Chapter 3: Trigonometric Identities

Transcription

1 Chapter 3: Trigonometric Identities

2 Chapter 3 Overview Two important algebraic aspects of trigonometric functions must be considered: how functions interact with each other and how they interact with their arguments (the expression inside the function e.g., π ( 4 x ) in sin π ( 4 x ) ). First, trigonometric functions are inextricably linked because of how they were defined via the coordinates of points on the terminal side of an angle. Since the coordinates are related by the Pythagorean theorem, there are some relationships that will be true regardless of the angle. These are referred to as identities and will act as formulas. THESE MUST BE MEMORIZED. Second, since the trigonometric function is the operation and no multiplication is present, rules on how the functions work with their arguments will be provided. Without multiplication, the associative, commutative, and distributive rules do not hold. In particular, sin a + b sina + sinb and sinx sin x. New rules for the algebra of trigonometric operations will also be developed. 95

3 3-: Quotient, Reciprocal, and Pythagorean Identities REMEMBER: sinθ y r cosθ x r tanθ y x cscθ r y secθ r x cotθ x y Based on these definitions, here are the basic identities. The Reciprocal Identities: cscθ sinθ secθ cosθ cotθ tanθ The Quotient Identities: tanθ sinθ cosθ tanθ secθ cscθ cotθ cosθ sinθ cotθ cscθ secθ The Pythagorean Identities*: sin θ + cos θ tan θ + sec θ + cot θ csc θ *Recall: sin θ + cos θ ( sinθ ) + ( cosθ ) 96

4 The Pythagorean identities have alternative versions as well: sin θ + cos θ cos θ sin θ sin θ cos θ tan θ + sec θ sec θ tan θ sec θ tan θ + cot θ csc θ csc θ cot θ csc θ cot θ These alternative forms are very useful because they are difference of squares binomials that can be factored. For example, sin θ ( sinθ )( + sinθ ) LEARNING OUTCOME Prove basic trigonometric identities. The proofs of the identities are algebraic in nature, meaning that the use of multiplication, addition, and common denominators will cause one side of the equation to simplify to the other. Similar to proofs in Geometry where one can work both top down and bottom up, in these proofs the two sides can simplify to the same thing. EX Prove csc x tan xcos x csc x tan xcos x sin x i sin x cos x i cos x Notice that the answer is the process, not the final line; the final line was given. 97

5 EX Prove cot xsec x + csc x csc x This one can be done quickly if the substitution cot x csc x sec x This would give: cot xsec x + csc x csc x sec x + csc x sec x csc x + csc x csc x is used. Another, longer way, would be to turn the whole problem into sin x and cos x. cot xsec x + csc x cosx sin x cosx + sin x sin x + sin x sin x csc x EX 3 Prove csc A+ cot A + csc A cot A csc A csc A+ cot A + csc A cot A csc A cot A csc A+ cot A csc A cot A csc A cot A+ csc A+ cot A csc A+ cot A ( csc A cot A) csc A csc A cot A csc A csc A csc A+ cot A + csc A+ cot A csc A cot A 98

6 3- Free Response Homework Prove the following identities.. cos x + tan xcos x. cosθ cosθ tanθ + sinθ tanθ ( tanσ cotσ ) sec σ + csc σ ( cosθ )( cosθ ) ( sinθ )( sinθ ) cos θ ( sin x) ( sin x) ( cos x) ( cos x) sin x ( 6. sinθ ) secθ tanθ cos θ 7. cosϕ cscϕ cotϕ sin ϕ sec θ csc x 8. ( cosa + sina) cosa + sina cosa sina 9. tanw ( cotw cosw + sinw ) secw 0. sec x sin x sin x cos x cot x. sin x cos x + csc x. cos cotθ cscθ tanθ sinθ θ cos x sin x cscθ 3. tanb + cotb cscb cosb 4. cosx cosx cot xcsc x + cot x 5. + sin x sin x sec x + sec x tan x 99

7 6. State some strategies used for approaching the previous identities. 3- Multiple Choice Homework. Where defined, csc x sin x (a) sin x (b) sin x (c) csc x (d) csc x (e) cos x. If the ratio of sin x to cos x is :, then the ratio of tan x to cot x is (a) :4 (b) : (c) : (d) : (e) 4: 3. If ( sec x) ( tan x) < 0, which of the following must be true? I. tan x < 0 II. csc x cot x < 0 III. x is in the third or fourth quadrant (a) I only (b) II only (c) III only (d) II and III (e) I and II 4. Simplify ( tan + cot ) csc (a) sin (b) sec (c) tan (d) (e) cos 00

8 5. If 90 <α <80 and 80 < β < 70, then which of the following must be FALSE? (a) tanα tanβ (b) cosα cosβ (c) sinα tanβ (d) tanα sinβ (e) secβ secα 6. To which of the following is ( sin y cosy) cosy equal? (a) sec y (b) sec y sin y (c) sec y tan y (d) sin y (e) tan y 0

9 3-: Trigonometric Identities and Factoring Hopefully the answer to #6 in the previous homework looked something like this: Techniques for Approaching Trigonometric Proofs:. Get down to two or less trig functions. These might be Sin and Cos, but be aware of how Sec and Tan, or Csc and Cot, work together.. Do Algebra: a. Common Denominators b. Adding like terms c. Distribute and/or FOIL 3. Squares indicate possible Pythagorean identities Another technique under the Do Algebra heading would be Factoring. LEARNING OUTCOME Prove trigonometric identities that involve factoring. It might be best to go back and review factoring at this point. There are three basic modes of factoring:. Factoring by rules. Factoring by guess and check 3. Factoring by splitting the middle term I. Factoring by Rules The first technique for factoring is called monomial factoring. It is the most common and most often forgotten. Monomial factoring is the reverse of distribution, and involves pulling out factors common to each term. 0

10 Distribution and Monomial Factoring a( b ± c) ab ± ac ab ± ac a( b ± c) EX Factor x y + xy x y + xy xy( x + y) What would this look like in a trigonometric identity proof? EX Use factoring to prove cos x + tan xcos x cos x + tan xcos x cos x + tan x cos x isec x cos x i cos x Factoring Rules for the Sum or Difference of Equal Powers:.* x y x + y. x 3 + y 3 x + y 3.** x 3 y 3 x y 4.** x 4 y 4 x + y ( x y) x xy + y x + xy + y ( x y ) *NB. x + y does not factor when working with real numbers. The rule for x + y involves imaginary numbers, which are not of interested at this point. **The trinomials in rules 4 and 5 are not factorable in the real numbers either. 03

11 EX 3 Simplify u3 v 3 ( u v) 3 u 3 v 3 ( u v) 3 ( u v) u + uv + v u v ( u v) ( u v) u + uv + v u v ( u v) EX 4 Prove cot 4 w csc 4 w csc w cot 4 w csc 4 w ( cot w csc w) cot w+ csc w ( ) cot w+ csc w csc w + csc w csc w EX 5 Prove cos3 x sin 3 x cosx sin x cosxsin x cos 3 x sin 3 x cosx sin x ( cosx sin x) cos x + cosxsin x + sin x cosx sin x cos x + cosxsin x + sin x cosxsin x 04

12 II. Factoring by Guess and Check. The second technique is trinomial factoring, and is a guess and check process. It involves finding numbers that are factors of the first and third terms of a trinomial and checking the FOILing. EX 6 Factor y 4y 5 If y 4y 5 is factorable, the first terms must multiply to y. So, y 4y 5 ( y )( y + ) The two numbers must multiply to the last number ( 5) and add to the middle number ( 4). In this case, the numbers must be 5 and +. So, y 4y 5 ( y 5) ( y +) EX 7 Factor 4x 0x 6 This problem is more complicated than the last because there is more than one possibility for the factors of the first and the third terms. 4x 0x 6 ( 4x )( x ) or 4x 0x 6 ( x )( x ) The second numbers might be and 3, and 3, and 6, or and 6. The guess and check process takes some time, but ultimately yields 4x 0x 6 4x + ( x 3) ( x 3) x + 05

13 III. Factoring by splitting the middle term and/or by grouping. EX 7 Factor 4x 0x 6 (EX 4 again) In order to factor by grouping, 0 needs to be split into two terms. How? The proper split will be two factors of the product of first and last coefficients and that add to 0. 4 times 6 is 4. The factors of 4 are The pair that adds to 0 are and. So, 4x 0x 6 4x + x x 6 x( x +) 6 x + x 6 ( x +) ( x +) x 3 ± and 4 ± and ±3 and 8 ±4 and 6 EX 8 Prove 3sin θ 8sinθ + 4 9sin θ 4 3sin θ 8sinθ + 4 9sin θ 4 3sin θ 5sinθ 9sin θ 3sinθ 3sinθ 3sinθ + 3sin θ 5sinθ 9sin θ 3sinθ 3sin θ 8sinθ + 4 9sin θ 4 ( sinθ ) ( 3sinθ + ) 3sinθ 3sinθ 3sinθ 3sinθ + ( 3sinθ ) ( sinθ ) 3sinθ + 3sinθ + 9sin θ 3sinθ 3sin θ 5sinθ 06

14 EX 9 Prove 5sec θ 4tanθ + 3 5sec θ 8tanθ 9 5tanθ 4 5tanθ + 5sec θ 4tanθ + 3 5sec θ 8tanθ 9 5( tan θ +) 4tanθ + 3 5( tan θ +) 8tanθ 9 5tan θ +5 4tanθ + 3 5tan θ +5 8tanθ 9 5tan θ 4tanθ +8 5tan θ 8tanθ 4 5tanθ 4 ( tanθ ) ( 5tanθ + ) ( tanθ ) 5tanθ 4 5tanθ + 07

15 3- Free Response Homework Factor and simplify the following expressions.. x 6xy + 8y. 3x 3 y +0x y 8xy x 5x x 3 4. x 3 x y + xy y 3 5. x 4 3x + 6. k 4 r 4 ( k + r ) k + r 7. x 3 + x + x + x 4 8. x 3 + y 3 ( x + y) 3 Prove the following identities. 9. sin x + cot xsin x 0. sec 4 β tan 4 β + tan β cos w + cosw + cosw + cos w 3cosw 4 cosw 4 cos 3 A sec 3 A cos A sec A sin A+ tan A+ 3 tan w secw 5 tan w + 3secw + 3 secw 3 secw + sin w + 5cosw + 6sin w + 5cosw cosw 3 3cosw 4 3sec t 8tant + sec t tant 3 3tant tant + 08

16 6. csc y 7cot y 6 6csc y 5cot y 0 cot y 4 3cot y 4 Simplify the following expressions. 7. x + 5x x + 6x + 5 x3 3x + 3 i x x + 8. x 4 6 ( x + ) x x 3 x x ( r + rx) ( r rx) x + r x r r3 + x 3 r 3 x 3. x + 3wx 3w x i x 4wx + 3w x + 3w w x x + w. x + 4xy + 3y x + 5xy + 6y i x + y x + y x + 4xy + 4y Prove the following identities cot B + cot B + tan 3 B tan B + sec B ( 8 + cos 3 x) cosx i cos x 4cosx + 4 cos x cosx + 4 ( sec x + 3sec x 0) sec x 7sec x cos x 4 ( i sec x + sec x 3) ( sec x + sec x 6) sec x + 5 sec x 6 cot x 4csc x csc x 6 cot x 3 csc x sin B sin B + + sin3 B sin B sin B 8. csc 6 x cot 6 x + 3csc xcot x 09

17 sin 3 x sin 4 x 8 6 sin x sin x + sin x 3 sin x + 3 ( sin x + sin xcosx) sin x + cosx sin x sin xcosx sin x cosx cos3 x + sin 3 x cos 3 x sin 3 x sin x + sin xcosx + cos x sin x sin xcosx + cos x 3- Multiple Choice Homework. cot x sec x + csc x (a) (b) (c) (d) (e) cot x sec xcsc x cot x sec x + csc x cot x sec x + csc x cot xsec x + cot xcsc x sec xcsc x cot xsec x + cot xcsc x. Divide x 3 x + 3x 8 by x 3. (a) x + x + 6 (b) x x 6 (c) x 5x +8 (d) x + 5x 8 (e) x 5x 8 0

18 3. Find 6 ( tan x ) tan x + 4 tan x + tan x + tan x + 4 tan x tan x( tan x ) (a) tan x + (b) 3tan x tan x ( tan x + tan x +) (c) tan x tan x tan x + (d) 3tan x 3 tan x + 3tan x tan x (e) tan x + 4. Simplify cos x 3cosx cos x 5cosx + 6 cosx ( cosx ) (a) cosx ( cosx 3 ) (b) cosx (c) cosx cosx cosx 3 (d) cosx (e) cosx 5. Simplify 5x3 y + 0x y + 0xy 3 ( x + y) (a) x + (b) (c) x + y (d) x + 4y (e) x 4xy + 4y 5xy

19 3-3: Composite Argument and Even/Odd Rules As stated in the chapter overview, sin( a + b) sina + sinb. This is easily demonstrated with a pair of the special angles. If a 30 and b 60, then sin( ) sin90. But sin30 + sin Clearly these are not equal. So how does a trigonometric operation apply to its argument when the argument is a composite (i.e. a + b )? The Composite Argument Identities: cos Acos B + sin Asin B cos Acos B sin Asin B cos A B cos A+ B sin Acos B cos Asin B sin Acos B + cos Asin B sin A B sin A+ B tan( A B) tan( A+ B) tan A tan B + tan Atan B tan A+ tan B tan Atan B

20 Some functions are called odd functions because they act like odd powers they preserve negative signs: x 3 ( x) 3. Others are called even functions because ( x). they act like even powers and cancel negative signs: x Odd Functions: Even Functions: sin A cos ( A) cos A csc A sec ( A) sec A sin A csc A tan A cot A tan A cot A LEARNING OUTCOMES Find exact trigonometric values for composite arguments. Solve equations involving composite argument rules. Prove identities involving composite rules. There are basically three tasks that can be completed using these rules:. Find exact values,. Solve equations, and 3. Prove other identities. 3

21 EX Given sinβ 5 in Quad II, and 6, 8 3 the exact value of sin α + β. is on the terminal side of α, find If sinβ 5 3 in Quad II, then by the Pythagorean identity sin θ + cos θ, cosβ 3. If ( 6, 8) is on the terminal side of α, then sinα and cosα sin( α + β ) sinα cosβ + cosα sinβ

22 EX Solve for x { All Reals} if cosxcos π 3 sin xsinπ 3 3. cosxcos π 3 sin xsinπ 3 3 cos x + π 3 3 x + π 3 cos 3 x + π 3 π 6 ± π n π 6 ± π n x π 6 ± πn π ± πn The final answer here is known as the general solution. It is a condensed way to express and infinite number of coterminal answers. x π 6 ± π n π ± π n... 5π 6, 3π 6, π 6, π 6, 3π π, 5π, π, 3π, 7π... Vocabulary:. General Solution all (the infinite number of) coterminal solutions. Particular Solution the specific solutions in a given domain 5

23 Sometimes the request is for the particular solution. EX 3 Solve for x 0, π if sin xcos π 3 cos xsinπ 3. sin xcos π 3 cosxsinπ 3 sin x π 3 x π 3 sin x π 3 x π 6 ± π n 5π 6 ± π n π ± π n 7π 6 ± π n Of all the answers summarized by the general solution above, only two fall within the domain x 0, π : x π, 7π 6 6

24 EX 4 Prove cos( x + 30 ) sin( x + 60 ) sin x cos( x + 30 ) sin( x + 60 ) cosxcos30 sin xsin30 ( sin xcos60 + cosxsin60 ) 3 cosx sin x sin x 3 cosx sin x 7

25 3-3 Free Response Homework. Given tanα 5 8 the exact values of a) sin µ α b) cos µ +α c) tan α µ. Given secλ 5 4 find the exact values of a) sin λ +ω b) cos λ ω c) tan ω + λ in Quad III, and ( 3, 5) is on the terminal side of µ, find d) sec( α µ ) e) csc( α + µ ) f) cot( α + µ ) in Quad III, and (, 5) is on the terminal side of ω, d) sec( ω + λ) e) csc( ω λ) f) cot( λ ω ) Find the general solutions to the following equations. 3. sin3θ cos cos3θ sin for x { All Reals} 4. sin xsin60 cosxcos60 for x 0, sinφ cos5 cosφ sin5 for x { All Reals} 6. tan3x + tan x tan3x tan x for x 0, π 7. sec x π 4 + sec x π 4 for x π, π 8. ( cos Acos B sin Asin B) + ( sin Acos B + cos Asin B) for x { All Reals} 8

26 Prove the following identities. 9. sin( A+ B)+ sin A B sin( A+ B) sin A B tan Acot B 0. cos( a + b)cosb+ sin( a + b)sinb cosa.. 3. sin Acos A+ cos Asin A cos Acos A sin Asin A tan A tan A sin Acos B + cos Asin B tan A+ tan B cos Acos B sin Asin B tan Atan B cot Acot B cot( A+ B) cot A+ cot B 3-3 Multiple Choice Homework. Which of the following is equivalent to sin( A + 30 )+ cos A + 60 values of A? for what (a) sin A (b) cosa (c) 3sin A + cosa (d) 3sin A (e) 3cosA. Which of the following is equivalent to sin( α + β )+ sin( α β )? (a) sinα (b) sin α β (c) sinα sinβ (d) sinα cosβ (e) cosα sinβ 9

27 3. If sin A 3 5, 90 A 80, cosb, and 70 B 360, sin( A + B) 3 (a) (b) (c) 0.83 (d) (e) tan( x x ) (a) tan x tan x tan x tan x (b) tan x tan x + tan x tan x tan x (c) + tan x tan x tan x tan x (d) + tan x + tan x tan x (e) None of these 5. cos 7π 8 cosπ 8 sin 7π 8 sinπ 8 (a) (b) (c) 0 (d) (e) 0

28 3-4: Double Angle Rules As stated in the chapter overview sinx sin x, and, as with the composite rules, this is easily demonstrated with a pair of the special angles. If a 30, then sin60 3 sin i30. But sin30 So if sinx sin x, what does sinx equal? sinx sin( x + x) sin xcosx + cosxsin x sin xcosx. Clearly these are not equal. The Double Angle Argument Identities: tan( A) tan A tan A sin( A) sin A cos A cos( A) cos A sin A sin A cos A or tan( A) sin A cos A LEARNING OUTCOMES Find exact trigonometric values for double angle trigonometric functions. Solve equations involving double angle rules. Prove identities involving double angle rules.

29 The same three types of problems will be done with the double angle rules as were done with the composite rules. EX Given sinβ 5 3 tan( β ). in Quad II, find the exact value of sin( β ), cos( β ), and If sinβ 5 3 sin( β ) sinβ cosβ in Quad II, then by the Pythagorean identity, cosβ cos( β ) cos β sin β tan( β ) sin β cos β

30 EX Solve for x if 4sin xcosx. 4sin xcosx sin xcosx sin x x sin π x 4 ± π n 3π 4 ± π n π x 8 ±π n 3π 8 ±π n EX 3 Prove tan x sinx + cosx It would be appropriate in this problem to work both sides of the equation toward the same goal. tan x sinx + cosx sin x cosx sin xcosx + cos x sin xcosx cos x sin x cosx 3

31 3-4 Free Response Homework. Given sin A 3 the exact values of a) sin A b) cos B c) tan A in Quad II, and ( 7, 4) is on the terminal side of B, find d) sec( B) e) csc( A) f) cot( B) Find the general solutions to the following equations. Use exact values wherever possible.. 4cosθ sinθ for θ { All Reals} 3. cos x sin x cos x for x 0, π 4. sin( x 30 )cos( x 30 ) 3 for x { All Reals} 5. cscx + 6 for x { All Reals} sin xcosx 6. tan x tan x for x π, π 7. tan x tan x 3 for x { All Reals} Prove the following identities. 8. cosb tan B + tan B 9. cosβ cos 4 β sin 4 β 4

32 0. cot A+ tan A csc( A). cot φ cosφcot φ + cosφ. tan x tan x + cos x sin x sin xcosx 4csc x 4 3. ( + tanδ )tanδ tanδ tanδ 3-4 Multiple Choice Homework. Which of the following is NOT equal to cosx cos x? (a) cos x sin x tan x (b) sec x sin x cot x (c) cos x sec x (d) sec x sin x tan x (e) All are equal to cosx cos x. Which expression equals cotx? (a) (b) (c) (d) (e) cot x + cot x cot x cot x sin x tan x tan x tan x cot x tan x 5

33 3. If cos x 4 5 and 3π x π, then tan x (a) 7 4 (b) 4 5 (c) 3 4 (d) 4 7 (e) For what value between 0 and 360 does cosx cos x? (a) 68.5 or 9.5 (b) 68.5 only (c) 03.9 or 56. (d) 90 or 70 (e).5 or If 3sin θ cosθ and 0 θ 80, then θ (a) 0 (b) 0 or 80 (c) 80.5 (d) 0 or 80.5 (e)

34 6. If sin A 3 5, 90 < A <80, cosb, and 70 < B < 360, the value of 3 tan( A + B) is (a) 0.30 (b) 3.9 (c) 3.34 (d).05 (e) 0.3 7

35 3-5: Half Angle Rules Just as there are Double Angle Identities, there are half angle formulas. The Half Angle Argument Identities: sin A ± cos A cos A ± + cos A tan tan tan A A A ± cos A + cos A sin A + cos A cos A sin A Note the ± on three of the formulas. This does really mean plus OR minus, not plus AND minus as usually meant in mathematics. That is, in these formulas, one OR the other is correct, but not both. Determine the sign from the size and quadrant of A. The same three types of problems will be done with the half angle rules as were done with the double angle and composite rules. LEARNING OUTCOMES Find exact trigonometric values for half angles. Solve equations involving half angle rules. Prove identities involving half angle rules. 8

36 EX Given sinβ 5 3 and 540 < β < 630, find the exact value of sin β, cos β, and tan β. If sinβ 5 3 and 540 < β < 630, then cosβ 3. Also, since 540 < β < 630, then 70 < β < 35. This means that β is in QIV, and the cosine will be positive and the sine negative. sin β ± cosβ cos β ± + cosβ tan β sinβ + cosβ

37 EX Solve for x if sin x cos x 3. The left side of the equation is not one of the formulas, but it is the reciprocal of one of the formulas. sin x cosx 3 cosx sin x 3 tan x 3 x 3 tan x π 6 ±π n x π 3 ± π n EX 3 Prove tan x cot x + tan x This is an interesting problem because there are two places to use a half angle rule, and the same formula is not needed both times. tan x cot x + tan x sin x + cosx sin x + cosx cosx sin x + cosx sin x + cosx sin x 30

38 3-5 Free Response Homework Given the following values and quadrant ranges, find the exact values of a) sin x b) cos x c) tan x. cos x 4 5 and 0 < x < 90. cos x 4 5 and 90 < x <80 3. cos x 4 5 and 80 < x < 70 Given the following values and quadrant ranges, find the exact values of a) csc x b) sec x c) cot x 4. sin x 4 5 and 90 < x < 0 5. cos x 4 5 and 630 < x < 70 Find the general solutions to the following equations. Use exact values wherever possible. 6. cos x for x { All Reals} 7. ( cos x) 3 for x 0, 4π 8. + cos x for x 0, π 9. ( + cos x) 0 for x All Reals { } 3

39 0. sin x for x { All Reals} + cos x. cos x sin x 3 for x π, π Prove the following identities.. tan x + cot x csc x 3. tan x tan x sec x 4. tan x + tan x sin x 5. tan x csc x cot x 6. sin x + cos x + sin x 7. cos x + sin x cos x sin x sec x + tan x 8. cos x sin x cos x + sin x cos x + sin x 3

40 3-5 Multiple Choice Homework. If cosθ 3 sinθ 3, then θ (a) 5 (b) 30 (c) 45 (d) 60 (e) 75. tan θ + cot θ (a) cotθ (b) cscθ (c) cotθ (d) cotθ (e) cscθ 3. Which of the following is not equivalent to sin40? (a) cos 40 (b) sin0 cos0 (c) cos80 + cos80 (d) (e) All are equivalent to sin40 33

41 4. If sinθ 0 9, θ lies in QIV, find cosθ. (a) (b) 5 (c) (d) 5 (e) None of the above 5. tan x + tan x (a) sin x (b) cos x (c) tan x (d) sin x + (e) cos x 34

42 3-6: Solving Equations with Identities and Algebra One of the main reasons for studying the identities in the last several sections is to use them in solving equations. These trigonometric identities will sometimes occur in the midst of a calculus question, and they will often be mixed in with factoring or other algebraic operations. Much of the carryover from Trigonometry to Calculus is in the process of solving equations. Therefore, this process will be addressed specifically. Since so many trigonometric identities are used to solve equations involving trigonometric functions, THEY MUST BE MEMORIZED. The Reciprocal Identities: cscθ sinθ secθ cosθ cotθ tanθ The Quotient Identities: tanθ sinθ cosθ tanθ secθ cscθ cotθ cosθ sinθ cotθ cscθ secθ The Pythagorean Identities: sin θ + cos θ cos θ sin θ sin θ cos θ tan θ + sec θ sec θ tan θ sec θ tan θ + cot θ csc θ csc θ cot θ csc θ cot θ The Composite Argument Identities: cos A B cos Acos B + sin Asin B cos( A+ B) cos Acos B sin Asin B sin( A B) sin Acos B cos Asin B sin( A+ B) sin Acos B + cos Asin B tan A tan B tan( A B) tan( A+ B) + tan Atan B tan A+ tan B tan Atan B 35

43 Even Functions: cos( A) cos A sec( A) sec A Odd Functions: csc A tan A cot( A) cot A sin( A) sin A csc A tan A The Double Angle Argument Identities: sin Acos A sin A cos A sin A sin A cos A tan A cos A tan A tan A The Half Angle Argument Identities: sin A ± ( cos A ) cos A ± + cos A tan A ± cos A + cos A sin A + cos A cos A sin A LEARNING OUTCOME Solve equations involving the trigonometric identities. 36

44 EX Solve 4sin x 3 exactly for x 0, π This is a fairly straightforward isolate the variable problem. Be careful when square rooting both sides of the equation. 4sin x 3. sin x 3 4 sin x ± 3 sin x 3 x π 3 ± π n π 3 ± π n sin x 3 x 4π 3 ± π n 5π 3 ± π n Now, choose the particular answers that occur in the interval stated: x π 3, π 3, 4π 3, or 5π 3 37

45 EX Solve sec x sec x exactly for x 0, π ). sec x sec x sec x sec x 0 sec x ( sec x +) 0 sec x 0 sec x cos x x π 3 ± πn π 3 ± πn sec x + 0 sec x cosx x π ± π n So, x π 3, 5π 3, or π EX 3 Solve sec x sec x 3 exactly for x 0, π ). sec x sec x 3 sec x sec x 3 0 sec x 3 ( sec x +) 0 sec x 3 0 sec x 3 cosx 3 x ±.3± π n sec x + 0 sec x cosx x π ± π n So, x.3, 5.05, or π 38

46 As with proofs in Geometry or with trigonometric identities, there is no algorithmic approach to solving trigonometric equations. Each problem is different and only experience can really guide the approach to a particular problem. However, there is a helpful strategy sequence. Ask these questions:. Are the arguments different?. Are the trig functions different? 3. Is there any algebra to perform? Strategies for Solving Trigonometric Equations: I. If the arguments are different, make them the same. Usually use: Composite Rules Double or Half Angle Rules II. If the arguments are the same but the trigonometric functions are different, make them the same, by using: Reciprocal or Quotient Rules Pythagorean Identities tan x formulas Double Angle Argument Formulas III. Do the algebra: Set equation equal to 0 and factor (Remember: Do NOT divide by a variable!!!) Find common denominators (then recheck for Pythagorean identities) IV. Get to a place where there is one trigonometric function to inverse away, thus isolating the variable. As with proofs in Geometry or with trigonometric identities, there is no shortcut to understanding. Most people need a great deal of experience with these before enlightenment occurs. 39

47 EX 4 Solve cos3θ cos sin3θ sin exactly for θ 0, 360 ). The arguments in this equation are different from one another, so they must be made the same. Since there are no triple angle rules, there must be something else. Four trigonometric functions in these pairs look like a composite rule. Do some minor rearranging to see it more clearly. cos3θ cos sin3θ sin cos3θ cos sin3θ sin 0 cos 3θ + 0 3θ + 3θ θ 90 ± 360n 90 ± 360n 78 ± 360n 0 ± 360n 6 ±0n 34 ±0n θ 6, 86, 46, 06, 66 or 36 EX 5 Solve sin x + cos x 0 exactly for x 0, 360 ). This cannot be factored like EX because the trigonometric functions are different. But the Pythagorean identities can be used to make them the same. sin x + cos x 0 cos x + cos x 0 cos x + cos x 0 cos x cos x + 0 cos x 0 x ±90 ± 360n cos x + 0 cos x x 0 ± 360n x 0, 90 or 70 40

48 The interval will not always be a standard x 0, π ). EX 6 Solve cos x sin x exactly for x π, π ). cos x sin x cosx x ± π 4 ± πn x ± π 8 ± πn x π 8, 7π 8, π 8, or 7π 8 4

49 3-6 Free Response Homework Solve the following equations in the given interval. Use exact values when possible.. cos θ sin θ 3 in θ 0, sin x cos x 0 in x π, π 3. sin3θ cos cos3θ sin in θ 0, sec x π 4 + sec x π 4 in x ( 0, π 5. cos x 3cosx in x ( π, π ) 6. cos4θ cosθ 0 in θ 0, sin xcos x 3 in x 0, π 8. sin x cos x 3 in x 0, π 9. cosθ cos0 sinθ sin0 in θ 0, cos x 3 + sin x in x 80, 0. tan 4 x 4tan x in x π 4, π 4. sin 3t sin 3 tcos 3 t 0 in t 0, 360 ) 4

50 3. 3 3sin x cos x 0 in x { All Reals} 4. cot x 5csc x on x π, π ) 5. tan 4 x sec 4 x 3 in x 45, 0 6. tan x sec x sec x 0 in x π, π 7. sin A sin( 3A+ π ) cos A in A ( π, 3π ) 8. sin( x 30 )cos( x 30 ) 3 in x 0, sec ( x 4) + sin( 3x) + tan ( x 4) in x π, π 0. cosxsin x + sinxcosx sin3x in x 0, π ). cscx + sin xcosx 6 in x 0, π. cos x sin x 3 in x ( 0, π ) 3. tan x tan x + cos x sin x sin xcosx in x ( 0, π ) 4. cosx in x 0, π 5. 4tan x + 3 tan x in x ( 0, π 6. sin3acos π + cos3asinπ + tan a sec a in a π, π 43

51 7. ( cos x sin x) sin xcosx 0 in x π, π ) 8. sin x tan x sin x 0 on x π, π ) 9. csc θ +5cotθ 0 in θ 80, sin θ cos θ in θ ( 90, 70 ) 3. tan3x + tan x tan3x tan x in x 0, π 3. cos x sin x in x { All Reals} 33. sinθ sinθ in θ ( 90, 70 ) 34. tan θ 3secθ 0 in θ 80, sec 3x π 4 + sec 3x π 4 in x ( 0, π 36. cosx sin4x 0 in x ( π, π ) 37. cos x cosx in x ( π, π ) 44

52 3-6 Multiple Choice Homework. If sin x 3cosx and 0 x π, then x (a) 0.5 (b) 0.5 (c) 0.49 (d) 0.39 (e) If 4 sin x on 0 x π, then x (a) (b) (c) or (d) or (e) Solve the equation sin5x + cos5x 0. What is the sum of the three smallest positive solutions? (a) (b) (c) (d) (e) π 0 π 3 7π 0 π 0 π 4 45

53 4. For what positive value(s) x 80 of does tan x cot x? (a) 54.7 (b) 5 and 55 (c) 7.4 and 7.4 (d) 7.4, 6.6, 7.4, and 5.6 (e) None of the above 5. For all positive angles less than 360, if csc x + 30 sum of x and y is (a) 85 (b) 5 (c) 30 (d) 5 (e) 95 cos 3y 5, the 46

54 Trigonometric Identities Practice Group Test Page Find EXACT values (no decimals). Show all work. For problems #-6, ( 3, 4) is on the terminal side of A and 70 A 360, csc B 3 5 and 90 B 80, and tanc 7 4 and 80 C 90. Find the EXACT value of:. sin( A B). cosa 3. tan C 4. csc( A+ B) 5. tanb 6. cos A 47

55 7. Prove: tan x + cot x csc xsec x 8. Prove: + tan x tan x secx 9. Solve EXACTLY: sin x sin xcos x for x π, π ) 0. Solve EXACTLY: cos x sin x sin 3x + π for x 0, π ) 48

56 Trigonometric Identities Practice Group Test Page Find EXACT values (no decimals). Show all work. For problems #-6, ( 3, 4) is on the terminal side of A and 70 A 360, csc B 3 5 and 90 B 80, and tanc 7 4 and 80 C 90. Find the EXACT value of:. cos( A B). sina 3. cot C 4. sec( A+ B) 5. cotb 6. sin A 49

57 7. Prove: tan x + tan x sin x 8. Prove: cosβ tan β + tan β 9. Solve EXACTLY: sin x sin xcos x for x 3π, π ) 0. Solve EXACTLY: cos x sin x sin 3x + π for x π, 0 ) 50

58 Trigonometric Identities Practice Group Test Page 3 Find EXACT values (no decimals). Show all work. For problems #-6, ( 3, 4) is on the terminal side of A and 70 A 360, csc B 3 5 and 90 B 80, and tanc 7 4 and 80 C 90. Find the EXACT value of:. sin( A+ B). seca 3. tan B 4. csc( A B) 5. tana 6. cos C 5

59 7. Prove: tan x + cot x csc x 8. Prove: ( + tan ) ( tan ) tan tan 9. Solve EXACTLY: sin x sin xcos x for x π, 0) 0. Solve EXACTLY: cos x sin x sin 3x + π for x 3π, π ) 5

60 Trigonometric Identities Practice Group Test Page 4 Find EXACT values (no decimals). Show all work. For problems #-6, ( 3, 4) is on the terminal side of A and 70 A 360, csc B 3 5 and 90 B 80, and tanc 7 4 and 80 C 90. Find the EXACT value of:. cos( A+ B). csca 3. cot B 4. sec( A B) 5. cota 6. sin C 53

61 7. Prove: sin A+ B + sin( A B) sin( A B) sin A+ B tan Acot B 8. Prove: cos a + b cosb + sin(a + b)sinb cosa 9. Solve EXACTLY: sin x sin xcos x for x 0, π ) 0. Solve EXACTLY: cos x sin x sin 3x + π for x π, π ) 54

62 3- Free Response Homework. cos x + tan xcos x cos x + cos x sin x cos x cos x + sin x. cosθ tanθ + sinθ tanθ cosθ tanθ + sinθ tanθ tanθ cosθ + sinθ tanθ cosθ + sinθ sinθ cosθ cosθ + sinθ cosθ sinθ cosθ + cosθ cosθ Trigonometric Identities Homework Answer Key ( tanσ cotσ ) 4 + tan σ tanσ cotσ + cot σ 4 + tan σ ( ) + cot σ + tan σ + cot σ + sec σ + csc σ sec σ + csc σ ( cosθ ) cosθ ( sinθ )( sinθ ) cos θ cos θ + sin θ cos θ cos θ sec θ ( sin x) sin x ( cos x) ( cos x) sin x sin x cos x sin x sin x + cos x sin x csc x ( 6. sinθ ) sin x cos θ sinθ cosθ cosθ tanθ secθ 7. cosϕ sin ϕ cosϕ sinϕ sinϕ cscϕ cotϕ 55

63 8. ( cosa + sina) ( cosa + sina) cosa cos a + sinacosa + sin a cosa cos a + sin a + sinacosa cosa + sinacosa cosa sinacosa cosa sina 9. tanw ( cotw cosw + sinw ) cosw + tanw sinw cosw + sinw cosw sinw 0. cos W cosw + sin W cosw cosw secw sec x sin x sin x cos x cos x cos x sin x cos xsin x cos x cos xsin x cos x sin x cot x sec x sin x sin x cos x sin x sin x.. sin x cos x + cos x sin x sin x sin x sin x cos x ( cos x) sin x + cos x cos x sin x + cos x + cos x sin x cos x + cos x sin x cos x cos x sin x cos x cos x sin x cos x sin x csc x cotθ cscθ tanθ sinθ csc θ cscθ sinθ cscθ sinθ cscθ sin θ cos θ 3. tanb + cotb sinb cosb + cosb sinb sinb sinb sinb cosb + cosb cosb sinb cosb sin bcosb cscb cosb 56

64 4. cot xcsc x + cot x cos x sin x sin x + cos x sin x cos x + cos x cos x cos x( + cos x) cos x ( + cos x) cos x cos x 5. sec x + sec x tan x cos x + sin x cos x cos x + sin x cos x cos x + sin x ( sin x) sin x + sin x ( sin x) sin x ( + sin x) + sin x + sin x sin x ( + sin x) ( + sin x) ( + sin x) ( sin x) ( + sin x) + sin x sin x 6. Answers vary 3- Multiple Choice Homework. D. A 3. C 4. B 5. A 6. B 3- Free Response Homework ( x y). x 4y. xy( 3x y) ( x + 4y) 3. x( x ) ( x + 3) 4. ( x + y ) ( x y) 5. ( x ) ( x +) ( x ) k r k + r x x xy + y ( x + y) 9. sin x + cot xsin x sin x + cot x sin x csc x 0. sec 4 β tan 4 β ( sec β + tan β ) sec β tan β sec β + tan β + tan β + tan β + tan β 56

65 .. cos w + cosw + cos w 3cosw 4 ( cosw +) ( cosw + ) ( cosw +) ( cosw 4) cosw + cosw 4 cos 3 A sec 3 A cos A sec A cos A sec A cos A sec A cos A++ sec A ( sin A)++ ( + tan A) sin A+ tan A cos A+ cos Asec A+ sec A sin w + 5cosw + 6sin w + 5cosw ( cos w)+ 5cosw + 6( cos w)+ 5cosw cos w + 5cosw + 6 6cos w + 5cosw cos w + 5cosw + 3 6cos w + 5cosw + 4 cos w 5cosw 3 6cos w 5cosw 4 ( cosw +) ( cosw 3) ( cosw +) ( 3cosw 4) cosw 3 3cosw 4 3. tan w secw 5 tan w + 3secw + 3 sec w sec w secw 5 + 3secw + 3 sec w secw 6 sec w + 3secw + secw + secw + ( secw 3) ( secw +) secw 3 secw sec t 8tant + sec t tant 3 3 ( tan t +) 8tant + ( tan t +) tant 3 3tan t + 3 8tant + tan t + tant 3 3tan t 8tant + 4 tan t tant ( 3tant )( tant ) ( tant +) ( tant ) 3tant tant + 56

66 csc y 7cot y 6 6csc y 5cot y 0 ( cot y +) 7cot y 6 6( cot y +) 5cot y 0 cot y + 7cot y 6 6cot y + 6 5cot y 0 cot y 7cot y 4 6cot y 5cot y 4 ( cot y +) cot y 4 cot y + ( 3cot y 4) cot y 4 3cot y 4 3 x x + x x x + x 3 x r + rx + x r rx + x. x tan 3 B tan B + sec B + tan B tan B + ( + tan B) tan B + tan B + tan B ( + tan B) cot B + cosx cot B + cot B cot B + sin 3 B sin B + sin B + sin B ( sin B + sin B) ( sin B) sin B sin B + sin B ( 8 + cos 3 x) cosx ( i cosx )( cosx ) ( cosx ) ( cos x cosx + 4) i cos x 4cosx + 4 cos x cosx + 4 cos x cosx + 4 ( cosx ) + cosx cos x 4 6. sec x + 3sec x 0 ( ( sec x 7sec x + 6) i sec x + sec x 3) ( sec x + sec x 6) ( sec x ) ( sec x ) sec x + 5 sec x 6 sec x + 5 sec x 6 ( sec x + 3) ( sec x ) sec x i sec x

67 7. cot x 4csc x cot x 3 4csc x csc x csc x 3 csc x 4csc x csc x 4 ( csc x + ) csc x 6 csc x + ( csc x ) csc x 6 csc x 8. csc 6 x cot 6 x 3 ( cot x) 3 ( csc 4 x + csc xcot x + cot 4 x) csc x csc x cot x csc 4 x + csc xcot x + cot 4 x + csc xcot x + cot x( csc x ) csc x + cot x csc x cot x + 3csc x cot x + 3csc x cot x sin 3 x sin 4 x 8 6 sin x sin 3 x ( ( sin x 9) ( sin x + 9) i sin x + 9) 6 ( 3 sin x) ( 9 + 3sin x + sin x) ( ( sin x 3) ( sin x + 3) ( sin x + 9) i sin x + 9) sin x + sin x 3 sin x

68 30. ( sin x + sin xcosx) sin x + cosx sin x sin xcosx sin x cosx cos3 x + sin 3 x cos 3 x sin 3 x sin x sin x + cosx sin x sin x cosx i sin x cosx sin x + cosx cos x + sin xcosx + sin x cos x sin xcosx + sin x cos x + sin xcosx + sin x cos x sin xcosx + sin x cosx sin x i cosx + sin x 3- Multiple Choice. D. A 3. E 4. C 5. B 3-3 Free Response Homework a. b. c. d. e. f a. b. c. d. e. f θ 4 ±0n 54 ±0n 4. x { 60, 80 } 5. φ 5 ±80n 56

69 6. x π 6, 5π 6, 9π 6, 3π 6, 3π 6, 7π 6, π 7. x π, 5π 8. A and B { All Reals} 6, 5π 6. sin Acos B + cos Asin B cos Acos B sin Asin B sin( A+ B) cos A+ B tan A+ B tan A+ tan B tan Atan B 9. sin( A+ B)+ sin A B sin( A+ B) sin A B sin Acos B + cos Asin B + sin Acos B cos Asin B sin Acos B + cos Asin B sin Acos B cos Asin B sin Acos B cos Asin B tan Acot B 0. cos( a + b)cosb+ sin( a + b)sinb ( cosacosb sinasinb)cosb+ ( sinacosb+ cosasinb)sinb cosacos b sinasinbcosb+ sinacosbsinb+ cosasin b cosacos b+ cosasin b. cosa cos b+ sin b cosa sin Acos A+ cos Asin A cos Acos A sin Asin A sin( A+ A) cos A+ A tan A+ A tan A+ tan A tan Atan A tan A tan A 3. cot Acot B cot A+ cot B tan Atan B cot Acot B tan Atan B cot A+ cot B tan Atan B tan B + tan A tan A+ B cot A+ B 3-3 Multiple Choice Homework. B. C 3. D 4. E 5. A 3-4 Free Response Homework a b c. 7 d

70 e. f θ π 8 ± πn 3π 8 ± πn 3. x π, 3π 4. x 0 ± 90n 75 ± 90n π 5. x ± πn 5π ± πn 6. x 7. x π ± π n 8. π 8, 5π 8, 9π 8, 3π 8, 3π 8, 7π 8, π 8, 5π 8 tan B + tan B tan B sec B sec B tan B sec B cos B sin B cosb 9. cos 4 β sin 4 β ( cos β + sin β) cos β sin β cos β 0. cot A+ tan A cos A sin A + sin A cos A cos A+ sin A sin Acos A sin Acos A sin Acos A csc A. cosφ cot φ + cosφ. ( csc φ) cosφ cot φ + cos φ sin φ cos φcsc φ cos φ sin φ cot φ tan x tan x cotx + cosx sinx cot x 4cot x 4 csc x 4csc x 4 + cos x sin x sin xcos x 56

71 3. ( + tanδ )tanδ tan δ tanδ ( + tanδ ) ( tanδ )( + tanδ ) + tanδ tanδ tanδ tanδ 3-4 Multiple Choice Homework. D. E 3. D 4. E 5. C 6. B 3-5 Free Response Homework a. sin x 0 b. cos x 3 0 c. tan x 3 a. sin x 3 0 b. cos x 0 c. tan x 3 3a. sin x 7 5 b. cos x 5 c. tan x 7 4a. csc x 5 3 b. sec x 5 4 c. cot x 4 3 5a. csc x 0 b. sec x 0 3 c. cot x 3 π 6. x 3 ± 4πn 5π 3 ± 4πn 7. x 8π 3, 0π 3 8. x π 3, 0π 3 9. x ±π ± 4πn 56

72 0. x π ± πn. x π 3, 4π 3. tan x + cot x cos x sin x sin x csc x 3. tan x tan x + cos x + sin x sin x cos x cos x sin x cos x cos x sec x 4. tan x + tan x tan x sec x sin x cos cos x x sin x cos x sin x sin x 5. csc x cot x sin x cos x sin x cos x sin x tan x 6. sin x + cos x sin x + sin xcos x + cos x sin x + cos x + sin x + sin x 56

73 56 7. cos x + sin x cos x sin x cos x + sin x cos x sin x cos x + sin x cos x + sin x cos x + sin x cos x + sin x cos x sin x + sin x cos x + sin x cosx cosx + sin x cosx sec x + tan x 8. cos x sin x cos x + sin x cos x sin x cos x + sin x cos x + sin x cos x + sin x cos x sin x cos x sin x cos x + sin x cos x + sin x cosx + sin x 3-5 Multiple Choice Homework. D. E 3. D 4. A 5. C 3-6 Free Response Homework. θ 7.5, 8.5, 97.5, 7.5, 87.5, 6.5, 77.5, 35.5

74 3π. x, π, 7π 6, π 6, π 6, 5π 6 3. θ 4, 54, 34, 74, 54, x π, 9π 5. x ± π 3 6. x 30, 90, 50, 0, 70, x π 6, π 3, 7π 6, 4π 3 8. x 5π, 7π, 7π, 9π 9. θ { 5, 95 } 0. θ { 65, 5 }. x π 4. t { 90, 0, 330 } 3. x 4. x 5. x { 45 } π 6 ± πn 5π 6 ± πn π ± πn 5.553, 3.87, 0.730,.4, 3π, π 6. x { ±4.373, ±.9} 9π 7. A 8, 3π 8, 3π 8, 35π 8, 43π 8, 47π 8 8. x 9. x ± π 3, 0 0, 75, 90, 65, 80, 55, 70, 345, 360, 435, 450, 55, 540, 65, 630, 705, 70 7π 0. x 8, π 8, 9π 8, 3π 8, 3π 8, 35π 8. x π, 5π 56

75 . x π 3 3. x { 0.554,.5, 3.696, 5.66} 4. x π, 3π 5. x 5π 6, π 6 6. a π 3 ±5π 7. x 8, ±3π 8, ±π 8, ±9π 8, ±7π 8, ±5π 8, ±3π 8, ±π 8 8. x π, π ) 9. θ , 6.565, 6.55, θ { ±67.5,.5, 47.5 } 3π 3. x 6, 7π 6, π 6, 5π 6, 9π 6, 3π 6, 7π 6, 3π 6 3. A π ± π n 33. θ { ±60, 0, 80 } 34. θ { ±60 } π 35. x 36, 35π 36, 59π 36, 9π 36, 43π 36, 67π x 37. No solutions π, 7π, π, 5π, ± π 4, ± 3π Multiple Choice Homework. C. C 3. C 4. D 5. A 56

76 Trigonometric Identities Practice Test Answer Key Page tan x + cot x 8. + tan x tan x sin x cos cos + x x sin x sin x + cos x sin xcos x sin xcos x csc xsec x 9. x 3π 4, π 4 0. x 7π 8, π 8, 9π 8, 3π 8, 3π 8, 35π 8 + sin x cos x sin x cos x + sin x cos x sin x cos x i cos x cos x cos x + sin x cos x sin x cos x sin x cosx secx 56

77 Page tan x + tan x tan x sec x sin x cos xsec x sin x cos xsec xsec x sin x sec x sin xcos x sin i x sin x 8. tan β + tan β sin β cos β + sin β cos β cos β sin β cos β cos β + sin β cos β cos β sin β cos β i cos β cos β + sin β cos β cos β cos β sin β cosβ 9. x 5π 4, 9π 4 π 0. x 8, 5π 8, 3π 8, 7π 8, 5π 8, 9π 8 56

78 Page tan x + cot x sin x + cos x + + cos x sin x ( sin x )( sin x) + + cos x sin x + cos x ( + cos x) sin x + + cos x + cos x sin x + cos x + cos x sin x + cos x + cos x sin x + cos x sin x csc x 8. ( + tanδ) ( tanδ) + tanδ tanδ tan δ tanδ( + tanδ) ( tanδ) ( + tanδ) tanδ tanδ 9. x 5π 4, π 4 5π 0. x 8, 9π 8, 37π 8, 4π 8, 49π 8, 53π 8 56

79 Page sin A + B 5. + sin( A B) sin( A B) sin A + B sin Acos B + cos Asin B + sin Acos B cos Asin B sin Acos B + cos Asin B sin Acos B + cos Asin B sin Acos B cos Asin B sin A cos A i cos B sin B tan Acot B 8. cos( a + b)cosb + sin( a + b)sin b cosb cosacosb sin asinb 5 + sinb sin acosb + cosasinb cosacos b sin asinbcosb + sin asinbcosb + cosasin b cosa cos b + sin b cosa 9. x 3π 4, 7π 4 0. x 7π 8, π 8, π 8, 5π 8, 3π 8, 7π 8 56