Download MATHEMATICS

MATHEMATICS
in English
2nd year
R. O’Donovan
[email protected]
«
“Take some more tea”, the March Hare said to Alice, very earnestly.
“I’ve had nothing yet”, Alice replied in an offended tone, “so I can’t take more”.
“You mean you can’t take less”, said the Hatter: “it’s very easy to take more than nothing”.
Lewis Carroll, Alice’s Adventures In Wonderland
I
1
2
How Things Are Said
Sign
Example
=
Spoken form
equal or
equals or
are
+
1+2
one plus two
−
3−1
three minus one
×
2×3
two multiplied by three or
six)
: or ÷
six divided by two
6=
6:2
6
2
2 6= 3
>
3>2
3 is greater than 2
≥
4≥3
4 is greater than or equal to 3
<
2<3
2 is less than 3
≤
2≤5
2 is less than or equal to 5
±
±4
plus or minus 4
−
−4
minus 4
a
ba
b to the power of a
2
32
√
4
3 to the two or
23
√
3
2 to the three or
2 cubed
the third root or
the cubic root of 8
√
3
√
3
8
or
is or
or
three take away one
two times three
or
or
or
two into six (goes three)
or
2 is not equal to 3
or
makes
or
2 does not equal 3
one from three
two threes (are
six over two
negative 4
or
b to the a
3 squared
the square root of 4
2 to the power 4 or
10−4
10 to the minus 4
.
0.1
zero point one
.
.2
point two (= 0.2)
3.105
three point one oh five
−4
make
one and two
24 = 16
4
or
or
2 power 4 or
2 to the 4 = 16
(American) naught point one
3
4
Part I
FUNDAMENTALS
Mathematics is yet another language. When spoken orally, it can be pronounced using English,
French, German or any other natural language. It can also be seen as a specific dialect of these
languages.
Mathematics is a language such that absolute certainty can be achieved within.
In order to achieve this, mathematics is a strongly restricted language. Words are redefined
as to become univocal (i.e: one and only one non ambiguous meaning accepted by all)
Symbols
Exercise 1
What sort of symbols have you already used in mathematics? Could you classify them in different
categories?
Statements
Definition 1 (Statement)
A statement is a declarative sentence that is either true or false.
“Pink ice creams taste better” is not a mathematical statement because too many words have
ambiguous interpretations.
No statement is meaningful without a context.
Exercise 2
Which of the following are statements?
(1) Lausanne is a Swiss town.
(8) Some pretty red roses.
(2) The train will probably be late.
(9) The sky is blue.
(10) The rainbow has fifty colours.
(3) 25 is the square of 5 and -5.
(11) A dog.
(4) It is raining.
(5) What time is it?
(12) All the King’s horses and all the King’s
men.
(6) 20 is the square of 10.
(13) This statement is True.
(7) He may have forgotten his key.
(14) This statement is False.
5
Letters
Exercise 3
Letters are used in mathematics. What is their use?
Communication, convenience and good manners are why the context is not always explicitly
given. If for justifiable reasons the context is changed, it must be done explicitly.
1
LOGIC
1.1
True/False
The two fundamental rules of classical logic will always be assumed.
Consistency – or soundness : A statement cannot be simultaneously True and False.
The Excluded Middle Principle : A statement is either True or False (no third or
middle possibility).
The two rules above define what we accept as a statement in the context of classical logic.
The excluded middle principle does not always apply in everyday life.
Definition 2 (negation)
A statement can be true for a certain number of instances of a given context. It is false
for all other instances in the same context.
If a statement is false, then its negation is true.
The negation of a statement is a statement which is the description of what would make the
original statement wrong.
Exercise 4
Give the negation of the following statements:
(1) The night sky is black.
(3) The decimal form of a rational number is
of finite length or it has a repeating sequence of digits.
(2) I am young and handsome.
We will talk of the Truth value of a statement. A truth value is 1 or 0 if considered numerically,
otherwise: True (T) or False (F).
Examples
“2 + 3 = 5” is a true statement (truth value is 1).
“2 + 4 = 3” is a false statement (truth value is 0).
This does not mean that we actually know whether the statement is true or false. It means
that the answer would only be one of the two: for instance “The billion billionth decimal of π is
5” is a statement, but no human nor machine has the answer yet...
6
The words right and wrong are for answers, thus 3 is the wrong answer for the computation
of 2 + 4. These words have nothing to do with logic. Also, they have nothing to do with the
moral concepts of right (good) and wrong (bad).
1.2
Truth Tables
A truth table is a simple way to solve simple logical problems.
For the truth table of “A AND B” put one column for A, one column for B and put all
possible combinations of truth for A and B.
You can think in terms of examples and then generalise. Consider the statement “It is sunny
and it is Monday”
A B A&B
2
1
1
1
1
0
0
0
1
0
0
0
0
Logical operations
Apart from consistency and the excluded middle principle, the basic logical connectors are
(1) ∧ (or &) meaning and:
(2) ∨ meaning or:
A&B reads A and B
A ∨ B reads A or B
(3) ¬ meaning not.
(4) ⇒ meaning implies
¬A reads not A or negation of A
A ⇒ B reads A implies B or if A then B.
Exercise 5
Write the truth table for “A ∨ B”
Exercise 6
Write the truth table for “¬A”.
Exercise 7
Write the truth tables for ¬(A&B) and for (¬A) ∨ (¬B)
Compare the results and interpret with the example: A =“It is sunny”, B = “it is Monday”.
Exercise 8
Write the truth tables for ¬(A ∨ B) and for (¬A)&(¬B).
Compare the results and interpret with the example: A =“It is sunny”, B = “it is Monday”.
7
Exercise 9
Write the truth table for “A XOR B” – where XOR stands for “exclusive OR” which is used when
both cannot be simultaneously true.
2.1
Implication
Exercise 10
Consider A ⇒ B in the case A =“It is raining”, B = “the roads are wet”
If it is raining, then the roads are wet
There are four possible combinations of truth values for A and B. In which cases is the
statement A ⇒ B true?
Exercise 11
What observation on A and B would show that A ⇒ B is false?
What observation on A and B would show that B ⇒ A is false?
Exercise 12
Write the truth table for A ⇒ B.
Exercise 13
Use a truth table to answer: If A ⇒ B and C ⇒ A are both true, and if A and C are true, is
B necessarily true?
Exercise 14
Give an example of a statement of the form A ⇒ B which is true but for which B ⇒ A is false.
Give an example of a statement of the form A ⇒ B which is true and for which B ⇒ A is
also true.
Exercise 15
In the following four situations, the two statements must be accepted. The question is the same
for each: “What is the conclusion” (From Lewis Carrol 1896)
(1) All cats understand French.
(3) All hard working students succeed.
Some chicken are cats.
All ignorant students fail.
(2) None of my sons is dishonest.
(4) All lions are fierce.
Honest men are always respected.
Some lions don’t drink coffee.
Exercise 16
(1) Knowing that if my neighbour is late in the morning she runs to catch the bus, and that
this morning my neighbour was running to catch the bus, is it possible to conclude that
she was late?
8
(2) When it rains the roads are wet. What is the conclusion if the road is wet? What is
the conclusion if the road is dry? What is the conclusion if it is raining? What is the
conclusion if it is not raining?
Exercise 17
Let A be the statement: ’I am tired’ and B the the statement: ’I am happy’.
Translate in English the following statements:
(1) ¬A
(5) ¬(A ∧ B)
(2) ¬¬A
(6) ¬A ∨ ¬B
(3) ¬A ∨ B
(7) ¬(A ∨ B)
(4) ¬(¬A ∨ B)
(8) ¬A ∧ ¬B
2.2
Converse and equivalence
The converse of A ⇒ B is B ⇒ A.
In general, the converse of an implication is not True. In the exceptional case when two
statements imply each other, we say that they are equivalent, and symbolize this with a twoway arrow.
In other words, A ⇔ B means A ⇒ B and B ⇒ A.
2.3
Contrapositive
The contrapositive is the equivalence between A ⇒ B and ¬B ⇒ ¬A.
Equivalence means that proving one of the two is also a proof for the other one.
Exercise 18
Prove, using truth tables, that A ⇒ B and ¬B ⇒ ¬A are equivalent.
Exercise 19
Show that if
“if circles have angles then a square is a circle”
is true then
“if a square is not a circle then circles have no angles”
is also true.
Exercise 20
Show that if the statement “if x > 1 then x2 > 1” is true, then the statement “if x2 < 1 then
x < 1” is also true.
Do not prove the second one directly! Use that “if x > 1 then x2 > 1” is true.
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2.4
Proving/Disproving
To prove a statement means to prove that it is true.
To prove that a statement is not true is sometimes said: to disprove.
There are two kinds of statements: those that state that under certain conditions a given
property can be satisfied and those that state that under certain conditions a given property is
satisfied.
In the context of a zoo:
• All polar bears have perfect teeth.
• Some elephants are white.
To prove the first statement, it will be necessary to check all the teeth of all the polar bears.
To find a single polar bear with a faulty tooth would be enough to disprove it.
To prove the second statement, it would be enough to find a white elephant (even if there
are more than one, to find one is enough.) To disprove it would need checking all the elephants.
In the context of geometry:
• All rectangular triangles with sides of length a, b, c – with c being the longest side –
satisfy a2 + b2 = c2
• Some multiples of 7 can be divided by 3.
Exercise 21
For each of the following, state what would be necessary to prove it and what would be necessary
to disprove it. (It is not necessary to know if the statements are true or not.)
(1) No dog is smaller than a cat.
(2) Some dogs are smaller than some cats.
(3) There is a peace Nobel prize every year.
(4) It never rains when the temperature is over 30◦ C
There are 3 main ways to prove a statement of the form A ⇒ B
(1) Direct proof: assume A is true and use only that to prove that in that case B is necessarily
true.
(2) Proof by contrapositive: assume ¬B is true and use only that to prove that in that case
¬A is necessarily true.
(3) Proof by contradiction (also called reductio ad absurdum): Assume A and ¬B to be both
true and deduce a contradiction (of the from 1 = 0 for instance).
The two last methods rely on the excluded middle principle.
10
Exercise 22
The following example seems to shows that 2 = 1! What went wrong and where?
a−a
=
a−a
multiply both sides by a
a(a − a)
=
a(a − a)
expand the left hand side
a2 − a2
=
a(a − a)
factorise the left hand side using
a2 − b2 = (a + b)(a − b) [Here, a = b]
(a + a)(a − a)
=
a(a − a)
simplify both sides by (a − a)
a+a
=
a
collect similar terms
2a
=
a
divide both sides by a
2
=
1
Exercise 23
Consider the statement
If unicorns exist, then I am the King of Switzerland.
If unicorns existed, then I would be the King of Switzerland.
Is there a difference in their truth values?
2.5
Axioms
In any logical system, some statements have to be accepted without possible proof. They are
called axioms. Euclid (300 BC) was the first to introduce this concept, such as: ’any two points
can be joined by a straight line’. He took his axioms to be self evident.
It is important to understand that it is not possible to get away without axioms when building
a language in which certainty can be achieved. The axioms which are chosen define the context
of a particular logical system. Once the axioms and the rules of logic are accepted, theorems
can be proved and they are absolutely certain within this system. Truth therefore means: truth
relative to a specific list of axioms and rules of logic.
We will consider that integers, rational numbers and real numbers exist and that addition
and multpilication are defined. These are taken as axioms.
11
2.6
Equality
The equal sign “=” relates something on the left hand side with some other thing on the right
hand side by asserting that they are the same. Obvious?
The hidden assumption is that sometimes you have to perform several computations before
the truth value is obvious.
An equal sign does not mean that the two are equal, but the whole object becomes a statement
– which is either true or false.
2 + 4 = 5 + 1 is true because if you compute both sides, you get 6 = 6
A statement is meaningless if it equates mathematical objects which are not of the same
kind. If a statement is incomplete in such a manner that its truth value cannot be decided, then,
formally, it is not a statement!
2.7
Solving Equations
An equation is a conditional statement, it is an equality where one or more letters appear as
unknown values. It is a statement which gives a condition to be satisfied by the unknown(s).
For equations with one unknown:
To solve an equation (unknown written as x) is to find values such that if these values are
substituted for x they produce a true statement.
We conclude that an equation has no solution if we can prove that there is no way to make
the statement true.
Whether true or false, the truth value of a statement is not changed if the same operation
is performed on each side of the equal sign – with two strong restrictions: It is not possible
to divide both sides by 0 because the division by zero is not defined, and it is not possible to
multiply both sides by zero because the truth value is not preserved.
Exercise 24
Why is the division by zero undefined?
Could it be possible to “invent” a value for, say 5/0?
Exercise 25
Why is the multiplication by zero not truth preserving?
An equation is satisfiable if there is (at least) one value which substituted
for the unknown makes it a true statement.
The set of these values are is the solution set.
An equation is unsatisfiable if there is no value which substituted for the
unknown can make it a true statement.
It has no solution. The solution set is empty: { } or ∅.
An equation is an identity if is true for any value substituted to the
unknown.
12
Because an equation contains at least one unknown value, it is neither True not False. All
depends on the value taken by the unknown.
The equation 2x = 6 does not mean that x must take the value 3. It states that if x = 3
then the statement is True and if x 6= 3 then the statement is False. Both are important!
2.8
Quantifiers
∀ means “for all”.
If P is a statement about a property of mathematical objects,
∀x ∈ A
P (x)
reads “for all x in A, x has the property P ” or: “all x in A have property P ” or “all x in A
satisfy P
∀x ∈ N
2·x∈N
means that any natural number multiplied by 2 is a natural number.
∃ means “there exists (at least one)”.
If P is a statement about a property of mathematical objects,
∃x ∈ A
P (x)
reads “there is an x in A such that x has the property P ” or: “there is an x in A which has
property P ” or: “there exists an x in A which satisfies P ”.
∃x ∈ N
x:2∈N
means that there is a natural number which if divided by 2 yields a natural number. ∃ does
not mean that there is only one.
∃ and ∀ are called “quantifiers”.
To disprove
∀x ∈ A
P (x)
it is sufficient to show
∃x ∈ A
¬P (x)
where ¬P (x) is the negation of P , this means “there is an x which does not satisfy P ”
To disprove
∃x ∈ A
P (x)
it is necessary to show
∀x ∈ A
¬P (x)
“All x satisfy non-P ” which by the excluded middle principle is equivalent to “No x satisfies P ”
13
Exercise 26
Let x and y ∈ R. Prove or disprove:
(1) ∀x
∃y
(x + y = 3)
(3) ∀x
∀y
(x + y = 3)
(2) ∃y
∀x
(x + y = 3)
(4) ∃x
∃y
(x + y = 3)
Exercise 27
Prove or disprove the following statements. State whether you are using a direct or indirect
proof.
Make the context explicit.
(1) All prime numbers are odd numbers.
(2) The square of an even number is an even number.
(3) The square of an odd number is an odd number.
(4) If x2 is even and if x is a natural number, then x is even.
(5) If x2 is odd and if x is a natural number, then x is odd.
p √
p2
= 2 then 2 = 2 and thus p is an even number.
q
q
p √
(7) If
= 2 and if p is an even number then q is also an even number.
q
√
(8) No fraction can be equal to 2
(6) If
Exercise 28
(1) Consider the statement “x < 5 and y = 2”. What is its negation?
(2) Consider the statement “∀x ∈ N
2.9
f (x) = 0”. What is its negation?
Theorems
A theorem has three parts:
(1) A context – not necessarily given explicitly in the theorem. It is about geometry, or
functions or numbers, etc.
(2) Initial assumptions; the hypothesis
(3) A conclusion; the thesis
A theorem is a statement which has been proved which relates the hypothesis and the
conclusion in the following manner:
14
Theorem 1 (Example Theorem)
For x satisfying conditions C
P (x)
If x satisfies conditions C then it satisfies the property P . This does not give any information
about x if it does not satisfy condition C.
Theorem 2 (Sum of Angles)
The sum of the internal angles of a triangle is a flat angle
Context: Plane (Euclidean) geometry.
Hypothesis: the figure we are talking about is a triangle.
Thesis: its angles sum up to a flat angle (notice that no numerical value is given)
A theorem is true (because it has a proof) – not sometimes true, nor always true: just True.
2.10
Proof
axiom 1
All flat angles are the same
axiom 2
If same values are added to same values, the results are the same.
If same values are subtracted from same values, the results are the same.
Exercise 29
Prove that opposite angles α and β are equal.
The following sketch is an illustration. The proof is more general.
α
β
axiom 3
Two lines are parallel if and only if the two angles of any pair of corresponding angles of any
transversal are equal in measure.
Exercise 30
Prove theorem 2, page 15.
15
Reasoning and using intelligence are more important than finding the actual answer! This
is why, throughout this mathematics class,
calculators are banned !
16
Part II
INDEXES and SUMS
(The plural of index is indexes or indices.)
Expressions using letters are called literal expressions.
Exercise 31
(1) Write the list of the five successive numbers after n, and the three successive numbers
before u6
(2) Write the first four natural numbers.
(3) Write the first four integers.
(4) Write the first three fractions after
1
4
(5) Write a list of 5 numbers which are not fractions.
(6) Write a list of four fractions, the first being
1
2
and the difference between each is
3
3
Adding all positive integers from 1 to 10 is the sum 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10.
A shorter writing can be used with the concept of a variable: n, i, j or k are often used when
the variable takes only integer values. Add all values of n from 1 to 10 by steps of 1. This is
written
10
X
n
n=1
Exercise 32
Expand each of the following sums and give the result:
(1)
4
X
(5)
j
j=1
(2)
4
X
(4)
k(k − 1)
k=0
2
i
(6)
2
X
2 · 3i
i=0
i=1
(3)
5
X
0
X
2j
(7)
2
X
j=−5
i=0
4
X
4
X
u
(8)
i=0
k=0
17
2xi
ak · xk
Exercise 33
Show that:
a·
10
X
j=0
bj =
10
X
a · bj = a · b0 +
j=0
10
X
a · bj
j=1
3
Exercise 34
X
Which of the following statements about the sum
ak are true? (Explain why).
k=0
(1) The sum has three terms.
(2) The first term is 0.
(3) The last term is three.
Exercise 35
Calculate
3
X
i=1


4
X

i · j
j=2
18
Part III
FUNCTIONS
Exercise 36
(1) What is a function? (definition)
(2) Give examples of functions – from the mathematical world and from the “real world”. Try to
give examples which are very different from one another: difference in the type of function,
difference in the presentation.
(3) Do functions and equations have anything in common? (what? why? where? justify
everything!) give examples or counter-examples.
(4) Give examples of things which “look like functions” but are not – and explain why they
are not.
(5) How can functions be represented? Give examples of each.
(6) On a graphical representation of a curve, which observations would contradict that it can
represent a function?
(7) What is the difference between variable and unknown?
(8) Which of these are functions?
(a) temperature with respect to time.
(b) time with respect to temperature.
(c) y = 2x2 + 5
(d)
f:
R −→ R
x
7→
1−x
(e) u : x 7→ 3x − 3
(f) g(x) = 1
(g) 3x2 + 1
Exercise 37
Specify the domain of the following functions:
(1) f
(2) f
(3) f
(4) f
(5) f
1
: x 7→
x
√
: x 7→ x
√
: x 7→ x − 2
√
: x 7→ x + 3
√
: x 7→ 12 − x
(6) f : x 7→
√
20 + x
1
x+3
√
x−2
(8) f : x 7→
x−4
√
x−2
(9) f : x 7→
x+1
√
4
(10) f : x 7→
(7) f : x 7→
19
x2
x−2
− 3x − 4
√
3
(11) f : x 7→
x2
x−2
− 3x − 4
√
x−1
(12) f : x 7→ √
1−x
Exercise 38
Which of the following graphs represent functions?
Exercise 39
The absolute value, denoted |a| is defined by:
(
a if a ≥ 0
−a if a < 0
|a| =
Sketch the graph of and specify the range of
f : [−1; 4] −→ R
x
7→
|2x − 3|
(start by writing the above function without the absolute value)
Exercise 40
A piecewise-defined function is a function defined by more than one expression:


−4 if x < −1


x − 2 if − 1 ≤ x < 2
g : x 7→


 1 x if x ≥ 2
2
Sketch the graph of g(x).
20
Exercise 41
A step function is a function defined by steps: The cost of parking at an airport is 1CHF for the
first hour or any part of the first hour, and 2CHF for each additional hour or any part of the
additional hour. Sketch its graph.
Exercise 42
Sketch the graph of
(1) f : x 7→ |x + 2| − 3
(2) g : x 7→ |x2 − 4|
(3) h : x 7→ |x − 2| + |x + 3|
Exercise 43
Sketch the graph of
f : [−2; 4] −→ R
7→
x
3x − 2
Exercise 44
Sketch the graph of
g : [0; 10] −→ [−5; −4]
7→
x
2x + 5
Exercise 45
Sketch the graph of
h : [−2; 4] −→ R
7→
x
3x − 200
Exercise 46
Sketch the graph of
h : [2000; 2010] −→ R
x
7→
3x − 200
Exercise 47
Let x 7→ 23 x + 1 be a rule. Sketch the graph of the corresponding function such that the the
point < 2; 4 > is the centre of the drawing and the horizontal interval is 4 units. The vertical
interval must be the corresponding output interval. The axes must be drawn passing through
this point.
21
Exercise 48
If h1; −4i and h2; 3i are two points of a line, what is the slope of the line?
Give the equation of the line.
Exercise 49
Let h3; 5i be a point on a line of slope 12 Give the equation of a point hx; yi on the line. (This
is the equation of a line with respect to h3; 5i).
Exercise 50
Give the equation of the line of slope -2 through Ah100, −650i Sketch this line, centred on A,
horizontal interval of length 0.01
Exercise 51
What kind of function is f :
R∗
x
→R
n
X
7→
ak · xk
k=0
Is it the same as f :
R
x
→R
7→ a0 +
n
X
ak · xk
k=1
22
Part IV
POLYNOMIALS
Exercise 52
Do the following expressions represent polynomials? If so, indicate the degree and coefficients.
Specify which is the leading coefficient.
(1)
2x2 − 5x + 1
√
3
(4) 1000
(5) (t + 5)2 − (t2 + 25) − (10t + 1)
3x7 + 5x − 4
x
√
(3) x2 + 1
(2)
(6)
√
3x2 +
√
5x +
√
7
Exercise 53
Evaluate the following polynomials for the given value of x:
(1) P (x) = x3 + 3x2 + 5x + 6 when x = −2
(2) Q(x) = 2x3 + 5x2 − 7 when x = 1
Exercise 54
Expand and collect similar terms for the following polynomials:
(6) (x + 1)(x2 − 1) − x(1 − x2 )
(1) (x2 + 1)(5 − 2x)
(2) (x2 + 1)(5 − 2x) + 2x3
(7)
x2
+ (x − 1)
2
(4) (4x + 1)2 − (4x − 1)2
(8)
2
(t − 1) − (2t + 6) − 4(t − 5)
3
(5) (x + 1)3 − x(x + 1)2 − 2x2
(9) x2 (x + 2)(2x − 3)
(3) (4t + 1)2 + (4t − 1)2
Exercise 55
Let A(x) = −x4 + x3 + 3x2 + x; B(x) = x4 − x2 + 2; C(x) = 2x2 + x − 3.
(1) Determine, without expanding, the coefficients of x3 and x4 in A(x) · B(x). Same question
for B(x) · C(x).
(2) Determine, without expanding, the degree of:
(a) [A(x) + B(x)]
(b) [A(x) − B(x)]
(c) C(x)[A(x) + B(x)]
(d) [A(x)]2
(e) [B(x)]3 + 3[C(x)]
(3) If n = deg[P ] and m = deg[Q], what can you say about deg[P · Q], deg[P + Q] and
deg[P − Q]?
23
Exercise 56
Can you find two polynomials A and B satisfying the following conditions? If so, specify A and
B and if not, justify why.
(1) The degree of A · B is 10 and the degree of A + B is 3.
(2) The degree of A · B is 11 and the degree of A + B is 3.
(3) The degree of A and the degree of B is 4 and the degree of A ± B is 4.
(4) The degree of A and the degree of B is 3 and the degree of A + B is zero.
3
POLYNOMIAL FUNCTIONS
Exercise 57
What is the difference between a polynomial, a polynomial equation and a polynomial function?
Exercise 58
Consider a square piece of cardboard 12cm× 12cm. Cut a small square of xcm out of each
angle. A box without top can then be made.
(1) Express the volume V of the box with respect to x. Is it a polynomial function?
(2) Compute the zeroes of V . What do these zeroes represent?
(3) Compute others images then sketch the plot of V
(4) Use the plot to show that there are 2 ways of making a box of 100cm3
(5) Estimate the value of x which makes a box of maximal volume.
Exercise 59
Same box as above.
(1) This time express the outside area A of the cardboard with respect to x. Is it a polynomial?
(2) For what values of x is there a maximal area? a minimal area?
Exercise 60
A box is made from a piece of wood 12 meters longs and 6 × 6cm
The ends of the box must be a square with a side of x meters. The total volume is 1m3 . For
which values of x is this possible?
Exercise 61
Chebyshev Polynomials
The polynomial function of Chebyshev of order 4 is f : x 7→ 8x4 − 8x2 + 1 and is used in
statistics. For what values of x does it yield f (x) > 0?
Hint: write z = x2 and solve for second order in z)
24
Exercise 62
A tank is made to stock gas. It is a cylinder with two hemispherical ends. The length of the
cylindrical part is 10m. The volume is V and the radius is r.
Write V (r)
3.1
Polynomial Equations and Roots
Exercise 63
For each of the following polynomials, determine the degree, number of distinct roots and their
multiplicity.
(1) P (x) = (2x + 4)(x + 2)(x − 1/2)(x − 1)(2x − 2)
(2) Q(x) = −3(x + 3)(x + 3)(x + 3)(x − 3)
(3) R(x) = (x − 1)2 (x − 2)2
(4) S(x) = (x2 − 2)(x − 2)2
(5) T (x) = (x2 + 2)(x2 + 4)
(6) U (x) = (x2 + 2) + (x2 − 4)
Exercise 64
Write a polynomial function (in factored form) which has zeroes at x = 2 and x = 1. Is there
only one possible correct answer?
Exercise 65
Determine a polynomial P (x) satisfying the following conditions, where S is the solution set of
P (x) = 0, i.e, the set of roots of P :
2
(1) degree 2; S = −3;
7
(2) degree 3; S = {−2; 0; 12}
(3) degree 1000; S = {1; 2; 3; ...; 998; 999; 1000}
(4) degree 2; S = {1; 2} and P (0) = 10
(5) degree 3; S = {−3; 1; 2} and P (3) = 48
(6) degree 4 having 2 distinct roots of multiplicity 2.
(7) degree 5 having 1 distinct root. This root is 3 of multiplicity 3.
Exercise 66
True or False? Justify.
(1) Two polynomials having the same roots are equal.
25
(2) A polynomial which has exactly two distinct roots is of degree 2.
(3) A polynomial of degree ≥ 2 can always be factored in factors of degree one.
(4) A polynomial whose roots set is S = {1, 2, 3, 4, 5} cannot be of degree 4.
(5) A polynomial whose roots set is S = {1, 2, 3, 4, 5} is of degree 5.
(6) Two polynomials having the same roots and the same multiplicities are equal.
(7) A polynomial of an odd degree has at least one root.
3.2
Dividing polynomials
The division of a polynomial by another polynomial can be made by supposing that the result
exists, that its order of the resulting polynomial is no higher than the difference of the two initial
orders (why?) and writing it in general form. Then multiply this polynomial with the divisor and
work out the coefficients.
Exercise 67
Divide 5x3 − 8x2 − 6x + 4 by x − 2
Exercise 68
Divide 2x3 + 4x − 6 by 2x − 2
Exercise 69
Divide x4 − 5x3 − 2x2 + 20x − 50 by x − 5
Exercise 70
Give the quotient of the following polynomials:
(1) 3x3 + 7x2 + 3x + 7 by x2 + 1
(2) x2 − x5 by x3 − x2
(3) x2 − 9 by x + 3
(4) x4 − 1 by x − 1
Exercise 71
As with the division of integers by integers, the long division will stop when the remainder is
less than the divisor. Here, it will be when the order of the remainder is less than the divisor.
For A and B in the following, give the result of the long division (Division with remainder)
of A by B (C is the quotient and r is the remainder.)
26
(1) A = 3x4 + 2x3 − 8x2 − x + 8
(3) A = 2x3 + x + 2
B = x2 − 1
B = x2 + x − 1
(2) A = x4 − 3x3 − 11x2 + 15
(4) A = 2x3 − 3x2 − 8x + 12
B = 5x2 − x + 6
B =x−5
For each of the above, rewrite the rational function
A(x)
r(x)
= P (x) as Q(x) +
B(x)
B(x)
Exercise 72
P (x)
Show that for any polynomial P , the value P (a) is the remainder of the division
.
(x − a)
Give a condition for which P can be factored.
Justify the following:
Theorem 3 (Roots of polynomials)
If P is of degree n, then P has at most n factors (counting multiplicity).
Show that this implies that if P is of degree n, then P has at most n roots (counting
multiplicity)
Exercise 73
Factorising a polynomial of higher orders: Find a root by trial and error guessing. When you
have found a root x0 , divide the polynomial by x − x0 . Do the same thing with the quotient.
When the quotient is down to order 2, find the roots in the usual way (maybe there are none).
(If you cannot guess an initial root, there is nothing that can be done...)
Factorise completely the following polynomials:
(1) 2x3 − 3x2 − 8x + 12
(3) 2x5 + 3x4 − 3x3 − 2x2
(2) x4 − 9x2 + 4x + 12
(4) x4 + x3 − x − 1
Exercise 74
Solve the following equations.
x3
4
+ 2x = x2 +
6
3
(1) 35t2 + 7t = 0
(7) x3 − 1 = 0
(2) 121x2 − 25 = 0
(8) x3 + 8 = 0
(3) x2 − 4x − 21 = 0
(9) 3x3 = 12x2 + 63x
(4) 2x2 + 2x − 40 = 0
(10) t2 (t − 14) + 13t = 0
(5) 2y 2 + 11y + 9 = 0
(11) 2y 3 + 11y 2 + 9y = 0
(16) x4 + 4x + 12 = 9x2
(6) x3 + 1 = 0
(12) x3 + 3x2 + 3x + 1 = 0
(17) x4 − x2 − 6 = 0
27
(13)
(14) x4 − 1 = 0
(15) x4 + 1 = 0
3.3
Polynomial Inequalities
Exercise 75
Fill in the dots with < or >.
(1) a < b
⇔
a + 1......b + 1
(2) a < b
⇔
a − 1......b − 1
(3) a < b
⇔
(−a) . . . . . . (−b)
(4) a < b
⇔
(5) a < b
⇔
1
1
......
a
b
1
1
−
...... −
a
b
Exercise 76
Solve the following inequalities.
(1) x2 − 4x − 21 < 0
(6) x4 − 1 < 0
(2) 2y 2 + 11y + 9 ≥ 0
(7) x4 + 4x + 12 − 9x2 ≥ 0
(3) −2u2 + 12u + 14 < 0
(8) (x + 3) · (x + 1) · (x − 1) ≥ 0
(4) 2y 3 + 11y 2 + 9y ≤ 0
(9) 3x + 2 < 5
(5) x3 + 3x2 + 3x + 1 > 0
(10) 5 − 2x > 3x + 2
Exercise 77
Draw a sign table for the polynomial for each of the following polynomials.
(1) x2 − 4x − 21
(5) x3 + 3x2 + 3x + 1
(2) 2y 2 + 11y + 9
(6) x4 − 1
(3) −2u2 + 12u + 14
(4) 2y 3 + 11y 2 + 9y
(7) x4 + 4x + 12 − 9x2
Exercise 78
Solve the following inequalities and write the solution set using interval notation:
(1)
x 1
x x
+ ≥ +
2
3
4 2
(4) (x − 7)(x − 3) + 1 ≥ (x − 2)2
(5) 16x2 ≥ 9x
x−4
7x
(2)
<
− (3x − 2)
2
2
(6) x3 > x
(3) (x2 − 1) − (x + 1)2 ≤ 0
(7) −5 ≤ 1 − 3x ≤ 10
28
(8) −6 < 2x − 4 < 2
(9) −x + 2 < 2x − 1 < −2x + 1
Exercise 79
A certain experiment requires that the temperature be between 30 and 40 degree Celsius.
Farenheit and Celsius degrees are related by the formula
5
C = (F − 32)
9
what are the permissible temperatures in Farenheit?
Exercise 80
A construction company needs to purchase a new crane and is hesitating between two models.
Model A costs 50’000CHF and has a yearly maintenance fee of 4’000 CHF. The model B costs
40’000 CHF and has a yearly maintenance fee of 5’500 CHF. For how many years will the
company have to use model A before its yearly cost is cheaper than model B?
4
RATIONAL EXPRESSIONS
Exercise 81
Simplify the expressions and specify the domain:
4
2
+
x2 − 3x + 2 x2 − 1
(1)
x2 − 4
2
·
x + 2 4x − 2x2
(8)
(2)
2y + 6y y 2 − 2y
·
y−2
8
(9)
(3)
a2 a2 − 25 4a2 + a
·
16a3 − a a2 + 5
(10)
3
8z
2
+ 2
−
2z − 1 4z − 1 2z + 1
(4)
25x3 − x x − 1
10x
·
· 2
2
5x
5x − 1 x − 1
(11)
a2 − 4 a2 − 6
−
a2 + 6 a2 + 4
(5)
a2 − a a + 2
÷
(4a)2
2a
(12)
2 − 3x 2 + 3x
−
2 + 3x 2 − 3x
(6)
9x2 − 1
3x + 1
÷ 3
x2 − x
x − x4
(13)
z+3 z+1
−
z+2
z
(7)
a2 + a − 2
a2 + 7a + 10
÷
a2 + 2a − 15 a2 + 10a + 25
(14)
Exercise 82
Solve the following equations.
(1)
13 + 2x
3
=
4x + 1
4
29
2x
4x
+
− 1 2x + 1
4x2
t2
t
1
−
− 25 2t + 10
(2)
3
9
=
7x − 2
3x + 1
(3) 2 −
5
=2
3x − 7
(4)
2
4
7
+
=
5 10x + 5
2x + 1
(5)
2x + 3 1 − 2x
4
2x + 3
+
=
+ 2
x
x+1
x+1 x +x
(6)
3x + 2
1
x2 − 6
−
= 2
+2
x
x−3
x − 3x
(7)
x2
5
4
1
+ 2
= 2
−x x +x
x −1
30
5
ASYMPTOTES
Consider the curve of f : x 7→
x2
x+2
y
x
-2
Exercise 83
For values of x close to −2 the function has large positive or negative values. Why?
For large positive or negative values, the function is close to the oblique dashed straight
line. Why?
Exercise 84
x−7
Let f : x 7→
.
x+7
(1) What is the domain of the function f ?
(2) Determine the y-intercept and the x-intercept.
(3) Draw a sign table for the function f .
(4) What happens to f (x) when x becomes very close to -7?
(5) What happens to f (x) when x becomes very large (in absolute value)?
(6) Sketch a graph of function f .
31
Exercise 85
For each of the following functions:
(1) Determine its domain.
(4) Determine the equations and the nature
of all the asymptotes, if any.
(2) Determine the y and x-intercepts.
(3) Draw its sign table.
(5) Draw its graph.
• f1 : x 7→
3x + 5
4−x
• f5 : x 7→
x−2
(x − 3)(5 − x)
• f9 : x 7→
• f2 : x 7→
2x + 1
x−2
• f6 : x 7→
(x + 1)(x − 2)
3x + 1
• f10 : x 7→
• f3 : x 7→
x+4
(x + 1)(x − 3)
x−1
• f7 : x 7→ 2
x −4
• f4 : x 7→
x−2
(x − 3)(x − 5)
• f8 : x 7→
x2 − 6x + 9
x−3
32
1−x
2−x
√
x2 + 4
√
• f11 : x 7→
• f12 : x 7→
x2 + 4
x
x
x2 + 1
6
COMPOSITION OF FUNCTIONS
f (g(x)) or f ◦ g(x) (f circle g) is a composite function.
First g then f on the result. If f and g have domain R, we have:
g
f
R −→ R −→ R
Exercise 86
g : x 7→ x − 1 and f : x 7→ x2 . Write f ◦ g(x) and g ◦ f (x)
Exercise 87
Let f : x 7→ x2
g : x 7→
1
x+1
h : x 7→
(1) Find the domain of each function.
(2) Give f ◦ g, f ◦ h, h ◦ f , g ◦ f , g ◦ h, h ◦ g, f ◦ f .
Exercise 88
Write the following as composites of elementary functions:
√
(1) f : x 7→ 1 − x2
(4) j : x →
7 sin2 (x + 1)
(2) g : x 7→ (x + 5)3 + 1
(3) h : x 7→
x3
(5) k : x 7→ (5 +
2
−1
(6) l : x 7→
Exercise 89
Let
f : x 7→ x2 + 3x + 1
Give in expanded form:
(1) f (x + 1)
(2) f (2x)
(3) f (x + h)
33
√
p
3
(x + 2)2 )2 + 18
x+
√
4
x+1
√
x
Exercise 90
1
x2 + 2x + 1
For f (x) = |x|, g(x) = x − 1, h(x) = and k(x) =
x
x+1
(1) Find the domains of f and k.
(2) Determine the y and x-intercepts of f and k.
(3) Write their sign tables.
(4) Find all the asymptotes of f and k, if any.
(5) Sketch f and k.
(6) Determine f ◦ g and g ◦ f , h ◦ f as well as their domains. Find all the asymptotes of f
and k, if any.
(7) Write the sign tables for f ◦ g, g ◦ f and h ◦ f
(8) Sketches the graphs for f ◦ g, g ◦ f and h ◦ f
(9) Write the sign table for h ◦ g ◦ f
(10) Find the asymptotes of h ◦ g ◦ f .
(11) Sketch the graph of h ◦ g ◦ f .
6.1
Recursive Composite Functions
It is sometimes possible to define a function by using the function itself in the definition. This
will not lead to a contradiction or impossibility to solve if, after finitely many steps, a defined
value of the function is reached and if the definition of the function enables to calculate back
to the initial value. An object which is described by using its own description is recursive.
For example, when a video camera films the screen on which the camera output appears, its
generates a recursive picture: a picture of the screen on which there is a picture of the screen
on which there is... etc.
Exercise 91
The Fibonacci Sequence:
f : N −→ N
∀n ≥ 2
f :1
7→
1
f :2
7→
1
f :n
7→
f (n − 1) + f (n − 2)
Calculate f (10)
34
Exercise 92
Using the Fibonacci sequence:
g : N −→ N
g:n
7→
f (n)
f (n − 1)
Calculate g(10)
Exercise 93
More tricky:
h : N −→ R
(
x − 10
h : x 7→
h(h(x + 11))
if x > 100
if x ≤ 100
Give fullest possible description of the values taken by the function.
By stating that the output set is R we are sure that this set is big enough. What is the
smallest possible output set; i.e: the set of all outputs: the range of h.
35
7
INVERSE FUNCTIONS
Exercise 94
Let f : x 7→ x2
g : x 7→
Find F, G, H such that
(1) f ◦ F (x) = x
[= F ◦ f (x)]
(2) g ◦ G(x) = x
[= G ◦ g(x)]
(3) h ◦ H(x) = x
[= H ◦ h(x)]
1
x+1
h : x 7→
√
x
Exercise 95
Let f : x 7→
3x + 2
.
2x − 5
(1) Determine the domain, intercepts, asymptotes, draw the sign table and sketch the graph
of f . Specify its range.
(2) Determine f −1 (x) and its domain.
(3) Show that (f ◦ f −1 )(x) = (f −1 ◦ f )(x) = x
Exercise 96
Let f be defined as follows:
f:
[0; +∞[ −→ B
x
7→
−x2 + 3
Specify the largest possible set B so that f is one-to-one and determine f −1 .
Sketch the graph of f and f −1 on the same yx−plane.
Exercise 97
For each of the following functions, find the largest possible restriction of its domain where it is
one-to-one, therefore invertible, and find its inverse for that restricted input set. The restricted
function and its inverse will have their input and output sets (restricted domain and range)
defined explicitly.
(5) e : x 7→ x5
(1) a : x 7→ 2x + 5
(2) b : x 7→
5x + 1
2x − 3
(6) f : x 7→
1
x−2
(7) g : x 7→
3−x
2x + 1
(3) c : x 7→ 6x − 4
(4) d : x 7→ x4
36
(8) h : x 7→ x2 − 5x + 6
(9) j : x 7→ 10x
Exercise 98
Here is the graph of a one-to-one function.
(1) Sketch the graph of its inverse.
(2) Give the domain and range of f .
(3) Give the domain and range of f −1 .
37
8
SHIFTING
Exercise 99
Given the function f and its graph. Draw approximately, the graph of
(1) f (x) + 1
(4) 2f (x)
(2) f (x + 1)
(5) f (−x)
(3) f (2x)
(6) −f (x)
f
f
1
1
1
1
f
f
1
1
1
1
f
f
1
1
1
1
38
Part V
EXPONENTIAL and LOGARITHM
Exercise 100
ax
ax · ay = ax+y , y = ax−y and (ax )y = axy
a
By extending these rules, prove that ∀a, b ∈ N
(1) x−a =
1
xa
1
√
a
x
a
√
b
(3) x a =
(2) ∀x 6= 0 x0 = 1
(4) x b =
xa
Definition 3 (The logarithm function)
The inverse function of the exponential function ax = y is loga (y) = x
It reads: logarithm in base a of y
Exercise 101
Calculate
(1) log10 (100)
(3) log10 (105 )
(2) log10 (1)
(4) log10 (10n )
Exercise 102
Prove the following:
Theorem 4 (Logarithms)
loga (x) + loga (y) = loga (x · y)
n · loga (x) = loga (xn )
Useful bases for logarithms are the base 10, written log, base 2, written Log and base e ≈ 2.7
written ln.
Exercise 103
For base 2, calculate Log(1/2), Log(1), Log(2), Log(4), Log(8) and draw a reasonable approximation for the curve of x 7→ Log(x)
Exercise 104
For base 10, calculate log(1), log(0.1) and log(0.000001).
39
Exercise 105
Assuming log(2) ≈ 0.3, calculate
(1) log(20)
(2) log(0.2)
(3) log(0.5)
(4) log(8)
Exercise 106
Changing base:
Express loga (x) in terms of logb
(Use the definitions and the fact that exponential and logarithm are inverses.)
Exercise 107
Use the following graph to find log(6), log(0.2), log(6.2)
1
y
0
x
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
-1
-2
-3
The Richter Scale
The Richter Scale is a system used to measure the magnitude of an earthquake. It enables
one to compare the energies which are released at the epicentre by different earthquakes. The
magnitude M is related to the energy E by the following empirical formula:
log(E) = 4.4 + 1.5 · M
Charles Richter established this measure in California in 1935.
40
Energy is measured in Joules (J), 1J is equal to the work done when the point of application
of a force of 1 Newton moves through a distance of 1 metre. (Or approximately, the energy
needed to move an object of 100g by 1m, on earth)
An earthquake of magnitude 3 is hardly felt and on a small range.
Small damages occur with a magnitude of 4.5
Magnitude of the Skopje Earthquake in 1963 was 6
Magnitude of the San Fernando Earthquake in 1971 was 6.6
Magnitude of the Lisbon Earthquake in 1963 was 9 (This was the worst earthquake ever
registered.)
Exercise 108
Find the energy of an earthquake of magnitude 2.2
Exercise 109
Find the magnitude of an earthquake which releases 1010 J
Exercise 110
The formula shows that an earthquake of magnitude 8.5 releases one billion times more energy
than an earthquake of magnitude 2.4. Explain why.
Exercise 111
Suppose that one earthquake releases one million times more energy than another. Find their
difference in magnitude on the Richter scale.
Exercise 112
Find the energy released with a magnitude of 1.
Exercise 113
Find the energy released with a magnitude of 0.
Exercise 114
Is it possible to give a magnitude when there is no earthquake, i.e: when there is no energy
released? Explain your answer.
The Richter scale is an open scale because it has no theoretical upper bound.
41
DECIBELS
Decibels (dB) is a unit for expressing the ratio between two amounts of acoustic power or
for measuring the relative loudness of sounds. The name comes after Graham Bell, inventor of
the telephone.
Intensity is objectively measured.
Loudness is a subjective phenomenon and can be measured only comparatively. Loudness is
the human sensation of intensity.
Originally, the unit was the bel, but it was too big for practical reasons. It is now the decibel:
10dB = 1B
If I1 and I2 are two intensities, then the loudness difference (in dB) is
I1
log10
I2
The decibel measure is therefore a comparison:
0dB is the limit of audition of a young and healthy listener. It is the basis for all measurements.
5dB is a very silent room
30dB is a whisper et 50cm
60dB is a normal conversation
86dB is city traffic from inside the car
100dB is a pneumatic hammer at 2m distance (very loud)
120dB is the threshold of pain
140dB is intolerable
Some Rock concerts reach the 120dB limit for long durations...
Exercise 115
Find the increase in dB when the intensity doubles.
Exercise 116
If 10 singers in a choir can sing to 80dB, how many singers would it take to sing to 82dB?
Exercise 117
What is the ratio of intensities between the two extremes: 0dB and 120dB?
Exercise 118
If the threshold of pain is 120dB, how many dB is half the intensity? How much more intensity
is 121dB?
Exercise 119
When a conversation starts in a very silent room, what is the ratio of intensities between before
and after the beginning of the conversation?
42
Exercise 120
Consider a population of bacteria which doubles every hour. At t = 0 (initial time), the population
contains 1000 bacteria.
(1) Give the formula expressing the population P as a function of time t.
(2) Sketch the graph this function from[−5; 5].
(3) After how many hours will the population be multiplied by 4, 8, and 16?
(4) After how many hours will the population be multiplied by 3? (give your best approximation).
(5) Write the equation corresponding to the above question. Can you solve it? If so, do it. If
not, why?
(6) After how many hours will the population be multiplied by 5? (give your best approximation).
(7) Write the equation corresponding to the above question. Can you solve it? If so, do it. If
not, why?
(8) At what time did the population contain 500 bacteria? 250? 150?
(9) Write the equation corresponding to the above question (number of bacteria = 150).
(10) Looking at the graph, specify the domain and the range, draw a sign table and specify the
y- and x-intercepts.
(11) Is this function one-to-one? If so, sketch its inverse.
(12) Looking at the graph of its inverse, specify the domain and the range, draw a sign table
and specify the y- and x-intercepts.
43
44
Part VI
TRIGONOMETRY
Angles
An angle is defined by two lines meeting at a point called the vertex of the angle. It can be
regarded as the measure of the rotation involved in moving from one line to coincide with the
other line. It has a direction represented by the sign.
The measure of the rotation is performed by drawing a circle centred at the vertex and
observing what amount of the circle is covered by the rotation.
Up till now, the measure for angles was done in degrees. One degree (1◦ ) is the angle
1
defined by 360
th of a circle. i.e: one complete turn is 360◦
!
4
This measure does not depend on the size of the circle.
Exercise 121
Another measure is based on an arc of a circle with centre at the vertex of the angle. If r is the
radius of the circle and l the length of arc subtending the angle, then the angle is rl i.e: it is the
proportion of a complete turn measured by the length along the circle compared to its radius.
This is called the radian measure.
Considering a circle of radius r:
(1) What is the measure of a complete turn (a round angle)?
(2) What is the unit?
(3) What is the measure of a right angle (a quarter turn)?
(4) What is the measure of a flat angle?
(5) Why is it that a radius equal to 1 will be used to simplify things?
Exercise 122
Find the formula for transforming a degree measure to a radian measure.
Exercise 123
Transform the following into radian measure:
(1) 60◦
(6) 120◦
(2) 22.5◦
(7) 10◦
(3) 30◦
(8) 1◦
(4) 15◦
(5) 7.5◦
(9) 452◦
45
Exercise 124
Find the formula for transforming a radian measure to a degree measure.
Exercise 125
Transform the following into degree measure:
(1) 1
(2)
(5)
π
3
7π
4
(6) 15π
(7) 16π
(3) 0.1
(8) 3
2π
(4)
3
(9) 1.5
Exercise 126
The minute hand of a clock measures 12cm. What is the distance covered by the tip of the hand
in 20 min? What is the total distance covered in one day?
(use π ≈ 3)
Exercise 127
Given that the circumference of the earth is (theoretically) 40 000km, what is the radius of the
earth?
Exercise 128
One nautical mile is the length of arc of 1 minute at the surface of the earth.
What is one minute of arc in radian measure?
What is the length of one nautical mile? 1
Exercise 129
If a bridge were built at a constant height of 10m all around the earth’s equator – with its
theoretical measure of 40 000km, how much longer than the equator would the bridge be?
Exercise 130
A circular sector is the part of a disc lying between two radii.
Find the area of a sector of angle θ (in radian measure).
Exercise 131
A circle has a radius of 2.5m. Find the area of a sector of angle
3π
4
1
Because the earth is not a perfect sphere, the official length of the nautical mile is now slightly different and
does not depend on which part of the earth you are.
46
Exercise 132
In a unit circle, what is the angle of a sector of area 1? (radian measure).
Angles are a measure of rotation. The positive direction is anticlockwise!
Because radian measures have no units and as in mathematics units are considered an
unnecessary nuisance:
!
4
From now on till the end of college, in mathematics,
only radian measure will be used.
Exercise 133
On the following unit circle, divided into 24 sectors, write on the outside, the corresponding
angles in radian measure.
47
Circular functions
In a right-angled triangle the following ratios have been defined:
opposite
sin(α) =
hypotenuse
hypotenuse
adjacent
cos(α) =
opposite
hypotenuse
α
opposite
adjacent
tan(α) =
adjacent
We now redefine these ratios in the Trigonometric Circle which has a radius equal to 1
centred on the origin and angle direction is anti-clockwise.
The original definitions work only for positive values (lengths are always positive), i.e: only
for the first quarter circle. The definition is now extended to the whole circle: the sine is the
vertical coordinate of a point, the cosine is its horizontal coordinate, the position of the point
itself depends on the arc-length. If one considers the ration between arc-length and complete
turn, the the position of the point depends on the angle measures on the outside circle of radius 1.
y
1
θ
sin(θ)
cos(θ)
1 x
As the radius is 1, we then have
opposite
= opposite
1
adjacent
cos(α) =
= adjacent
1
opposite
sin(α)
tan(α) =
=
adjacent
cos(α)
sin(α) =
Exercise 134
Using the figure of exercise 133, give an approximation of
(1) sin(π/3)
(2) sin(π/4)
(3) cos(π/12)
(4) cos(1)
48
If you unroll the circle, the circular functions appear as in the following
The Sine curve
y
θ
-1·π
-1
0·π
1·π
2·π
3·π
The Cosine curve
x
θ
-1·π
-1
0·π
1·π
2·π
3·π
The Tangent curve
y
θ
-1·π
-1
0·π
1·π
2·π
3·π
Exercise 135
Use the definition of the trigonometric ratios to prove the following theorem (explain why it is
true).
Theorem 5 (Relation Between Sine and Cosine)
for any θ
sin2 (θ) + cos2 (θ) = 1
Exercise 136
On the circle of exrecise 133, draw with best possible approximation, an angle of π/5.
Where are sin(π/5), cos(π/5) and tan(π/5).
Give their approximate values.
49
Exercise 137
Calculate precisely – give a precise algebraic expression – and fill in the following table:
θ
sin(θ)
cos(θ)
tan(θ)
0
π/6
π/4
π/3
π/2
Exercise 138
Considering the graph of the functions and the definitions, find the relations between
(1) sin(x) and sin(−x)
(2) cos(x) and cos(−x)
(3) tan(x) and tan(−x)
(4) x and y if sin(x) = cos(y)
Periodicity
A function is periodic if its curve repeats exactly the same pattern over and over again.
Definition 4 (Periodic Function)
A function f (x) is said to be periodic if there is a real number α such that
∀x f (x + α) = f (x)
Exercise 139
Show that for any periodic function, if ∃α
f (x) = f (x + α) then ∀n ∈ N f (x) = f (x + n · α)
50
Definition 5 (Period of a function)
The fundamental period or simply the period of a function, is the least possible p such
that ∀x f (x + t) = f (x)
Exercise 140
What are the periods of the trigonometric functions?
Exercise 141
Are the trigonometric functions defined on all the interval of a period?
Exercise 142
Determine the domain, the zeroes and the period of:
(1) f : x 7→ sin(2x)
(5) f : x 7→ sin(−x)
(2) f : x 7→ cos(π − 2x)
x
(6) f : x 7→ cos
(3) f : x 7→ sin(x + π/2)
(7) f : x 7→ tan
(4) f : x 7→ cos(x − π)
3
x
π
Exercise 143
Solve the following equations:
(1) cos(x) = −
(2) sin(x) =
√
1
2
(5) cos(x) = −
2
2
√
1
2
(6) sin(x) =
(3) cos(x) = −1
√
2
(4) sin(x) =
2
3
2
(7) sin(x) = cos(x)
(8) sin(x) = tan(x)
Exercise 144
1
(1) cos(2x + π) =
2
√
x π
3
(2) sin( − ) =
2
3
2
(3) tan( πx + π) = 1
Exercise 145
Solve using exercise 133:
(1) cos(x) = 0.4
(2) sin(x) = −0.3
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Definition 6 (Arc-functions)
The inverse function of the trigonometric function are
• arcsine arcsin(x)
the arc (angle) whose sine is x
• arccosine arccos(x)
the arc (angle) whose cosine is x
• arctangent arctan(x)
the arc (angle) whose tangent is x
As sin(x) ∈ [−1; 1] (same for cos(x)) the input for arcsin(x) must be restricted to [−1; 1]
The input for arctan(x) does not need any restriction.
Exercise 146
Give the domain and range of the arcsine function. Same question for arccosine and arctangent.
Exercise 147
Use exercise 133 to approximate
(1) arcsin(x) = .8
(2) arccos(2)
(3) arctan(4)
Exercise 148
Observe the following drawing where the angle β has been drawn on top of the angle α.
(1) Explain why the angle right at the top is equal to α
(2) Express the lengths of a, b and c in terms of sin(α), cos(α), sin(β) and cos(β).
y
α
b
c
β
α
0
a
52
x
1
Exercise 149
Finish the proof of
Theorem 6 (α + β)
sin(α + β) = sin(α) cos(β) + cos(α) sin(β)
cos(α + β) = cos(α) cos(β) − sin(α) sin(β)
Exercise 150
Prove also the following theorem
Theorem 7 (tan(α + β))
tan(α + β) =
tan(α) + tan(β)
1 − tan(α) tan(β)
Measuring the Earth
Exercise 151
The Greeks assumed that the earth was spherical and even measured its diameter over 2500
years ago. This exercise basically reproduces this method – only we will use modern units:
metres, kilometres, radians.
In two cities 500km apart on a North/South axis, somewhere in Egypt along the Nile, a 1m
pole is planted vertically. On a given day, in each city, a person measures the length of the
shade every half hour or so. The person notes the shortest length on a piece of paper. [The
measure of the shortest shade for two cities on a North/South axis ensures that the shades are
measured at the same time.]
In city A (to the North) the shortest shade of the 1m pole measures 0.932m
In city B (to the South) the shortest shade measures 1.089m
The sun rays are assumed to be parallel.
What is the diameter of the Earth?
(Sketch the situation)
Law of Sines
Solving a triangle consists of finding each part of the triangle (all the angles and sides) given
only three pieces of information.
53
Exercise 152
Consider the sketch of following triangle. give:
(1) its area;
(2) the other two angles
(3) the length of the segment AC
C
h
A
3cm
π/6
5cm
B
Exercise 153
Consider a triangle with angles α, β and γ. The opposite sides are, in the same order, a, b and c.
(1) Define the area of the triangle in terms of:
(a) sin(α)
(b) sin(β)
(2) Show then that
sin(α)
sin(β)
sin(γ)
=
=
a
b
c
These last equalities are known as the Law of Sines.
Exercise 154
(find exact values)
2π
π
, and β =
, what is the value of γ?
5
3
π
β = , a = 5 and b = 3, what is the value of c?
2
π
π
α = , β = and c = 4, what is the value of b?
3
6
π
α = β = and a = 5, what is the value of c?
4
π
π
α = , β = and a = 10, what is the value of b?
4
6
π
a = 3, b = 2 and γ = , what is the value of c?
6
(1) If α =
(2) If
(3) If
(4) If
(5) If
(6) If
54
(c) sin(γ).
Exercise 155
(find approximate values)
π
π
and β = , what is the value of b?
5
6
π
(2) If a = 3, b = 2 and γ = , what is the value of c?
5
π
(3) If a = 4, c = 2 and β = , what is the value of b?
6
(1) If c = 4, γ =
(4) If a = 4, b = 5 and c = 6, what is the value of cos(γ)?
(5) If a = 4, b = 5 and α =
π
, what is the value of sin(β)?
3
Exercise 156
Show that
sin(α) − sin(β)
a−b
=
sin(α) + sin(β)
a+b
55