Unit 6 Trigonometric Identities Prove trigonometric identities Solve trigonometric equations

Transcription

1 Unit 6 Trigonometric Identities Prove trigonometric identities Solve trigonometric equations

2 Prove trigonometric identities, using: Reciprocal identities Quotient identities Pythagorean identities Sum or difference identities (restricted to sine, cosine and tangent) Double-angle identities (restricted to sine, cosine and tangent)

3 TRIG IDENTITIES You should be able to explain the difference between a trigonometric equation and a trigonometric identity. An identity is true for all permissible values, whereas an equation is only true for a smaller subset of the permissible values. This difference can be demonstrated with the aid of graphing technology.

4 For example: This can be solved by using the graphs 1 of y sin x and y The solutions to are x = 30 and x = 150, which are the x-values of the intersection points. 1 sin x,0 x sin x,0 x Thus this is not an identity because it is only true for certain values of x.

5 Solve: sin x = tan x cos x. This can be solved by using the graphs of: y = sin x and y = tan x cos x These graphs are almost identical. The only differences in the graphs occur at the points (90, 1) and (270, 1). Why? They are non-permissible values of x.

6 Therefore, sin x = tan x cos x is an identity since the expressions are equivalent for all permissible values.

7 Note: There may be some points for which identities are not equivalent. These non-permissible values for identities occur where one of the expressions is undefined.

8 In the previous example, y = tan x cos x is not defined when x = n, n Ι since y = tan x is undefined at these values. Non-permissible values often occur when a trigonometric expression contains: A rational expression, resulting in values that give a denominator of zero Tangent, cotangent, secant and cosecant, since these expressions all have non-permissible values in their domains.

9 Practice: Determine graphically if the following are identities. Use Technology Identify the non-permissible values. ( i ) sin cos tan 2sin ( ) tan 1 sec cos ( iii ) sec sin 2 2 ii

10 ( i ) sin cos tan 2sin y y y sin cos tan y 2sin x x Non-permissible values? Is this an identity?

11 ( ) tan 1 sec 2 2 ii y y 2 2 y=t an 1 y sec x x Non-permissible values? Is this an identity?

12 cos ( iii ) sec sin y y cos sin y y sec x x Non-permissible values? Is this an identity?

13 We can also verify numerically that an identity is valid by substituting numerical values into both sides of the equation. Example: Verify whether the following are identities. A) sin cos B) (use degrees) 2 2 sin cos 1 (use radians)

14 Example: Verify whether the following are identities. 2 2 C) tan 1 sec D) cot 1 csc 2 2 (use degrees) (use radians) NOTE: This approach is insufficient to conclude that the equation is an identity because only a limited number of values were substituted for θ, and there may be a certain group of numbers for which this identity does not hold. To prove the identity is true using this method would require verifying ALL of the values in the domain (an infinite number). This type of reasoning is called inductive reasoning.

15 Proofs! A proof is a deductive argument that is used to show the validity of a mathematical statement. Deductive reasoning occurs when general principles or rules are applied to specific situations.

16 Deductive reasoning is the process of coming up with a conclusion based on facts that have already been shown to be true. The facts that can be used to prove your conclusion deductively may come from accepted definitions, properties, laws or rules. The truth of the premises guarantees the truth of the conclusion.

17 Find the fifth term in the sequence Inductive Reasoning 1. 3, 5, 7, 9,... Deductive Reasoning 1. t n =2n , 12, 27, 48, t n = 3n , 14, 21, 28, Dates of in year

18 What is the next number in this sequence? 15, 16, 18, 19, 25, 26, 28, 29,

19 Trig Proofs Trig proofs (and simplifications of trig expressions) are based on the definition of the 6 trigonometric functions and the Fundamental Trigonometric Identities.

20 Definition of the 6 trigonometric functions 1 x y P(x, y) Sine fn: Cosine fn Tangent fn Cotangent Secant fn Cosecant fn y sin 1 x cos 1 y tan x x cot y 1 sec x 1 csc y

21 Fundamental Trigonometric Identities. Reciprocal Quotient Pythagorean 1 sin csc 1 cos sec 1 tan cot 1 cot tan 1 sec cos 1 csc sin tan sin cos cot cos sin 2 2 sin cos 1 tan 1 sec 2 2 cot 1 csc 2 2 Note: These identities can be proven using the definitions of the trig functions.

22 Caution The Pythagorean identities can be expressed in different ways: 2 2 sin cos 1 tan 2 1 sec 2 cot 2 1 csc 2 1 cos sin sin cos 2 2

23 Simplify expressions using the Pythagorean identities, the reciprocal identities, and the quotient identities Strategies that you might use to begin the simplifications: Replace a squared term with a Pythagorean identity Write the expression in terms of sine or cosine For expressions involving addition or subtraction, it may be necessary to use a common denominator to simplify a fraction Factor Multiply by a conjugate to obtain a Pythagorean identity

24 You may also be asked to determine any nonpermissible values of the variable in an expression. For example, identify the non-permissible values of θ in sin cos cot, and then simplify the expression. Solution: The non-permissible values are when sin 0. Why? cot cos sin kk,

25 Simplify : sin cos cot In this case we write the expression in terms of sine or cosine

26 NOTE: Students often find simplifying trigonometric expressions more challenging than proving trigonometric identities because they may be uncertain of when an expression is simplified as much as possible. However, developing a good foundation with simplifying expressions makes the transition to proving trigonometric identities easier.

27 Simplify the following A) sin x secx In this case we write the expression in terms of sine or cosine

28 B) Simplify the following 2 1 cos sin In this case we use a Pythagorean Identity and simplify

29 C) sec sec sin 2 In this case we factor and then use a Pythagorean Identity

30 2 sec x D) tan x In this case we have choices We could use a Pythagorean Identity and simplify Or we could change each term to sin and cos

31 sin cos E) 1 cos sin What do we do here? Multiply by a conjugate to obtain a Pythagorean identity

32 F) tanx sinx 1 cosx What do we do here? First we change everything to sin x and cos x sinx sinx cosx 1 cosx Now we have choice. 1. We can add the numerator by finding a lowest common denominator and then simplify. 2. We can multiply both the numerator and denominator by the LCD of all of the fractions WITHIN the overall fraction.

33 1. We can add the numerator by finding a lowest common denominator and then simplify sinx sinx cosx 1 cosx

34 2. We can multiply both the numerator and denominator by the LCD of all of the fractions WITHIN the overall fraction. sinx sinx cosx 1 cosx

35 Page 296 # 1 a) d) 3b) c) 4, 7, 8c), 9, 10

36

37 Warm UP Factor and simplify 1. sin sin cos 2 x x x sin 2 x

38 Factor and simplify 2. 2 tan x 3tanx 4 sinx tanx sinx

39 Proving Identities The fundamental trigonometric identities are used to establish other relationships among trigonometric functions. To verify an identity, we show that one side is equal to the other side. Left Side = Right Side LS = RS

40 Each side is manipulated independently of the other side. It is incorrect to perform operations across the equal sign, such as: multiplying or dividing, adding or subtracting each side by an expression or cross multiplying or raising both sides to an exponent. These operations are only possible if the equation is true. Until we verify, or prove the identity to be true, we do not know if both sides are equal

41 Prove that the following are Identities using the definitions of the trig function on the unit circle A) cos 1 sec B) tan sin cos

42 2 2 C) sin cos 1 D) tan 2 1 sec 2

43 Guidelines for Proving Trigonometric Identities We usually start with side that contains the more complicated expression. If you substitute one or more fundamental identities on the more complicated side you will often be able to rewrite it in a form identical to that of the other side.

44 Rewriting the complicated side in terms of sines and cosines is often helpful. If sums or differences appear on one side, use the least common denominator and combine fractions In other cases factoring is useful. It may be necessary to multiply a fraction by a conjugate to obtain a Pythagorean Identity

45 There is no one method that can be used to prove every identity. In fact there are often many different methods that may be used. However, one method may be shorter and more efficient than another. The more identities you prove, the more confident and efficient you will become. Practice! Practice! Practice!

46 DON T BE AFRAID to stop and start over again if you are not getting anywhere. Creative puzzle solvers know that strategies leading to dead ends often provide good problem-solving ideas.

47 Prove the following A) secx cot x cscx Which side is the more complicated side? Left Lets work on the left side and change to sines and cosines.

48 B) sinx tan x cosx secx Which side is the more complicated side? Left Lets work on the left side and change to sines and cosines.

49 3 2 C) cos cos cos sin Which side is the more complicated side? Right Lets work on the right side and factor out the Greatest Common Factor

50 D) 2sec Which side is the more complicated side? Right cos 1 sin 1 sin cos Lets work on the right side add the fractions by using the LCD

51 sin 1 cos E) 1 cos sin Which side is the more complicated side? same Lets work on the left side and multiply by the conjugate

52 2 2 2 F) cos sin 2cos 1 In this case we change the sine into cosine using a Pythagorean Identity

53 sint cost G) sint cost tant cott In this case we change everything to sin and cos

54 sint H) cott csct 1 cost

55 1 1 2 I) 2 cot t 1 cost 1 cost

56 J) 2 1 sec sec 1 2 3sec 5sec 2 5sec 2

57 Text Page #1 b) c), 2-4, 7b), 9, 10b), 11 b)c)

58 OTHER TRIG STUFF Even-Odd Identities (Negative Angle): sinx sin x cscx cscx cos x cosx secx sec x tan x tan x cotx cot x

59 OTHER TRIG STUFF Addition and Subtraction Rules: sin a b sina cosb sinb cosa sin a b sina cosb sinb cosa OR sin a b sina cosb sinb cosa

60 Addition and Subtraction Rules: cos a b cosa cosb sina sinb cos a b cosa cosb sina sinb OR cosa b cosa cosb sina sinb

61 PROOF: cos a b cosa cosb sina sinb This one of those interesting proofs. We need to use the: Law of Cosines And the distance formula between 2 points

62 PROOF: cos a b cosa cosb sina sinb

63 PROOF: cos a b cosa cosb sina sinb

64 PROOF: cos a b cosa cosb sina sinb Replace b by b in cos a b cosa cosb sina sinb

65 PROOF: sin a b sina cosb sinb cosa Replace a by 2 a in cos a b cosa cosb sina sinb

66 PROOF: sin a b sina cosb sinb cosa Replace b by b in sin a b sina cosb sinb cosa

67 Addition Formula for Tan tan PROOF: tana tanb a b 1 tanatanb

68 Subtraction Formula for Tan tan a tana tan b 1 tanatan b b

69 Applications of the Angle Addition Formulae Finding exact values Deriving double and half angle formula Proving Identities In Calculus: Trig derivatives (3208) Trig substitution in integration. (1001)

70 EXAMPLES: 1. Find the exact values of: A) cos 15 o B) sin 75 o

71 C) sin 12

72 D) 7 tan 12

73 E) sin 60 o cos 30 o + sin 30 o cos 60 o How can we verify that this is true?

74 F) o tan15 tan30 o 1 tan15 tan30 o o

75 G) Aand B are both in Quadrant II, cos A 5 13 and sinb 3 5. Determine the exact value of cos A B.

76 2. Simplify A) sin sin 2 2

77 B) tan

78 Identities 3. A)Prove: sin x cos x cosx 6 3

79 B) Prove: cos cos 2cos cos

80

81 PROVE: sina b A) tana tanb cosacosb B) cos a b cos a b 2sina cosb C) sin x cos x sin x cos x sin x Find the exact value of: 9 23 D) sin E ) cos F )sin o

82 B) cos a b cos a b 2sina cosb

83 Double Angle Formulae sin2

84 cos2

85 tan2

86 Examples 1. Find the exact values of: A) 2sin15 o cos15 o 2 2 B) cos sin 8 8

87 A 2. Simplify: x x 4tan ) sin cos B) tan 2

88 C ) cos2x sin sin2x 2 x

89 3. PROVE: 1 cos2a A) tan 1 cos2a 2 A

90 tan2btan B B) sin2b tan2b tanb

91 C ) sin sin

92 D) 2sin cos sin

93 4 2 E x x x ) cos4 8cos 8cos 1

94 sin2x F )Show that can be 1 cos2x simplfied tocotx

95 Suppose: sin and Find the exact value of: A) sin 2 B) cos 2 C) tan2

96 Half Angle Formulae Not on Public but good to know Consider: cos2 = 2cos 2-1 x Let x 2 2

97 Half Angle Formulae Consider: cos2 = 1 2sin 2 x Let x 2 2

98 Examples: Find the exact value of: A) sin 15 o B) cos 75 o

99 Page #7A), 8, 9, 10A)C), 11A) 12, 13, 15,16,17

100 Last Section for Chapter 6 (6.4) SOLVE, ALGEBRAICALLY AND GRAPHICALLY, FIRST AND SECOND DEGREE TRIGONOMETRIC EQUATIONS

101 The identities encountered earlier in this unit can now be applied to solve trigonometric equations.

102 Examples: 1. Find the solutions of for 0 x < 360. sin2 x 3cosx Solution: Graphically A) Identify each curve B) What are the points of intersection?

103 1. Find the solutions of sin2 x 3cosx for 0 x < 360. Solution: Algebraically What are the solutions with an unrestricted domain, in radians?

104 2. Solve cos2x 1 cos x for 0 x 360, giving exact solutions where possible. Write the general solution in degrees and radian measure.

105 : 3. Solve the trigonometric equation shown below for : 0x 2 sin3x cosx cos3x sinx 3 2

106 4. Solve: cos 2x + sin 2 x = , for the domain 0 x < 360.

107 Identifying and Repairing Errors 1. Identify and repair the mistake A solution has been lost as a result of dividing both sides of the equation by sin x.

108 2. A student s solution for tan 2 x = sec x tan 2 x for 0 x < π is shown below: Identify and explain the error(s). How many mark should the student get if this question was worth 4 marks?

109

110 Page 320 #1, 2A) B) C), 3, 5 Page 321 #6,9,11,14,