Unit 6 Trigonometric Identities

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1 Unit 6 Trigonometric Identities Prove trigonometric identities Solve trigonometric equations 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) TRIG IDENTITIES You should be able to explain the difference between a trigonometric and a trigonometric. An is true for, whereas an is only true for a of the permissible values. This difference can be demonstrated with the aid of graphing technology. 1 P a g e

2 1 For example: sin x,0 x This can be solved by using the graphs of 1 The solutions to sin x,0 x are x = and x =, which are the. Thus this is because it is. 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 identical. The only differences in the graphs occur at the points. Why? Therefore, sin x = tan x cos x is an since the expressions are for all permissible values. 2 P a g e

3 Note: There may be some points for which identities are. These values for identities occur where one of the expressions is In the previous example, y = tan x cos x is not defined when since is undefined at these values. Non-permissible values often occur when a trigonometric expression contains: A, resulting in values that give a denominator of, since these expressions all have non-permissible values in their domains. Practice: Determine graphically if the following are identities. Use Technology Identify the non-permissible values. ( i ) sin cos tan 2sin y x Non-permissible values? Is this an identity? 3 P a g e

4 ( ) tan 1 sec 2 2 ii y x Non-permissible values? Is this an identity? cos ( iii ) sec sin y x Non-permissible values? Is this an identity? 4 P a g e

5 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) 2 2 sin cos 1 (use degrees) (use radians) 2 2 C) tan 1 sec D) (use degrees) 2 2 cot 1 csc (use radians) NOTE: This approach is to conclude that the equation is an identity because only a of values were substituted for θ, and there may be a certain group of numbers for which this identity. To prove the identity is true using this method would require verifying of the values in the domain ( ). This type of reasoning is called. 5 P a g e

6 Proofs! A proof is a that is used to show the validity of a mathematical statement. Deductive reasoning occurs when general are to specific situations. Deductive reasoning is the process of based on facts that have already been shown to be. The facts that can be used to prove your conclusion deductively may come from accepted. The truth of the premises the truth of the conclusion. Find the fifth term in the sequence Inductive Reasoning 1. 3, 5, 7, 9, , 12, 27, 48, , 14, 21, 28,... 6 P a g e

7 1. t n =2n + 1 Deductive Reasoning 2. t n = 3n 2 3. Dates of Wednesdays in 2015 year What is the next number in this sequence? 15, 16, 18, 19, 25, 26, 28, 29, Trig Proofs Trig proofs (and simplifications of trig expressions) are based on the definition of the 6 trigonometric functions and the Fundamental Trigonometric Identities. Definition of the 6 trigonometric functions Sine fn: Cosine fn Tangent fn Cotangent Secant fn Cosecant fn 7 P a g e

8 Fundamental Trigonometric Identities. Reciprocal Quotient Pythagorean Caution The Pythagorean identities can be expressed in different ways: 8 P a g e

9 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 Write the expression in terms of For expressions involving addition or subtraction, it may be necessary to use a to simplify a fraction Multiply by a to obtain a You may also be asked to determine any values of the variable in an expression. For example, identify the non-permissible values of θ in and then simplify the expression. Solution: sin cos cot Simplify : sin cos cot 9 P a g e

10 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. Simplify the following A) sin x secx B) 2 1 cos sin In this case we use a C) sec sec sin 2 In this case we 10 P a g e

11 2 sec x D) tan x In this case we have choices sin cos E) 1 cos sin What do we do here? 11 P a g e

12 F) tanx sinx 1 cosx What do we do here? Now we have choice P a g e

13 2. Page 296 # 1 a) d) 3b) c) 4, 7, 8c), 9, P a g e

14 Warm UP Factor and simplify 1. sin sin cos 2 x x x sin 2 x 2. 2 tan x 3tanx 4 sinx tanx sinx Proving Identities The fundamental trigonometric identities are used to establish other relationships among trigonometric functions. To, we show that one side is equal to the other side. Each side is manipulated of the other side. It is to perform operations, such as 14 P a g e

15 These operations are only possible if the equation is. Until we verify, or prove the identity to be, we do not know if both sides are. 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 2 2 C) sin cos D) tan 1 sec 15 P a g e

16 Guidelines for Proving Trigonometric Identities We usually start with side that contains the more expression. If you substitute one or more on the more complicated side you will often be able to rewrite it in a form identical to that of the other side. Rewriting the complicated side in terms of is often helpful. If sums or differences appear on one side, use and combine fractions In other cases is useful. It may be necessary to multiply a fraction by a to obtain a There is that can be used to prove every identity. In fact there are often different methods that may be used. However, one method may be and than another. The more identities prove, the more confident and efficient you will become. DON T BE AFRAID to over again if you are not getting anywhere. Creative puzzle solvers know that strategies leading to dead ends often provide good problem-solving ideas 16 P a g e

17 A) secx cot x cscx Prove the following Which side is the more complicated side? Lets work on B) sinx tan x cosx secx Which side is the more complicated side? Lets work on 17 P a g e

18 3 2 C) cos cos cos sin Which side is the more complicated side? Lets work on cos 1 sin D) 2sec 1 sin cos Which side is the more complicated side? Lets work on 18 P a g e

19 sin 1 cos E) 1 cos sin Which side is the more complicated side? Lets work on F) cos sin 2cos 1 19 P a g e

20 sint cost G) sint cost tant cott sint H) cott csct 1 cost 20 P a g e

21 1 1 2 I) 2 cot t 1 cost 1 cost J) 2 1 sec sec sec 5sec 5sec 2 21 P a g e

22 OTHER TRIG STUFF Even-Odd Identities (Negative Angle): Addition and Subtraction Rules: PROOF: This one of those interesting proofs. We need to use the: Law of Cosines And the distance formula between 2 points 22 P a g e

23 PROOF: 23 P a g e

24 PROOF: Replace b by b in PROOF: Replace a by in 2 a PROOF: Replace b by b in 24 P a g e

25 Addition Formula for Tan PROOF: Subtraction Formula for Tan 25 P a g e

26 Applications of the Angle Addition Formulae Finding exact values Deriving double and half angle formula Proving Identities In Calculus: Trig derivatives Trig substitution in integration. Find the exact values of: A) cos 15 o B) sin 75 o C) D) 7 sin tan E) sin 60 o cos 30 o + sin 30 o cos 60 o F) tan15 o tan30 o o 1 tan15 tan30 o How can we verify that this is true? 26 P a g e

27 G) A and B are both in Quadrant II, cosa 5 13 and sinb 3 5. Determine the exact value of cosa B. 2. Simplify A) sin sin 2 2 B) tan 27 P a g e

28 Identities 3. A) Prove: sinx cosx cosx 6 3 B) Prove: cos cos 2cos cos 28 P a g e

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30 Double Angle Formulae sin2 cos2 tan2 Examples 1. Find the exact values of: A) 2sin15 o cos15 o 2 2 B) cos sin P a g e

31 2. Simplify: x x 4tan A) sin cos B) tan C ) cos2x sin sin2x 2 x 3. PROVE: 1 cos2a A) tan 1 cos2a 2 A 31 P a g e

32 tan2btan B B) sin2b tan2b tanb C ) sin sin D) 2sin cos sin P a g e

33 E x x x 4 2 ) cos4 8cos 8cos 1 sin2x F)Show that can be simplfied tocot x 1 cos2x 33 P a g e

34 Suppose: sin and Find the exact value of: A) sin 2 B) cos 2 C) tan2 Half Angle Formulae Not on Public but good to know Consider: cos2 = 1 2sin 2 cos2 = 2cos 2-1 Let Let Examples: Find the exact value of: A) sin 15 o B) cos 75 o 34 P a g e

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36 Last Section for Chapter 6 (6.4) Solve, algebraically and graphically, first and second degree trigonometric equations The identities encountered earlier in this unit can now be applied to solve trigonometric equations. Examples: 1. Find the solutions of sin2 x 3cosx. for 0 x < 360. Solution: Graphically A) Identify each curve B) What are the points of intersection? Solution: Algebraically What are the solutions with an unrestricted domain, in radians? 36 P a g e

37 2. Solve cos2x 1 cos x for 0 x 360, giving exact solutions where possible. Write the general solution in degrees and radian measure. 3. Solve the trigonometric equation shown below for : sin3x cosx cos3x sinx 3 2 0x 2 37 P a g e

38 4. Solve: cos 2x + sin 2 x = , for the domain 0 x < 360. Identifying and Repairing Errors 1. Identify and repair the mistake 38 P a g e

39 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? Provide the correct solution 39 P a g e

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