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Mathematics C30 Module 2 Lesson 7 Mathematics C30 Mathematical Proof (Part 1) 79 Lesson 7 Mathematics C30 80 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. Mathematics C30 81 Lesson 7 Mathematics C30 82 Lesson 7 Objectives After completing this lesson, the student will be able to Mathematics C30 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. 83 Lesson 7 Mathematics C30 84 Lesson 7 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. Mathematics C30 85 Lesson 7 “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. Mathematics C30 86 Lesson 7 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? Mathematics C30 87 Lesson 7 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? Mathematics C30 88 Lesson 7 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. Mathematics C30 89 Lesson 7 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. Mathematics C30 90 Lesson 7 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. Mathematics C30 A is a counter example to “if p, then q.” 91 Lesson 7 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: Mathematics C30 If p, then q. If q, then p. 92 Lesson 7 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. Mathematics C30 93 Lesson 7 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: Mathematics C30 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. 94 Lesson 7 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. Mathematics C30 95 Lesson 7 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. Mathematics C30 96 Lesson 7 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. Mathematics C30 97 Lesson 7 Mathematics C30 98 Lesson 7 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? Mathematics C30 99 Lesson 7 Mathematics C30 100 Lesson 7 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. Mathematics C30 101 Lesson 7 Exercise 7.3 1. a. Consider the number patterns shown below and verify that they are correct. 1 13 135 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. Mathematics C30 102 Lesson 7 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. Mathematics C30 103 Lesson 7 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 y1 y If 1 , then 1 . x y x y 1 y1 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. Mathematics C30 104 Lesson 7 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? Mathematics C30 105 Lesson 7 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 Mathematics C30 106 Lesson 7 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 yx 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 c3 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 129 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 (8) 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. (5) 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 (5) 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