MA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically

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1 1 MA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically Definition Trigonometric identity Investigate 1. Using the diagram of the unit circle provided and the definitions of trigonometric functions, determine the following basic trigonometric identities. Pythagorean Identity: sin x + cos x = Quotient Identity: tan x = Verifying Identities. We can verify identities by two methods: graphically and numerically. The only way that we can prove that an equation is actually an identity is algebraically. MA40SP_C_TrigIdClassNotes.doc Revised:

2 a) Verify that tan x + 1 = sec x could be an identity by evaluating and testing to see π if makes the equation true. 6 b) Verify that tan x + 1 = sec x could be an identity by graphing. Numerically π Substitute into the equation for x. Is 6 the equation true? Graphically Plot y = tan x + 1 and y = sec x on the same coordinate plane. Why is this not a proof that the equation is an identity? Can we assume that this equation is an identity? Explain your answer.

3 3 Example 1: 1 cos x Verify the identity = sin x Numerically Substitute x = 1 into the equation for x. Is the equation true? Graphically 1 cos x Plot y = and y = sin x on the same coordinate plane. Is the equation an identity? Explain your answer.

4 4 Proving Trigonometric Identities Objective: To analyze and prove trigonometric identities algebraically 3. Can numerical substitution or graphing guarantee that an equation is an identity? 4. The most complete method for proving trigonometric identities uses algebra. This method can involve simplifying, factoring, and re-writing expressions. The trigonometric identities that have been learned thus far will need to be used here to prove other identities. Investigate sin x 1 5. Use the basic identities tan x =, sin x + cos x = 1, and sec x cos x = cos x to prove tan x + 1 = sec x. Work on the left side of this equation until it equals the right side. Helpful hints are given for each step. Step Hint L.S. (Left Side) tan x sin x use tan x = cos x. combine L.S. terms with a common denominator 3. sin x + cos x = 1 use cos x = 1 sin x R.S. (Right Side) sec x 4. use 1 sec x = cos x 6. Therefore, the identity List any restrictions on x. tan x + 1 = sec x is generally true for the variable x.

5 5 7. Some basic trigonometric identities Reciprocal Identities 1 csc x = sin x 1 sec x = cos x cot x = 1 tan x Quotient Identities sin x tan x = cos x cos x cot x = sin x Pythagorean Identities sin x + cos x = 1 tan x + 1 = sec x cot x + 1 = csc x Example 1: Use the basic identities to prove sin θ + tan θ = tanθ cosθ + 1 L.S θ + θ sin tanθ cos + 1 R.S tanθ

6 6 Several strategies that are often successful were used in this proof. Describe the strategies used. 8. There is often more than one correct way to prove an identity. Sometimes it helps to work on both sides of the equation until they simplify to the same expression. Based on experience the more complicated-looking side is the best place to start. Example : Prove cscθ 1 cotθ = cotθ cscθ + 1 Note: In this example the two sides appear symmetrical: there is no harder side to start on! Expressions such as (sinθ 1) and (sinθ + 1) are called the conjugates of each other. Multiplying them sometimes produces a Pythagorean Identity: ( sinθ 1)( sinθ + 1) = sin θ 1 = cos θ Use this idea as a hint. L.S θ cotθ csc 1 R.S cotθ θ + csc 1

7 7 9. Tips The use of the conjugate in some proofs (see Example 4) is based on the principle of multiplying by 1. Do this to one side of the proof only. Once an identity is established, it can be rearranged. For example cot θ = csc θ 1 is just another version of cot θ + 1 = csc θ, and any rearrangement can be used in future proofs. Do not combine more than one step in a proof on the same line. Your reasoning will not be clear and your proof will be confusing. If second degree terms are involved (ex. sin x ), consider using the Pythagorean Identities or factoring. Avoid using square roots. Reciprocal and Quotient Identities can be generalized; for example: 1 csc x = sin x 5 5csc x = sin x 3cos x 3cot x = or sin x 3 3cot x = tan x Avoid common mistakes, such as cos x cos x, if x + x = sin x + cos x 1, sin cos 1, sin x sin x sin x.

8 8 Sum and Difference Identities Objective: To use sum and difference identities for sine and cosine to verify and simplify trigonometric expressions and to prove other identities. Investigate 10. Consider the equation cos( A B) = cos A cos B a) Verify the equation cos( A B) = cos A cos B numerically, using B = π. 3 b) Is this an identity? A = π and

9 9 11. Consider the equations i. cos( A B) = cos Acos B sin Asin B and ii. cos( A B) = cos Acos B + sin Asin B a) Verify the equations numerically, using b) Which equation appears to be true? A = π and B = π. 3 c) Guess and check an identity for cos(a+b)

10 10 1. a) Verify sin( x + y) = sin x cos y + cos x sin y graphically by assigning then graphing each side of the equation. b) Guess and check a similar identity for sin( x y) π y = and 3

11 The above explorations lead to the following sum and difference identities. Sum and Difference Identities sin( A + B) = sin Acos B + cos Asin B sin( A B) = sin Acos B cos Asin B cos( A + B) = cos Acos B sin Asin B cos( A B) = cos Acos B + sin Asin B tan A + tan B tan( A + B) = 1 tan Atan B tan A tan B tan( A B) = 1 + tan Atan B Example 1: 13. Express the following as a trigonometric function of a single angle: π π sinπ cos cosπ sin. 5 5

12 1 Example : 14. Consider the identity sin( π x) = cos x. π a) Verify the identity numerically, when x = 6 b) Verify the identity graphically. c) Prove the identity algebraically

13 13 Example 3: If sin A = and cos B = and both A and B are in Quadrant, evaluate 3 5 cos( A B). (Use the sketches given to find the exact trig values.)

14 14 Example 4: π 16. Rewrite cos as a difference identity and use the identity to find the exact value 1 π of cos. 1

15 15 Example 5: 1 tan x 17. Graph the function f ( x) = 1 + tan x a) Use the graph to make a conjecture about the period and the amplitude of f ( x ). b) Make a conjecture stating what single trigonometric function f ( x) is equivalent to. c) Graph this function. d) Prove your conjecture algebraically.

16 16 Double Angle Identities Objective: To use double angle identities for sine and cosine to verify and simplify trigonometric expressions and to prove other identities. Investigate 18. Special cases of addition identities occur when the two angles are equal. a) Write an identity for sin( x + y). Derive the identity for sin x by substituting x for y and then simplifying. π b) Verify the identity derived in part a) numerically, for x =. c) Verify the identity derived in part a) and graphically.

17 a) Derive an identity for cos x by substituting x for y and then simplifying. b) Verify the identity derived in part a) numerically, for π x =. c) Verify the identity derived in part a) and graphically. d) The identity for cos x has three versions! Can you use the first version above and the Pythagorean Identity, sin x + cos x = 1, to produce the second and third versions of cos x?

18 18 0. Use the identity for sin x and cos x to find the identity for tan x. 1. The above investigations lead to the following double angle identities. Double Angle Identities sin x = sin x cos x cos x = cos x sin x = cos x 1 = 1 sin x tan x tan x = 1 tan x

19 19 Example 1:. Consider the identity 1 cos x = tan x. sin x a) Verify the identify numerically when b) Verify the identity graphically. c) Prove the identity algebraically. π x =

20 0 Example : 3. Write 10sin 3x cos3x as a single trigonometric ratio solution. Example 3: 4. Prove sin 3 3sin 4sin 3 x = x x

21 1 Example 4: 5. Find the amplitude and the period of the graph of y = 4sin x cos x + 3.

22 Example 5: 6. Solve ways. + = to 4 decimal places for 0 x π, in different 3cos x 3sin x 0