.18 or 224% after
rounding.
7
Ratio
The ratio of two numbers a and b is written a to b, a:b or b
a
We can think of this as a fraction in order to simplify it.
The ratio 10:5 is equivalent to 2:1.
The ratio 32.5 to 5 is equivalent to 65 to 10 since 10
65
2
2
5
5.32
5
5.32
► To show two ratios are equal use the following:
D
C
B
A if and only if BCAD
Example: Show that 12.5 : 5 is equivalent to the ratio 5:2
Solution: Since 552525.12 , 2
5
5
5.12 i.e. 12.5: 5 is
equivalent to the ratio 5:2.
Example: How much water should be mixed with 7 cups of salt to make a
12% saline solution?
Solution: We have the proportion 100
127
x
total
part
Multiply to get 70012 x
3
158x
So we need 3
1517
3
158 cups of water.
Exponents
►When working with algebraic expressions involving exponents, the following
properties of exponents are used:
1. baba xxx 4. aaa
yxxy
2. ba
b
a
xx
x 5. a
aa
y
x
y
x
3. abba xx 6. 10 x for any 0x
8
Exercise 9: Simplify each of the following expressions using the rules of
exponents
a. 243x b. 5
10
2
4
ab
a c.
322
25
25
16
xyx
yx
Exercise 10: Find the product 26073 125 zxyzyx
Algebraic Expressions
Most of Algebra involves manipulating expressions that contain variables.
Example: Evaluate the expression using the indicated value of the variable
2513 kforkk
Solution:
52123 substitute -2 in the expression for k
5216
37 = -21
Exercise 11: Evaluate the expression using the indicated value of the variable
a. 2362 2 xforxx b.
414
15
xforx
x
Exercise 12: Combine the terms below and simplify the result.
a. 2834 xx
b. baba 34216
c. 10
13
5
4
xx
Linear Equations in One Variable
A linear equation in one variable can have a unique solution, no solution or
infinitely many solutions. When it has no solution, the equation is called
inconsistent.
Example: Solve the following equations by isolating the variable.
214152 xx
Solution: xxxx 42144152 Add 4x to both sides
21156 x
152115156 x Add 15 to both sides
9
366 x
366
16
6
1
x Multiply both sides by
6
1
6x The equation has a unique solution
Example: Solve the equation 34416 xx
Solution:
121616 xx Distribute the 4
xxxx 1612161616 Add -16x to both sides
120 The equation is inconsistent.
Example: Solve the equation 510625310 xxxx
Solution:
5201215310 xxxx Clear the parentheses by
distributing
15131513 xx Combine like terms
0 = 0
Since the last statement is always true, every real number will be a
solution.
Linear Equations in Two Variables
A solution for an equation in two variables is a replacement for each of the
variables so that when substituted, the resulting statement is true. We generally
write the solution as an ordered pair. For example, 5,2 is a solution of the
equation 113 yx . When the variable x is replaced by 2 in the equation and
the variable y is replaced by 5, the resulting statement is true. The linear
equation in two variables has an infinite number of solutions. The graph of these
solutions is a line in the x-y plane.
Example: The formula in two variables gives the conversion between
Fahrenheit and Celsius temperatures. FytoCx
325
9 xy
Find the temperature in degrees Celsius that corresponds to F212
Solution: 325
9212 x Substitute the value 212 for y
10
x5
9180 Add -32 to both sides
x100 Multiply both sides by 9
5
A solution of the equation is 212,100
The Graph of a Line
We can graph the solutions of any linear equation on a set of axes called the
Euclidean Plane. In the case of the linear equation, the solutions form a line.\We
can measure how much and in which direction the line tips using the slope.
The slope-intercept form of the equation is : bmxy
If the equation is in this form, the line has slope m and y-intercept (0, b)
The y-intercept is the point where the line intersects the y-axis.
Example: In the graph below, find the x and y intercepts and the slope.
Example: Find the slope and y-intercept of the line with equation:
25105 yx
Solution: Write the equation in slope-intercept form: 2
5
2
1 xy
The slope is 2
1 and the y-intercept
2
5,0
Parallel lines have identical slopes. Lines are perpendicular if their slopes have
the relationship
2
1
1
mm
Example: Find the slope of a line that is perpendicular to the line
25105 yx
Solution: The line 25105 yx has slope 2
1 .
Any line perpendicular to it will have slope m= - 2.
Solution:
x intercept: (-12, 0)
y intercept: (0, 6)
Slope is 1/2
11
Algebra I
The Algebra I section of the test measures your ability to perform basic
algebraic operations and to solve problems that involve algebraic concepts.
Only those students attempting to place into Math 108, Math 112, Math 125,
Math 113, CS 112 will be given questions from the Algebra I section. In
addition to all material in the Basic Algebra, the following exercises and
examples provide a refresher of the material that is tested in the Algebra I
section.
Linear Inequalities in One Variable
Unlike linear equations, linear inequalities have an infinite number of solutions.
We can use a number line or interval notation to indicate the solutions. Any real
number can be added to or subtracted from both sides of the inequality. We can
multiply or divide both sides by any nonzero real number but, if we multiply or
divide by a negative number, the inequality sign must be reversed.
Example: Solve the inequality 3 £-9
5x+ 2
Subtract 2 from both sides: 1£-9
5x
Multiply both sides by -5/9: -5
9³ x
The solution is the interval:
]
Absolute Value Equations and Inequalities
The absolute value of a number is defined to be its distance from 0 on the
number line.
0
0
xforx
xforxx
Example: Solve the equation for x 75315 x
Solution: 75315 x or 75315 x
20x or 30x
Example: Solve the inequality 1552 x
Solution: 1552 x or 1552 x
10x or 5x
12
Distance Formula
► The distance between two points 11, yx and 22 , yx is given by
the formula: 212
2
12 yyxxd
The formula uses the Pythagorean Theorem
Exercise 1: Find the distance between the points 5,2 and 8,1
Systems of Linear Equations
We sometimes group equations together and investigate whether they have a
common solution. This is called a system of equations. The system of linear
equations has a solution if there is an ordered pair yx, that satisfies each
equation in the system. A system of linear equations can have exactly one
solution, no solution or infinitely many solutions. There are several methods
available to solve systems of linear equations. The next example illustrates the
Elimination Method.
Example: Solve the system
2489
4823
yx
yx
Solution:
4823 yx multiply by -3: 14469 yx
2489 yx 2489 yx
Add: 16814 y
Solve for y: 12y
The value 12 is substituted for y in either of the original equations.
481223 x Solve the resulting equation for x
48243 x
243 x
8x The solution to the system is (8, 12)
Exercise 2: Solve the systems
a.
1025
53
yx
yx b.
1026
23
yx
yx
13
Linear Inequalities in Two Variables
Linear inequalities in two variables can be solved by graphing. The solution is
an infinite set of ordered pairs in the plane. We can indicate the solution by
shading the region that contains these ordered pairs.
Example: Solve the inequality 5y ³ 3x-15
Solution: Graph the line that results from replacing the inequality sign with an
equal sign. Here, graph the line 5y = 3x-15 This line partitions the plane into
three sets:
The solutions of 5y > 3x-15 5y < 3x-15 5y = 3x-15
In the inequality, test an ordered pair from the plane that does not lie on the line.
Using (0,0), we get: 5(0) ³ 3(0)-15 which is a true statement.
Shade the half-plane of points that satisfy the inequality. See the graph below.
In a similar way we can solve a system of inequalities. The graph below shows
the solutions to the system
5y ³ 3x -15
y £ 0
x ³ 0
ì
íï
îï
Multiplying Algebraic Expressions
We use the Distributive Law to multiply algebraic expressions.
Example: Find the product 1026 xx
Solution: 1026 xx = 10626 xxx
= xx 6012 2
14
To multiply 2 binomial factors, we apply the Distributive Law twice and
combine like terms.
Example: Find the product 127 xx
Solution: 127 xx = 12712 xxx
= 172712 xxxx
= 7142 2 xxx
= 7132 2 xx
The result of this multiplication is called a quadratic expression. We “undo” the
multiplication in a process called factoring.
Factoring Algebraic Expressions
The next two examples show the factoring process. In the first, we factor out the
highest common factor among the terms. In the second, we write the quadratic
expression as a product of 2 binomial factors.
Example: Factor the expression xx 2416 2
Solution:
xx 2416 2 = 2416 xx The terms have the factor
x in common
= 328 xx They also have a factor of
8 in common
Example: Factor the expression 40182 tt
Solution:
40182 tt = 220 tt
Special Factorings:
))((22 bababa
))(( 2233 babababa
))(( 2233 babababa
Rational Expressions
Rational expressions are fractions where the numerator and/or denominator
involve variables. We can reduce, add, subtract, multiply and divide these
rational expressions using the rules of fractions. You may want to review the
rules of fractions in the Basic Algebra section on page 6.
15
Example: Reduce the rational expression 103
7352
xx
x
Solution: In order to reduce the expression we must factor both the numerator
and the denominator, then cancel any factors common to both.
103
7352
xx
x=
52
57
xx
x
=
52
57
xx
x Factor the numerator
= 2
7
x Cancel the common factor
Example: Find the sum 3
1
9
342
x
x
x
x
Solution:
3
1
9
342
x
x
x
x=
33
13
33
34
xx
xx
xx
x Use a common denominator
=
33
3434 2
xx
xxx
= 92
2
x
x
Example: Simplify the expression
ba
x
1
5 2
Solution: Using the rules of dividing fractions we get:
baxba
x
22 51
5
More Exponents
►The definition of exponents is extended to include all rational numbers:
a
a
xx
1
b ab
a
xx
Exercise 3: If x= 3
2 find the value of x-3
Exercise 4: Find the value of 2
1
3
2
927
16
Radicals
When simplifying algebraic expressions involving radicals, the following
properties of radicals are used:
1. nnn xyyx
2. n
n
n
y
x
y
x where 0y
3. nmm n xx
Example: Find the product 5156
Solution: 5156 = 55515616
= 551
Exercise 5: Find the sum: 7328
Exercise 6: Simplify each of the following expressions
a. 568 ba b. 2
37
7
28
uv
vu c. 3 1558 yx
Exercise 7: Rationalize the denominator to simplify: 23
22
Quadratic Equations
An equation of the form 02 cbxax is called a quadratic equation. The
equation will have two, one or no real number solutions. There are several
methods available to find the solutions of a quadratic equation. The following
examples illustrate these methods.
Example: Solve the equation 202 2 x
Solution: 202 2 x
102 x Isolate the term involving the radical
10x Take the square root of both side of the equation
17
Example: Solve the equation 25442 xx
Solution: 25442 xx
02142 xx bring all terms to one side
037 xx factor the left side
07 x or 03 x set each factor equal to zero
7x or 3x the equation has 2 solutions
Example: Solve the equation 0169 2 xx
Solution: In the quadratic formula: a
acbbx
2
42 we use
9a , 6b and 1c
92
19466 2
x =
18
6 since 042 acb
we get only one solution 3
1x to the equation.
►The expression acb 42 is called the discriminant. It tells us how many
solutions a quadratic equation will have.
If 042 acb the equation has 2 real number solution
If 042 acb the equation has 1 real number solution
If 042 acb the equation has no real number solution
Exercise 8: Solve the equations
a. 0494 2 x
b. 0592 2 xx
c. 0642 xx
Function Notation
Functions can be defined using a list of ordered pairs, a graph, a table, or an
equation.
The following equation defines a function: 12 xxf .
Each value of x is paired with the value xf .
8133 2 f so the pair 8,3 is in the function f.
9911010 2 f so the pair 99,10 is in the function f.
99110102
f so the pair 99,10 is in the function f.
18
Example: For the function 3
12
x
xxf find 2f and 1bf
Solution:
31
3
32
1)2(2
2
f
4
2
3)1(
1)1(1
22
b
bb
b
bbf
Complex Rational Expression
A complex rational expression is one in which a rational expression appears in
the numerator and/or denominator. Simplifying the complex rational expression
is equivalent to finding the quotient of the expression in the numerator with the
expression in the denominator.
Example: Simplify the complex rational expression
yx
xy
11
1
Solution: Combine the terms in the denominator first.
xy
xy
yx
11
Next, find the quotient of the numerator with the denominator
xy
xy
xy
1=
xy
xy
xy
1
= xy
1
Equations with Rational Expressions
Given an equation involving rational expressions we can find an equivalent
equation with no rational expressions. To find the equivalent equation we
multiply the entire equation by the least common denominator (LCD). After
solving the equivalent equation, we must check the solution in the original to see
if it makes any denominator equal to zero. If so, we discard that solution.
Example: Solve the equation xx 2
3
5
11
Solution: The LCD of the rational expressions is 10x
x
xxx
x2
310
5
110
110 Multiply equation by 10x
19
15210 x Solve the resulting equation
52 x
2
5x
Example: Solve the equation xx
3
1
3
4
Solution: The LCD of the rational expressions is 3x
xxxx
x 33
13
3
43 Multiply equation by 3x
234 xx Solve the resulting equation
043 2 xx Use the quadratic formula
32
)4(3411 2
x
32
491
x
3
41 andx
Exercise 9: Solve the equation 32
56
1
32
3
22
xx
x
x
x
x
Algebra II
The Algebra II section of the test measures your ability to solve problems that
involve more advanced Algebra than the previous section. Only those students
attempting to place into Math 113 and CS 112 will be given questions from the
Algebra II section. The following exercises and examples provide a refresher of
material that is tested in the Algebra II section.
More on Functions
A function is a set of ordered pairs where each first component is paired with
exactly one second component. The set of first components is called the domain
and the set of second components is called the range.
Unless otherwise stated, the domain of the function 12 xxf , is all real
numbers. Any real number a can be paired with another real number using the
equation 12 xxf . The pair can be written 1, 2 aa .
20
Suppose we have the function t
tg3
. We say tg is undefined at 0t
since the expression 0
3 is undefined. On the other hand, tg is defined at any
real number other than 0. The domain of tg is all real numbers except 0t .
Suppose we have the function 5 xxf . In order for the expression
5x to represent a real number, the expression 5x must not be negative.
For this function the domain is all real numbers 5x .
►The average rate of change for a function f(x) is the change in f(x) over the
change in x.
x
xfchangeofrateaverage
)(
Alternatively, it is the slope of the secant line passing through the two points
(x1, f(x1)) and (x2, f(x2))
Exercise 1: Let 62
1
xxf ; Find each of the following:
a. the domain of xf b. the range of f(x)
c. the average rate of change from x= -1 to x=5
Exercise 2: Let x
xg
3
1 ; Find each of the following:
a. the domain of xg b. the range of g(x)
c. the average rate of change from x= -2 to x=2
►We can compose two functions by applying one right after the other.
Example: For the functions f and g defined in the exercises 1 and 2, find
a. 6gf b. 2fg
Solution: 6gf =
)6(3
1f =
3
1f =
20
3
2fg =
6)2(2
1g =
2
1g =
2/5
1 or 5
2
Example: The function values of f and g are given in the table below. Use them
to find a. 0gf b. 3fg
21
x f(x) g(x)
0 -3 3
1 0 2
2 -2 1
3 1 0
Solution: a. 0gf = 3f =1
b. 3fg = 1g =2
►Some functions have an inverse. The inverse of a function will interchange
the real numbers in the ordered pair. If ba, is in the function f then ab, is in
1f , the inverse of f . Keep in mind, that not all functions have inverses.
The functions 53 xxf and 3
51 x
xf are inverses.
Notice f contains the pair (3,14) and f-1
contains the pair (14, 3). This is true
for all ordered pairs of the function f.
Graphs of Functions
We can graph the set of ordered pairs of a function on a set of axes just as we
do with linear equations in 2 variables.
The graph of a quadratic function cbxaxxf 2)( is a parabola.
The vertex of the parabola is located at
)
2(,
2 a
bf
a
b
This point on the graph is either a minimum or a maximum for the function. The
leading coefficient a, determines which. For a<0, the function takes on a
maximum at a
bx
2
and the range of the function is
a
bfyy
2.
For a>0, the function takes on a minimum at a
bx
2
and the range of the
function will be
a
bfyy
2.
Example: In the graph of 74)( 2 xxxf below, find the x and y
intercepts and the vertex.
22
► To graph the function baxfcy )( we shift the graph of f(x)
right a units (for a>0), left |a| units for a<0)
up b units for b>0, down |b| units for b<0
vertically stretch the graph of f(x) by a factor of c, c>1
vertically shrink the graph of f(x) by a factor of c, for 0<c<1
Example: The graph above can be viewed as a shift of 2xy
(2 units right and 3 units up). The function can be written in the form:
3)2()( 2 xxf
Transcendentals
The test measures your ability to work with exponential, logarithmic and
trigonometric functions of the right triangle and the unit circle. The graphs of
these functions and some common trigonometric formulas are given in the next
few pages.
Exponential Functions
An exponential function is one in which the variable occurs in the exponent.
The base determines whether the function is increasing or decreasing.
1)( aforaxf x 1)( aforaxf x
Domain: all real numbers Domain: all real numbers
Range: y>0 Range: y>0
Solution:
y intercept: (0, 7)
x intercept: none
vertex: (2, 3)
note: (2, 3) is the minimum of f(x)
23
Logarithmic Functions
A logarithmic function is the inverse of an exponential function.
Example: Evaluate: a. 8log2 b. 1log5
c. 001.log10
Solution: a. 38log2 since 823 b. 01log5 since 150
c. 3001.log10 since 001.10 3
Example: Solve the equation for x: 103 xe
Solution: Isolate x by taking the log of both sides. Here use ln the natural log.
10lnln 3 xe
10ln3 x
10ln3x 6974.
Example: Solve the equation for x: 6)13(log 4 x
Solution: Use the inverse function to isolate x 6)13(log
44 4 x
6413 x
4083x
Angle Measure
An angle can be measured in degrees or in radians. A full revolution is 360 or
2 radians. We use this information to convert from one measure to the other.
The following proportion shows the relationship between x degrees and y
radians. 2
360
y
x
Example: Find the degree measure of an angle whose radian measure is 3
2
Solution: 2
360
3/2
x
3/23602 x so 120x
This is the natural log
function, (the inverse of
y=ex)
f(x)= ln x
Domain: x>0
Range: all real numbers
24
Right Triangle Definitions
hyp
oppsin
opp
hypcsc
hyp
adjcos
adj
hypsec
adj
opptan
opp
adjcot
Definitions of Circular Functions
Special Angles
Degree Radian sin θ cos θ tan θ csc θ sec θ cot θ
0 0 0 1 0 -- 1 --
30 π/6 2/1 2/3 3/3
2 3/32
3
45 π/4 2/2
2/2
1 2 2 1
60 π/3 2/3
2/1 3 3/32
2 3/3
90 π/2 1 0 -- 1 -- 0
Sum and Difference Formulas
vuvuvu sincoscossinsin
vuvuvu sinsincoscoscos
25
Graphs of the Trigonometric Functions
f(x)=sin(x) period:
f(x)=cos(x) period:
f(x)=tan(x) period:
Exercise 1: Find the period of
a. x2sin b.
4cos
x c.
3tan
x
Exercise 2: Which has the shortest period?
a. xy cos b. xy cos2
1 c. )2cos( xy d. xy
2
1cos
26
Inverse Trigonometric Functions
The inverse sine function, written )(sin 1 x or )arcsin(x , has
domain [-1,1] and range
2,
2
. It is defined to be the number in the
interval
2,
2
whose sin is x.
In a similar way, )(cos 1 xor )arccos(x is defined to be the number
in the interval ,0 whose cosine is x. The inverse tangent function, written
)(tan 1 x or )arctan(x , is defined to be the number is the interval
2,
2
whose tangent is x.
Exercise 3: Find each of the following:
a.
3
3arctan b. )
3arccos(cos
c. )0arcsin(
27
Double Angle Formulas
uuu cossin22sin
1cos2sin21sincos2cos 2222 uuuuu
Law of Sines
c
C
b
B
a
A sinsinsin
Law of Cosines
Cabbac cos2222
Pythagorean Identity
1cossin 22 xx
Exercise 4: In the triangle below, find the length of the side marked with x.
Exercise 5: In the triangle below, find the length of the side marked with x.
Exercise 6: If 13
5sin A and
12
5tan A , find Acos
28
Answers to Exercises
Basic Algebra
1. 56 2. 167 3. $63.00
4. (a) (b) 16
1 (c)
20
173 (d)
3
210
5. (a) 8
17 (b)
20
91 (c)
24
11 (d)
3
21
6. (a) 25 % (b) %2
18 (c) 152 %
7. (a) .02 (b) .055 (c) .003 8. (a) 22.5 (b) 7.84 (c) 360
9. (a) 89x (b)
5
92
b
a (c)
45
2
y
10. 213460 zyx
11. (a) 23 (b) 21/80
12. (a) 144 x (b) ba 1020 (c) 10
15 x
Algebra I
1. 18 or 23
2. (a) (0, -5) (b) inconsistent system
3. 8
27
4. 33 or 27 5. 75
6. (a) bba 22 23 (b) vu32 (c)
3 252 xxy
7. 624
8. (a) 2
7,
2
7 (b)
2
1,5 (c) 102,102
9. 6,2
1
Algebra II
1. (a) all real numbers except 3x (b) all real numbers except y=0
(c) -1/32
2. (a) all real numbers (b) all real numbers y>0
(c)
√
√
3
21
3x
29
Transcendentals
1. (a) (b) (c) 3
2. c
3. (a) 30 (b)
3
(c) 0
4. 70sin2
1 5. 12 6.
13
12
8