Download Lesson 7 Mathematical Proof (Part 1)

Mathematics C30
Module 2
Lesson 7
Mathematics C30
Mathematical Proof (Part 1)
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Lesson 7
Mathematics C30
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Lesson 7
Introduction
Words and their meanings are the basis for communication between people. When you
participate in a debate, write an essay, or make a point in a discussion with someone, you
use some form of argument, which may be logical, emotional, invalid, or possibly
nonsensical. In this lesson you learn about some of the different forms of argument or
proof that are used in mathematics or in person to person communication.
The Mathematics A30, B30, and C30 Curriculum Guide from Ministry of Education is the
source for many of the examples and exercises used in this lesson.
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Objectives
After completing this lesson, the student will be able to




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define and illustrate by means of examples, inductive and
deductive arguments.
recognize and analyze conditional statements, and
correctly apply proof by counter example.
define and illustrate analogical statements.
prove trigonometric identities using deduction, or prove
the identity false by supplying a counter example.
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7.1 Arguments and the Need for Proof
We make choices about what to believe, and we prefer to believe that which is true. Some
of what we believe to be true may be a result of our intuition, which often serves us well,
but there are times when it will mislead us.
Other beliefs are a result of some form of logical argument, where the truth of a certain set
of beliefs leads to the truth of another. This logical argument may be an inductive
argument or a deductive argument. Unfortunately, even these arguments provide no
guarantee that the truth is arrived at, since the beliefs with which we begin the argument
are sometimes in error.
Let us begin by carefully defining the terms that will be used in this lesson and by
supplying examples.
A statement will be defined as a sentence that is either true or false.

Questions and imperatives (order or command) are not statements because they are
usually neither true nor false.
Each of these are statements.


A square has four congruent sides.
It is not raining now.
These are not statements.



Are the lines parallel?
Prove that the triangles are similar.
Please close the door.
An argument is two or more statements related to each other in such a way that one of the
statements, called the conclusion, is supported by the other statement, called the premise.
Premise
Conclusion
We can say that the conclusion is derived from, or inferred by the premises. A mere
collection of statements is not an argument since inference is supposed to hold between
them. That is to say, some words must be there stating that one statement implies the
other.
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“It is November. Oranges are on sale.”

This is a collection of statements with no indication which is the premise and which is
the conclusion.

“Since oranges are on sale, it must be November.” This is an argument in which the
premise is “Oranges are on sale,” and the conclusion is “It is November.” The word
“since” adds inference to the collection of statements.

“Since it is November, oranges are on sale.” This is another argument using the same
statements with the premise and conclusion interchanged.
The following example illustrates an argument.
All Terriers are dogs.
All Pitbulls are dogs.
Therefore, all Terriers are Pitbulls.


The first two statements are the premises, and the last statement is the conclusion.
The word “therefore” prevents this from being a mere collection of statements without
inference.
Note, however, that the conclusion is false even though the premises are true.
Such an argument is called invalid.
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Example 1
Combine each collection of statements to form an argument using the words “if ”
and “then.”
a. It is raining. The grass is wet.
b. The triangle is isosceles. The triangle is equilateral.
Solution:

In each case two arguments can be written.
a.
If it is raining, then the grass is wet.
If the grass is wet, then it is raining.
b.
If the triangle is isosceles, then the triangle is equilateral.
If the triangle is equilateral, then it is isosceles.
In each case the “if” statement is the premise and the “then” statement is the conclusion.
Note that not all of the conclusions are true. Nevertheless, each is an argument.
Exercise 7.1
1.
A motorist drove the 300 km from Saskatoon to Meadow Lake at a speed of 100
km/h. Poor visibility caused the motorist to make the return trip at 80 km/h. What
was the motorist’s average speed for the trip?
a.
b.
What is your intuitive answer to the question?
Let’s check it out.
How long does it take to drive there?
How long does it take to drive back?
What is the total time spent driving?
What is the total distance traveled?
What then is the average speed?
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2.
Two containers, one holding a litre (1000 mL) of cola, the other holding a litre of
coffee, are standing beside one another. A cup (250 mL) of the cola is transferred to
the coffee container and thoroughly mixed in with the coffee. A cup of the coffeecola mix is then transferred back to the cola container. Is there more coffee in the
cola container or is there more cola in the coffee container?
a.
b.
What is your intuitive answer?
Let’s check it out.
Cola Container
Amount
Amount
Cola
Coffee
Action taken
Coffee Container
Amount
Amount
Cola
Coffee
Original situation
1 cup from cola container to
coffee container
1 cup from coffee container to
cola container
3.
If it were possible to wrap the earth with a metal ring at its equator, you would
need a ring whose circumference was approximately 40 000 km. Suppose you
inserted an extra 2 m (0.002 km) into the ring so that it is now 40 000.002 km in
length. This ring would no longer be snug against the earth. Do you think there
would be enough room for you to crawl between the earth and the ring? Check your
intuitive answer. (Hint: C  d , use the value of  found on your calculator.)
4.
State whether each sentence is a statement or not. If it is a statement, tell whether
it is true or not.
a.
b.
c.
d.
e.
f.
Buy some milk at the store.
Are these lines parallel?
What do people with claustrophobia fear?
Every triangle is a polygon.
If x  5  19 , then x  15 .
Were some students late?
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5.
Use each collection of statements to write an argument. Underline the conclusion of
each argument.
a.
b.
c.
6.
A line has exactly one point of intersection with a circle.
This line is tangent to a circle.
It is raining.
It is cloudy.
When it is cloudy, it rains.
This polygon has six sides.
This polygon is not a pentagon.
A pentagon has five sides.
In each case write one argument using the words “if” and “then.”
a.
b.
This is a cat. This animal has four legs.
x  y 2, x 5  y 7
7.2 Conditional Statements and the Counter
Example
In the Section 7.1, Example 1, arguments were formed by combining two statements into
the “if – then” form. The “if – then” argument itself can appear as a statement in another
argument.
Example 1
Combine each of these statements into an argument.
Today is Monday.
If it is Monday, then Karen has music lessons.
Karen has music lessons today.
Solution:



If it is Monday, then Karen has music lessons.
Today is Monday.
Therefore, Karen has music lessons.
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Statements of the form, “if (premise or hypothesis) and then (conclusion),” are called
conditional statements, or just conditionals. They occur frequently and are essential to
many logical arguments. A Venn diagram is helpful for analyzing conditionals.
Example 2
Draw a Venn diagram and analyze the following conditional.
If it is a cat, then it is an animal.
Solution:


Let p be the circle that represents all cats, and q, the circle that represents all animals.
Circle p must be drawn within circle q.
q
A
B
p
This illustrates that any point in p is also in q. Any cat is an animal. Also, there are
points, such as A in q, that are not in p. The animal is not a cat. Any point outside of q,
such as B, is also outside of p, indicating that if it is not an animal, it is certainly not a cat.
Conditional statements may not always have the “if – then” form, but they can be
rewritten and still have the same meaning.
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Example 3
Rewrite each conditional statement into the “if – then” form.
a.
b.
c.
All parallelograms have opposite sides that are congruent.
The square of an even integer is even.
Eating candy will cause your teeth to decay.
Solution:
a.
b.
c.
If this is a parallelogram, then its opposite sides are congruent.
If n is an even integer, then n 2 is even.
If you eat candy, then your teeth will decay.
The Counter Example
A conditional statement can be proven to be false if an example can be found for which the
hypothesis (premise) is true, but the conclusion is false. Such an example is called a
counter example. It takes only one counter example to prove that a conditional is false.
In terms of a Venn diagram, a counter example to the statement, “if p, then q,” would
show the existence of a point in p, which is not in q. Therefore, p could not be drawn
entirely inside of q.
q
q
p
p
A
If p, then q.
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A is a counter example to
“if p, then q.”
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Example 4
For each statement provide a counter example which proves the statement false.
a.
b.
If x  0 , then x  x .
If the diagonals of a quadrilateral are perpendicular, then the quadrilateral
is a square.
Solution:
a.
If x  0 , then x  x .
1
1

If x  , then x  .
2
4
1 1

But  .
2 4

Therefore, x  x and the conditional is false.
b.
If the diagonals of a quadrilateral are perpendicular, then the quadrilateral is a
square.

For a rhombus the diagonals are perpendicular, but a rhombus is not a square.
Therefore, the conditional is false.
The Converse of a Conditional Statement
When you interchange the conclusion and the hypothesis of a conditional, you form the
converse of the conditional.
Original statement:
Converse statement:
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If p, then q.
If q, then p.
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Example 5
Write the converse of each of the following conditional statements. Determine if the
converse is true or false.
a.
b.
If a triangle is equilateral, then it is acute.
If a quadrilateral has four right angles, then it is a rectangle.
Solution:
a.
b.
If a triangle is acute, then it is equilateral.
This converse is false, since the counter example of a 35 – 60 – 85 degree triangle is
acute but is not equilateral.
If a quadrilateral is a rectangle, then it has four right angles.
This converse is true.
Example 6
Draw the Venn diagram that represents the following conditional statement.
If the figure is a square, then it is a polygon.
Use the diagram to help determine the conclusion, if any, to the given statements.
a.
b.
c.
d.
This is a square.
This is a polygon.
This is not a square.
This is not a polygon.
Solution:

The conditional says that all squares are polygons.
Polygon s
Squ ar es
a.
b.
c.
d.
The conclusion is that this is a polygon, since the circle of
squares is inside the circle of polygons.
No conclusion can be made here since a polygon may or
may not be a square
If this is not a square, it may be a polygon, or it may be outside the polygon circle.
No conclusion can be made.
The conclusion is that this is not a square, since a point outside the polygon circle is
certainly outside the smaller inside circle.
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The Inverse and the Contrapositive
For any conditional statement, there are three other statements that can be formed.
Conditional:
If p, then q.
Converse:
Inverse:
Contrapositive:
If q, then p.
If not p, then not q.
If not q, then not p.
Example 7
Write the converse, inverse, and contrapositive of the true conditional and decide if
the new statements are true or false.
If you are 10 years old, you only pay $5.
Solution:
The Venn diagram for this conditional looks like this.
All t hose
who pay $5.
All t hose who
are 10 years old.
Converse:
If you pay only $5, you are 10 years old.
This statement may not be true, since there may be other ages which pay
only $5, and this is not specified.
Inverse:
If you are not 10 years old, you do not pay $5.
This statement may not be true since other ages may have to pay only $5
also.
Contrapositive:
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If you do not pay $5, then you are not 10 years old.
This is true, since a point outside the $5 circle implies being outside
the 10 year old circle.
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A conditional and its contrapositive are always equivalent. They are both true, or
both false and mean the same thing. For example, these two statements mean
the same thing.


If an animal is a cat, then it has four legs.
If an animal does not have four legs, then it is not a cat.
Exercise 7.2
1.
For each of the following statements write the hypothesis, conclusion, and converse.
a.
b.
c.
If there’s a will, then there’s a way.
If  5 x  1  16 , then x  3 .
If AB and AC are opposite rays, then BAC is a straight angle.
Suppose that each of the following conditionals in numbers 2 to 4 are true. Assume
statement (a) is true. What conclusion follows? Assume statement (b) is true. What
conclusion follows? Assume statement (c) is true. What conclusion follows? Use a Venn
diagram to help you with your conclusions.
2.
If a car has anti-lock brakes, then it must be relatively new.
a.
b.
c.
3.
If you are my student, then you love math.
a.
b.
c.
4.
This car is relatively new.
This car does not have anti-lock brakes.
This car is not new.
You hate math.
You love math.
You are not my student.
If it rains tomorrow, I’ll pick you up for school.
a.
b.
c.
d.
It rains tomorrow.
I don’t pick you up for school.
It does not rain tomorrow.
I pick you up for school.
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5.
Give a counter example to disprove each of the following statements.
a.
b.
c.
d.
6.
All prime numbers are odd.
If a 2  b 2 , then a  b .
a a 1

for any positive integers a and b.
b b 1
j 2j

for any real non-zero numbers j and k.
k
k
State the converse, inverse, and contrapositive and decide if each is true or not.
a.
b.
If an angle has measure 35°, then it is acute.
If a line and a plane intersect, then they have at least one point in common.
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7.3 Inductive Arguments
An inductive argument is another type of argument



Data is collected.
Patterns are observed
Then a conclusion or a hypothesis is made.
An inductive argument uses a few particular examples to arrive at a general conclusion.
Science is a prime example where inductive arguments are used when scientists conduct
experiments to obtain experimental evidence which is needed to make a hypothesis. Other
scientists may perform the same experiment and obtain similar results. This may
strengthen the hypothesis. If a conclusion is made on the basis of very little evidence, it
may be described as “jumping to a conclusion.”
Example 1
Inductive arguments vary in strength. Which argument is a stronger inductive
argument?
a)
The first ten shoppers entering an ice cream shop were interviewed and 7
preferred vanilla flavor to chocolate. The other three shoppers preferred
chocolate. Therefore, 70% of the shoppers in this store will choose vanilla
over chocolate ice cream.
b)
Of the first fifty shoppers entering an ice cream shop, thirty-five preferred
vanilla flavor to chocolate. Therefore, 70% of the shoppers in that store will
choose vanilla instead of chocolate.
Solution:

Clearly, in part b) of the example, there is more support for the same conclusion as in
part a). Therefore, there is a stronger inductive argument in part b).
No inductive argument is absolutely perfect since not every possible case is in the premise.
Not every shopper entering the store is interviewed. An inductive argument claims that
the truth of the premises makes it probable that the conclusion is true.
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Activity 7.1
This activity is to be submitted with Assignment 7.
Consider the pattern suggested by the following figures.
Number of points
connected
Number of
resulting regions
2
3
4
5
2
a.
Count the number of regions resulting when 3, 4, and 5 points are
connected, and complete the appropriate portion of the table.
b.
Based on the data collected, predict how many regions can be formed
when 6 points are connected.
c.
Place 6 points on the circumference of the circle, and connect them in
every possible way. Count the number of resulting regions.
d.
What does this example illustrate about inductive reasoning?
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An inductive argument contains two key steps.


Observe that for every case checked, a certain property is true.
Conclude, or generalize, that the property is true for all cases.
The generalization in the second step may be correct, or it may not be correct, since not
every possible case was examined. Indeed, it may not be possible to examine every case.
In science, a generalization is tested further and, if no evidence appears to the contrary,
the hypothesis stands stronger. In mathematics a generalization would require nothing
less than a rigorous proof, which would establish the conclusion as true.
Analogical Argument
This is a type of inductive argument.
If two or more things are similar in some respects, it is concluded that they are similar in
some further respects. The argument is called argument by analogy.
Often, such comparisons are used to explain or illustrate certain concepts, rather than
used as an argument.
For example, concepts in electric current are explained with the aid of the familiar
properties of water pressure in a garden hose. Also, behavior of water waves is helpful to
illustrate certain behaviors of light and of sound.
Here are some examples where analogy is used as an argument.

Mary’s mother stated that she was poor in writing; therefore, Mary is likely to be poor
in writing.

Integers and rational numbers have similar properties of addition and multiplication;
therefore, they are likely to have the same rules of exponents.
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Exercise 7.3
1.
a.
Consider the number patterns shown below and verify that they are correct.
1
13 
135 
1 3 5 7 
1 3 5 7 9 
2.
12
22
32
42
52
b.
What are the next two lines in the number pattern if it is continued? Verify
these.
c.
According to the pattern above, what would be the sum of the first ten
positive odd integers?
d.
Verify your result in (c) by actually adding 1  3  5  7  . . .  17  19 .
e.
Do you think the pattern above continues indefinitely?
f.
What kind of reasoning are you using to answer e?
a.
Complete each of the following statements.
23  64 
32  46 
26  93 
62  39 
69  64 
96  46 
84  36 
48  63 
41  28 
14  82 
b.
Based on the pattern above one might be inclined to generalize that the
product of a pair of two-digit numbers is the same as the product of the
numbers formed by reversing their digits. Is this generalization true for all
two-digit numbers? Try a few of your own choosing.
c.
For what kinds of pairs of two-digit numbers does the generalization in part
(b) seem to hold? Search your data in part (a) carefully.
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3.
Consider the function f  x  and x 2  x  41 . The table below suggests that whenever
x is replaced by a positive integer, f  x  is a prime number. Certainly that is not the
case for the function, g  x   x 2  x , because g 2   6 and 6 is not a prime.
x
f x 
1
43
2
47
3
53
4
61
5
71
6
83
7
97
a.
Continue the table begun for f  x  until you reach the first value of x for
which f  x  is not prime. What is the x value?
b.
To what conclusion might inductive reasoning have lead you, had you not
done part (a) of this question?
c.
What then is the value of inductive reasoning?
7.4 Deductive Arguments
A deductive argument starts with certain premises that are assumed to be true and uses
the rules of logic to arrive at a conclusion.
The argument must be valid (correct) and must contain no errors in the use of the rules of
logic. A deductive argument differs from an inductive argument in that the logic of the
argument guarantees that if the premises are true, then the conclusion must be true. A
deductive argument is often called a proof.
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Example 1
Use a deductive argument to prove this statement.
For any two positive numbers x and y, if x > y, then
1 1
 .
x y
Solution:
Premise 1:
x and y are positive, and x  y .
Premise 2:
1
1
If x  y , then x    y  .
 x
 x
Premise 3:
Premise 4:
Premise 5:
y
1
1
If x    y  , then 1  .
x
 x
 x
1 y1
y
If 1  , then 1     .
x
 y x  y
1 y1
1 1
If 1     , then  .
y x
 y x  y
Conclusion: If x  y , then
1 1
1 1
 , or  .
y x
x y
The above argument starts by assuming that the hypothesis (Premise 1) of the statement
is true. It moves directly through a sequence of “if – then” statements, which are known
to be true from the rules of algebra, and arrives at the conclusion of the statement which
must be true. Such a proof is called a direct deductive proof.
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Activity 7.4


Try the following number trick.
Complete the chart using three different starting numbers. The first has been
done for you.
Directions
Choose any number
Multiply by 4
Add 10
Divide by 2
Subtract 5
Divide by 2
Add 3
Subtract the original
number
1st #
11
44
54
27
22
11
14
3
2nd #
3rd #

Inductive reasoning would suggest that the final result would always be 3.
We can use a deductive argument to prove what inductive reasoning
suggests. Let the starting number be x, and by a sequence of correct
algebraic steps, the conclusion will be reached which is true for all numbers.

Complete the chart below.
Directions
Choose any number
Multiply by 4
Add 10
Divide by 2
Subtract 5
Divide by 2
Add 3
Subtract the original
number

1st #
x
4x
4x + 10
What does this example illustrate about deductive reasoning?
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Indirect Poof (Proof by Contradiction)
To prove a statement “if p, then q,” by the indirect method, you may prove directly the
contrapositive , “if not q, then not p.” Earlier, it was stated that a statement and its
contrapositive are always either true together, or false together.
In an indirect proof, we assume that the conclusion of the statement is false. That is, we
begin by assuming that the opposite of what we are trying to prove is true. With this
assumption, continue with a sequence of “if – then” statements until it is shown that the
original premise is false.
It is helpful to begin with writing the conditional and the contrapositive.
Example 3
Suppose Dennis is the same age as Mark. Use an indirect proof to prove the
statement. “If Mark is younger than Fred, then Dennis is younger than Fred.”
Solution:


Let the initials D, F, and M represent the ages of the boys.
It must be proven that since D  M , if M  F , then D  F .
Conditional:
Contrapositive:
if p then q
if D  M , M  F then D  F
if D  F then M  F
Premise 1: D  M , and M  F
Premise 2: Suppose D  F .
Conclusion: M  F
Assume the conclusion is false.
Substitute D for M, using Premise 1.
Therefore, Premise 1 is false. The original premise has been contradicted. This proves the
original statement.
C
Example 4
Prove that in triangle ABC, if AB  AC and AB  BC , then
AC  BC .
A
B
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Solution:
Contrapositive:
If AB  AC , AB  BC
Then AC  BC
If AC  BC then AB  BC .
Premise 1.
Premise 2:
Conclusion.
AB  AC and AB  BC
AC  BC
Assume that the conclusion is false.
AB  BC
Use Premise 1 to replace AC with AB.
Conditional:
Therefore, Premise 1 is false. This proves the original statement.
Exercise 7.4
1.
a.
b.
c.
d.
e.
2.
State and prove the converse of the statement in Example 1.
1 1
For any positive numbers x and y, if x  y , then  .
x y
3.
Prove that the function f ( x)  x 2  8 x  7 will never yield a prime if x is replaced by
a positive integer.
4.
A bottle and a cork together cost $1.06. The bottle costs one dollar more than the
cork. How much does the cork cost? Prove your result.
5.
Use the following premises to prove (a) and (b).
If y, then a.
If r, then m.
If m, then y.
If x, then r.
If a, then b.
a)
b)
6.
Choose any two positive integers.
Find the square of the sum of the integers chosen in (a).
Find the sum of the squares of the integers chosen in (a).
How does the answer in (b) compare in size to your answer in (c)?
Use a deductive argument to prove that your conclusion in (d) will hold for
any two positive integers.
If y, then b.
If x, then y.
Use an indirect proof to prove the following.
If a  b and b  c , then a  c .
Mathematics C30
107
Lesson 7
7. In any triangle, an exterior angle is always larger than each opposite interior angle.
3
2
a.
A
1
Explain what is wrong with the following proof.
Angle 2 and Angle 3 are acute angles and Angle 1 is an obtuse angle. Every
obtuse angle is larger than any acute angle. Therefore, Angle 1 is larger than
Angle 2 and Angle 1 is larger than Angle 3.
b.
Give a correct proof of the statement.
7.5 Proofs in Trigonometry
In Lesson 4, various trigonometric identities were developed and used to simplify
trigonometric expressions, or to prove other trigonometric identities true. Once an
identity is proven to be true, it can be used as a true statement to prove other identities.
In this way a collection of commonly used “Basic Identities” is developed to be used for
more advanced work in trigonometry. This is analogous to a carpenter having a collection
of fine tools for use in making fine furniture, or for building houses.
Initially, a carpenter may start in the profession with only a hammer and a saw.
Certainly, much can be done with this. Analogously, in trigonometry, we started with only
the definitions of sine, cosine, and tangent. From these humble beginnings, with the help
of algebra techniques, the Pythagorean Identities were developed. See Section 4.1. The
Cofunction Identities were also developed, but only for acute angles. Using only the
definitions of sine, cosine, and tangent, the Cofunction Identities can be proven true for
any angle. This will now be done.
Mathematics C30
108
Lesson 7
Proof of the Cofunction Identities for any Angle 
The basic tools that will be used are listed in the “tool box.”
Tool Box
•
•
Any algebra techniques known to us.
The definitions of the trigonometric ratios for any angle  .
y
y
r
x
cos  
r
y
tan  
x
sin  

x
r
(x, y )
•
csc  
1
,
sin 
sec  
1
1
, cot  
cos 
tan 
Let  be any angle, and let A be its reference angle.
y
y= x

2
(a, b)

A

x
B

(b, a )
Mathematics C30
109
Lesson 7



If (a, b) is on the terminal ray of  , then (b, a) is on the reflection of the terminal ray
about the line y  x .

In the diagram (b, a) is on the terminal arm of   .
2
Since the resulting right triangles have congruent sides, angle A must be congruent to
angle B.
Therefore,   A    B ,
or

By definition
of the ratios
from the “tool
box,” we have,
b


 cos    
r
2

a


cos    sin    
r
2

sin  
tan  
a


 cot    
b
2

b


 tan    
a
2

r


sec    csc    
a
2

r


csc    sec    
b
2

cot  
Proof of Identities Containing Negative Angles
Only the definitions of the trigonometric ratios will be used to prove the negative angle
identities.
sin( )   sin 
•
cos( )  cos 
•
Mathematics C30
110
Lesson 7

If  a, b is on the terminal arm of  , then  a,  b is on the terminal arm of   .
y
(–a, b)
r

–
x
r
(–a, –b)
The definitions are then applied to the triangles
b
b
     sin 
 sin( ) 
r
r
a
 cos 
 cos( ) 
r
In Lesson 4, after the addition and subtraction identities, the list of available basic
identities expanded to include double-angle and half-angle identities. These can be used
as justification, or as a reason for any step taken in a proof that an identity is true.
The “tool box” or list of Basic Identities is given on the next page to be used for reference.
Mathematics C30
111
Lesson 7
Basic Identities
Reciprocal Identities
tan  
csc  
sin 
cos 
1
sin 
cos 
sin 
cot  
sec  
1
cos 
cot  
1
tan 
Pythagorean Identities
sin 2   cos 2   1
1  tan 2   sec 2 
1  cot 2   csc 2 
Cofunction Identities


sin      cos 
2



cos      sin 
2



tan      cot 
2



sec      csc 
2

Angle Addition and Subtraction Identities
cos  A  B   cos A cos B  sin A sin B
cos  A  B   cos A cos B  sin A sin B
sin  A  B   sin A cos B  cos A sin B
sin  A  B   sin A cos B  cos A sin B
tan  A  B  
tan A  tan B
1  tan A tan B
tan  A  B  
tan A  tan B
1  tan A tan B
Double-angle Identities
cos 2   cos 2   sin 2 
cos 2   2 cos 2   1
cos 2   1  2 sin 2 
sin 2  2 sin  cos 
2 tan 
tan 2 
1  tan 2 
Half-angle Identities
cos
tan
Mathematics C30

1  cos 

2
2
sin

1  cos 

2
1  cos 
112

1  cos 

2
2
Lesson 7
Example 1
Prove that tan A  cot A   tan
2
2
A  cot 2 A is false.
Solution:

All that is required is one example (counter example) for which the equation does not
hold.

Let A 

.
4



LHS   tan  cot 
4
4

2


 cot 2
4
4
2
2
RHS  1  1
RHS  tan
LHS  1  1 
2
RHS  1  1
RHS  2
LHS  2 2
LHS  4

2
Therefore, LHS  RHS , and the equation is false.
Example 2
Prove that tan A  cot A   sec 2 A  csc 2 A , and supply a reason for each step.
2
Solution:
LHS  tan A  cot A 
Given
Binomial multiplication
Tan and cot are reciprocals
Pythagorean identities
Algebraic simplification
2
 tan A  2 tan A cot A  cot A
 tan 2 A  2  cot 2 A
 sec 2 A  1  2  csc 2 A  1
 sec 2 A  csc 2 A
 RHS
2
Mathematics C30
2
113
Lesson 7
Exercise 7.5
1.
Prove each of the identities using the definition of the trigonometric ratios.
a.
sin   A   sin A
b.
sin   A    sin A
c.
cos   A    cos A
d.
cos   A    cos A
2.
Apply the addition and subtraction identities to prove the identities in Question 1.
3.
Prove each of the identities and state a reason for each step.
a.
sin A cot A  cos A
b.
sin
c.
1  2 sin
d.
sin A csc 2 A  cot 2 A
 sin A
cos A sec A
Mathematics C30
2
A  cos 2 A
 sec A
cos A

2
A  2 cos 2 A  1

114
Lesson 7
Answers to Exercises
Exercise 7.1 1.
b.
88.89 km/h
2.
b.
The amounts are the same.
3.
The ring is raised by 0.3 m
4.
a.
b.
c.
d.
e.
f.
not
not
not
True statement
False statement
not
Exercise 7.2 1.
a.
Hypothesis: There is a will.
Conclusion: There is a way.
Converse: If there’s a way, then there’s a will.
b.
Hypothesis:  5 x  1  16
Conclusion: x  3
Converse: If x  3 , then  5 x  1  16
c.
Hypothesis: AB and AC are opposite rays.
Conclusion: Angle BAC is a straight angle.
Converse: If angle BAC is a straight angle, then AB and AC
are opposite rays.
2.
new
car s
car s wit h
ant i-lock
br akes
a.
b.
c.
Mathematics C30
No conclusion
No conclusion
Conclusion: This car does not have antilock brakes.
115
Lesson 7
3.
my
students
a.
b.
c.
students
who love
math
Conclusion: You are not my student.
no conclusion
no conclusion
4.
It rains
tomorrow
5.
a.
b.
c.
d.
Conclusion: I’ll pick you up for school.
Conclusion: It doesn’t rain tomorrow.
no conclusion
no conclusion
a.
b.
2 is an even prime number.
a  1, b  1
2 2 1

1 1 1
j  1, k  2
c.
d.
Mathematics C30
I’ll pick you
up for school
116
Lesson 7
6.
a.
converse:
inverse:
If an angle is acute, it measures 35°. False
If an angle does not measure 35°, then it not
acute. False.
If an angle is not acute, then it does not
measure 35°. True
contrapositive:
b.
converse:
If a line and a plane have at least one point
in common, then they intersect. True.
If a line and a plane do not intersect, then
they do not have a point in common. True.
If a line and a plane do not have a point in
common, then they do not intersect. True
inverse:
contrapositive:
Exercise 7.3 1
c.
Note that in 23  64 , we have 2  6  3  4 .
2.
a.
b.
If x is 41, then f  x  terms have a common factor.
Inductive reasoning would suggest that f  x  is prime for every
integer x.
Inductive reasoning suggests things that may be proved by other
means.
c.
3.
Exercise 7.4 1.
2 19  524288
Prove that x 2  y 2   x  y 
2
 x  y 2
 x 2  2 xy  y 2


 x 2  y 2  2 xy  x 2  y 2
Mathematics C30
117
Lesson 7
2.
The converse is, if
Proof:
1 1
 , then x  y .
x y
1 1

x y
1
 y  1  y
x
y
y
1
x
y
 x   1 x 
x
yx
3.
f  x    x  7  x  1
f  x  is always a product of two numbers greater than or equal to 2.
4.
Cost of cork is c and cost of bottle is b.
c  b  106 ¢
c  100  b
Substitute c  100 for b in the first equation.
c  c  100   106
2c  6
c3
b  103
5.
Mathematics C30
a.
If y, then a.
If a, then b.
Therefore, if y then b.
b.
If x, then r.
If r, then m.
If m, then y.
Therefore, if x, then y.
118
Lesson 7
6.
Premise 1: a  b and b  c .
Premise 2: suppose a  c .
Conclusion: c  b , and this contradicts b  c in Premise 1.
7.
a.
The first premise is false. Although the diagram is drawn with
angles 2 and 3 acute and angle 1 obtuse, this is not always the
case.
b.
1  A  180 
2  3  A  180 
1  A  2  3  A
1  2  3
Therefore, 1  2 , and 1  3
1 and A are a linear
pair.
The angles of a triangle
add up to 180°.
Both equal 180°.
Subtract A .
A sum of positive terms is
greater than each term.
Exercise 7.5 1.
y
y
(–a, b)
(–a, b)
(a, b)
A
A
x
–A
+ A
For questions a) and d).
a.
b.
c.
d.
Mathematics C30
x
(a, –b)
For questions b) and c).
sin   A  
b
 sin A
r
b
b
sin   A  
     sin A
r
r
a
a
cos   A    
   cos A
r
 r 
a
a
cos   A    
   cos A
r
 r 
119
Lesson 7
3.
a.
b.
LHS = sin A cot A
cos A
= sin A
sin A
= cos A
= RHS
LHS =
sin
2
Given
Reciprocal Identity
Reducing
A  cos 2 A
cos A
Given
1
cos A
= sec A
= RHS
=
c.
LHS =
=
=
=
=
d.
LHS =
=
=
=
=
Mathematics C30
Pythagorean Identity
Reciprocal Identity
1  2 sin 2 A
1  2 1  cos 2 A
1  2  2 cos 2 A
 1  2 cos 2 A
RHS


Given
Pythagorean Identity
Algebraic Simplification


sin A csc 2 A  cot 2 A
cos A sec A
sin A 1  cot 2 A  cot 2 A
cos A sec A
sin A
cos A sec A
sin A
1
RHS

120
Given

Pythagorean Identity
Algebraic Simplification
Reciprocal Identity
Lesson 7
Mathematics C30
Module 2
Assignment 7
Mathematics C30
121
Lesson 7
Mathematics C30
122
Lesson 7
Optional insert: Assignment #7 frontal sheet here.
Mathematics C30
123
Lesson 7
Mathematics C30
124
Lesson 7
Assignment 7
Values
(40)
A.
Multiple Choice: Select the best answer for each of the following and place a
check () beside it.
1.
Select the sentence which is not a statement.
____
____
____
____
2.
It is not a conditional.
It is an imperative.
It is a question.
It is impossible to determine if it is true or false.
a.
b.
c.
d.
What is the moon made from?
Assign the value 2 to the variable x.
The baby’s name is Bob.
Prove that if x  y , then x  y  0 .
Select the one case which is not an argument.
____
____
____
____
Mathematics C30
a.
b.
c.
d.
Select the sentence that is a statement.
____
____
____
____
4.
Please lock the barn door.
The back door is closed.
All triangles are equilateral.
Some dogs are good hunters.
State why the following sentence is not considered to be a statement.
“This remark is false.”
____
____
____
____
3.
a.
b.
c.
d.
a.
b.
c.
d.
If x 2  x  0 , then x  0 , or x  1 .
x 2  x  0 , x  0 , or x  1
If x  0 or x  1 , then x 2  x  0 .
When ever x  0 or x  1 , the result is x 3 x 3  1  0 .

125

Lesson 7
5.
6.
State another way of writing the following argument. “The domain of
f  x   x  1 is x  1 .”
a.
____
b.
If f  x   x  1 , then x  1 .
____
____
c.
d.
If x  1 , then the domain of f is
If x  1 , then f  x   x  1 .
If n 2  1 , then n is a natural number.
If n is a natural number, then n 2  1 .
If n 2  1 , then n is not a natural number.
When ever n is a natural number, it follows that n 2  1 .
a.
b.
c.
d.
Find a case where p and q are false.
Find a case where p is false, but q is true.
Find a case where p is true, but q is false.
Prove that if q, then p.
Select the converse of this statement.
“If the straight line is horizontal, then its slope is zero.”
____
____
____
____
Mathematics C30
a.
b.
c.
d.
To prove that the given statement, “if p, then q,” is false, the following
must be done.
____
____
____
____
8.
x 1 .
“For all natural numbers n, n 2  1 .” Select the one statement which is
not equivalent to this statement.
____
____
____
____
7.
If f  x   x  1 , then the domain of f is x  1 .
____
a.
b.
c.
d.
If the line is not horizontal, then its slope is not zero.
If the slope of the line is not zero, then it is not horizontal.
Whenever the line is horizontal, its slope is zero.
If the slope of a line is zero, then the line is horizontal.
126
Lesson 7
9.
Select the valid conclusion for the following given hypothesis. If angle
A is congruent to angle B, then the measure of angle A is equal to the
measure of angle B. The measure of angle A is not equal to the
measure of angle B.
____
____
____
____
10.
Mathematics C30
a.
b.
c.
d.
if not q, then not p
if p, then not q
if not p, then q
if not q, then q
“If the triangle is equilateral, it has a 60° angle.” A statement that is
equivalent to this statement is the***.
____
____
____
____
12.
No conclusion can be made.
Angle A is not equal to angle B.
Angle A is congruent to angle B.
Angle A is not congruent to angle B.
The proof of “if p, then q” by contradiction is the same as the direct
proof of ***.
____
____
____
____
11.
a.
b.
c.
d.
a.
b.
c.
d.
converse
inverse
contrapositive
contradiction
A strong inductive argument ***.
____
____
a.
b.
____
____
c.
d.
uses a logical argument to arrive at a conclusion
has many particular examples to support a general
conclusion
starts by assuming the conclusion is false
relies on jumping to a conclusion
127
Lesson 7
13.
The solution to x  2 y  3 , 4 x  5 y  6 is  1, 2  . The solution to
4 x  5 y  6 , 7 x  8 y  9 is  1, 2  . Therefore, the solution to
11 x  12 y  13 , 14 x  15 y  16 is also  1, 2  . This is an example of ***.
____
____
____
____
14.
sin 70 
cos 70 
sin 20 
csc 70 
a.
b.
c.
d.
cos  30 cos 45   sin  30 sin 45 
sin  30 cos 45   sin 45  sin  30 
cos  30 sin  30   cos 45  sin 45 
cos 30  cos 45   sin 30  sin 45 
Tan   is equivalent to ***.
____
____
____
____
Mathematics C30
a.
b.
c.
d.
Cos 45   30  is equivalent to ***.
____
____
____
____
16.
deductive argument
analogical argument
inductive argument
proof by contradiction
sin 50   cos 20   cos 50   sin 20  is equivalent to***.
____
____
____
____
15.
a.
b.
c.
d.
a.
b.
c.
d.
tan 
 tan 
cot 
 cot 
128
Lesson 7
17.
The expression
____
____
____
____
18.
20.
a.
b.
c.
d.
 2 sin x cos x
2 sin x cos x
cos 2 x  sin 2 x
sin 2 x
The tan 15  is ***.
____
a.
____
b.
____
c.
____
d.
1  cos 30 
1  cos 30 
1  cos 30 

1  cos 30 
1  cos 30 
1  cos 30 
1  cos 30 

1  cos 30 
Of the following, which is not an identity is ***.
____
____
____
____
Mathematics C30
csc 
sin 
cot 
tan 
The expression sin 2 x   is equivalent to ***.
____
____
____
____
19.
a.
b.
c.
d.
sin   cos   sin 
1
is equivalent to ***.

1  cos 
cos 
a.
b.
c.
d.
sin 90     cos 
sec  cot   csc 
cos 2   1  2 sin 2 
sin 2 cot   2 cos 
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Lesson 7
Mathematics C30
130
Lesson 7
Answer Part B and Part C in the spaces provided. Evaluation of your
solution to each problem will be based on the following:
(8)
B.
•
A correct mathematical method for solving the problem is
shown.
•
The final answer is accurate and a check of the answer is shown
where asked for by the question.
•
The solution is written in a style that is clear, logical, wellorganized, uses proper terms, and states a conclusion.
1.
State the type of argument (analogical, inductive, deductive, or proof
by contradiction (indirect)) which is used in each case.
Mathematics C30
a.
Ben was asked to write the next three terms of the sequence
1, 1, 2, 3. Ben wrote 5, 8, 13.
b.
Mary’s mother stated that she was poor in writing so, therefore,
Mary was likely to be poor in writing.
c.
All horses eat hay.
Silver is a horse.
Therefore, Silver eats hay.
131
Lesson 7
d.
Joan ate brunch at Cody’s restaurant two Sundays in a row.
She saw Brie there both times. Joan told her friends that Brie
always eats lunch at Cody’s.
e.
Vertically opposite angles are congruent. Angles A and B are
vertically opposite. Therefore, angles A and B are congruent.
f.
 a  b 2
 a 2  b 2 . If a  1 , and b  2 , then 1  2   9 , but
2
12  2 2  5 .
Mathematics C30
g.
Every Tuesday, Molly plays bridge. Today Molly is not playing
bridge. Therefore, today is not Tuesday.
h.
Just as a rubber ball bounces so that the angle of incidence
equals the angle of reflection, sound waves bounce off a wall so
that the angle of incidence equals the angle of reflection.
132
Lesson 7
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2.
a.
Draw a fairly large equilateral triangle XYZ.
Draw the altitude from X to YZ.
Choose any point P inside the
X
triangle or on the triangle.
Draw perpendiculars from P
h
to the sides of the triangle.
u
b.
c.
Mathematics C30
t
P
Measure the altitude h and
s
the 3 perpendiculars s, t, and
Y
u to the nearest mm.
Repeat as many times as is necessary until you can state a
generalization concerning h, s, t, and u. (A minimum of 5
times). Show all your data.
Z
Do your experiments prove that your generalization is true?
133
Lesson 7
(8)
3.
For the following conditional
If you are my student, then you love math.
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4.
a.
draw a venn diagram.
b.
write the inverse, converse and contrapositive and indicate if
each leads to a true or false conclusion.
a.
Prove the identity is true and supply a reason for each step.
cot   tan   2 cot 2
(5)
b.
Prove the identity is true and supply a reason for each step.
sin A  cos A cot A  csc A
Mathematics C30
134
Lesson 7
(4)
c.
Prove the identity is false by supplying a counter example
sec 2 t 
5.
(7)
cos 2 t
2  cos 2 t
a.
What is the relationship between a conditional and the
contrapositive.
b.
Write the contrapositive for the following conditional.
If you own a Mustang then you own a Ford.
c.
Mathematics C30
Using a Venn Diagram, show why the conclusions for the
conditional and the contrapositive in b) are both true.
135
Lesson 7
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C
1.
Submit Activity 7.1
(5)
2.
Submit Activity 7.4
(5)
3.
Write a summary of this lesson in point form, suitable for review
purposes.
_____
(100)
Mathematics C30
136
Lesson 7