Math141HWAnswers4

Math141HWAnswers4 - (Answers for Chapter 5 Analytic Trigonometry A.5.1 CHAPTER 5 Analytic Trigonometry SECTION 5.1 FUNDAMENTAL TRIGONOMETRIC

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Unformatted text preview: (Answers for Chapter 5: Analytic Trigonometry) A.5.1 CHAPTER 5: Analytic Trigonometry SECTION 5.1: FUNDAMENTAL TRIGONOMETRIC IDENTITIES 1) Left Side Right Side Type of ID csc x 1 sin x Reciprocal ID tan x 1 cot x Reciprocal ID tan x sin x cos x Quotient ID cot x Cofunction ID tan 2 x cos x sin 2 x Cofunction ID ( x) sin x Even / Odd (Negative-Angle) ID cos ( x) cos x Even / Odd (Negative-Angle) ID tan ( x) tan x Even / Odd (Negative-Angle) ID sin sin 2 x + cos 2 x 1 Pythagorean ID tan 2 x + 1 sec 2 x Pythagorean ID 1 + cot 2 x csc 2 x Pythagorean ID (Answers for Chapter 5: Analytic Trigonometry) A.5.2 2) a) sec x , b) sec 2 , c) 1, d) csc 4 x , e) sin t , f) sin 3) a) 4 cos , b) 6 sec , c) 3 tan SECTION 5.2: VERIFYING TRIGONOMETRIC IDENTITIES 1) Solutions will vary. SECTION 5.3: SOLVING TRIGONOMETRIC EQUATIONS 1) a) All real solutions: Solutions in [ 0, 2 x= x ): 3 =± ): ( Solutions in [ 0, 2 e) All real solutions: Solutions in [ 0, 2 u= 3 2 = ( ) 5 +2 n n 4 3 22 , . ( ) 3 +2 n n 2 . u= u ): . , or, equivalently, 35 , 44 u ): ) 3 +2 n n 4 c) No real solutions; the solution set is No real solutions in [ 0, 2 ) . d) All real solutions: ) 2 33 3 + 2 n or 4 Solutions in [ 0, 2 ( 2 +2 n n 3 , b) All real solutions: = + 2 n or x = 2 . ( +nn ) . . . (Answers for Chapter 5: Analytic Trigonometry) A.5.3 Solutions in [ 0, 2 g) All real solutions: x= 3 Solutions in [ 0, 2 ( u= ): 7 11 , 6 6 + 2 n or x = ): ( Solutions in [ 0, 2 x= 7 66 , 3 22 , 6 Solutions in [ 0, 2 6 2 5 +2 n n 3 , or, equivalently, . ( ) . ( ) . +nn +nn 6 ( +nn 5 + n (n 6 5 7 11 . , , , 666 6 + n or ): ) +2 n n . =± k) All real solutions: = . . = ): ) 5 . 33 j) All real solutions: Solutions in [ 0, 2 ( +2 n n , x ): ) ( 3 h) No real solutions; the solution set is No real solutions in [ 0, 2 ) . i) All real solutions: 6 . x=± x ) 7 11 + 2 n or u = +2 n n 6 6 7 + 2 n or u = 6 u equivalently, x u= u f) All real solutions: = ) , or, equivalently, ) . , or, (Answers for Chapter 5: Analytic Trigonometry) A.5.4 = n or l) All real solutions: = n or Solutions in [ 0, 2 ): = 0, 4 = ( ( ) +nn 3 7 ,, 4 4 ) 3 +nn 4 , or, equivalently, . . m) All real solutions: x x= 6 Solutions in [ 0, 2 n) All real solutions: + 2 n or x = ): 2 5 62 6 , , x= x + 2 n or x = ( ) 5 +2 n n 6 . . + n or x = 2 ( ) 2n n 3 , where rotational symmetry is exploited. A less efficient way of writing the solution set would be: 2 x x = + n or x = 2 n or x = ± +2 n n . 2 3 ( Solutions in [ 0, 2 o) All real solutions: ): 0, 243 , , 2332 , x=± x be more useful for later: Solutions in [ 0, 2 p) All real solutions: Solutions in [ 0, 2 ): ( ) n n 2 x= x . 12 + . The following form may ( n n 5 or x = + n 12 2 2 5 7 11 13 17 19 23 , , , , , , 12 12 12 12 12 12 12 12 , x= x ): 12 + 6 53 , 662 , + ( 2n n 3 . ) ) . . ) . (Answers for Chapter 5: Analytic Trigonometry) A.5.5 x=± x q) All real solutions: 9 x= x be more useful for later: ( n n 3 + 9 + ) . The following form may ( n n 2 or x = + n 9 3 3 Solutions in [ 0, 2 ) : 2 4 5 7 8 10 11 13 14 16 17 , , , , , , , , , , , 999999 9 9 9 9 9 9 2) a) Solutions in [ 0, 2 {tan 1 ) : {arctan 2, ) . . + arctan 2} , or, equivalently, } + tan 1 2 . 2, b) Approximately: {1.107, 4.249} . (Make sure your calculator is in radian mode.) c) {x {x )} , or, equivalently, n (n )} . ( x = arctan 2 + n n x = tan 1 2 + 3) a) Solutions in [ 0, 2 cos arccos cos 1 1 ): arccos 1 , 5 + cos 1 ,2 5 1 5 1 cos 1 5 1 5 , or, equivalently, , or, equivalently, 1 5 1 + arccos , or, equivalently, arccos 1 ,2 5 1 , 5 . b) Approximately: {1.772, 4.511} . (Make sure your calculator is in radian mode.) c) ( ) 1 +2 n n 5 x = ± arccos x x x = ± cos x x = ± arccos 1 ( 1 +2 n n 5 ( , or, equivalently, ) ) (n 1 + 2n + 1 5 , or, equivalently, ) . (Answers for Chapter 5: Analytic Trigonometry) A.5.6 SECTIONS 5.4 and 5.5: MORE TRIGONOMETRIC IDENTITIES 1) Left Side Right Side Type of ID sin u + v ( ) sin u cos v + cos u sin v Sum ID ( ) cos u cos v sin u sin v Sum ID ( ) tan u + tan v 1 tan u tan v Sum ID sin u v ( ) sin u cos v cos u sin v Difference ID ( ) cos u cos v + sin u sin v Difference ID ( ) tan u tan v 1 + tan u tan v Difference ID () 2 sin u cos u Double-Angle ID cos 2u () cos 2 u sin 2 u , 1 2 sin 2 u , and 2 cos 2 u 1 Double-Angle ID (write all three versions) tan ( 2u ) 2 tan u 1 tan 2 u Double-Angle ID cos u + v tan u + v cos u v tan u v sin 2u sin 2 u cos 2 u sin cos () or 1 2 () or 11 + cos 2u 22 1 cos 2u 2 1 + cos 2u 2 () Power-Reducing ID (PRI) () Power-Reducing ID (PRI) 1 cos 2u 2 1 cos 2 (Choose the sign appropriately.) Half-Angle ID 1 + cos 2 (Choose the sign appropriately.) Half-Angle ID ± 2 ± 2 (Answers for Chapter 5: Analytic Trigonometry) A.5.7 2) 2+ 6 4 a) 6 b) 4 3 + 2 . (Remember to rationalize the denominator in c) 2 3) 3+3 .) 3 3 2 2 2+ 2 2 4) 5) a) 1 3 2 3 , b) , c) , d) 2 2 2 2 6) cos ( 2 7) 2 ) tan ( 4 x ) 6 8) a) Hint: Use a Sum Identity. b) Hints: Use a Double-Angle Identity and a Pythagorean Identity. c) Hints: Use the Sum Identities for sine and cosine, and then divide the numerator and the denominator by cos u cos v . 9) a) All real solutions: Solutions in [ 0, 2 x= x ): 12 + n or x = 5 13 17 , , 12 12 12 12 , ( 5 +nn 12 ) . (Answers for Chapter 5: Analytic Trigonometry) A.5.8 x= x equivalently, x=± x b) All real solutions: Solutions in [ 0, 2 ): Solutions in [ 0, 2 ) , or, ( x = n or x = ± x = n or x = 0, 1 2 cos ( 2 x ) + 3 ( ) +2 n n + 2 n or x = , or, ( 5 +2 n n 3 5 3 ): ,, 3 3 10) 2 x 1 x 2 3 8 11) cos 4 x = + 1 8 cos ( 4 x ) 12) a) 1 cos ( 2 2 b) 2 cos ( 4 ) + cos ( 8 ) , which is simplified from 1 cos ( 2 2 ) + cos ( 8 ) ) cos c) 2 sin ( 2 x ) cos x d) 1 sin (19 2 e) 1 cos 3x 2 f) g) h) ) () sin ()() 2 cos (5 ) sin ( 3 ) 2 sin 4 x sin 3x () 1 sin 9 2 , which is simplified from () cos 5x sin ) +2 n n 2 4 ,, 3 3 x equivalently, ( +2 n n 2 4 + 2 n or x = + 2 n or x = 3 3 x c) All real solutions: 2 + 2 n or x = 3 1 sin (19 2 ) + sin ( ) ) . . (Answers for Chapter 6: Additional Topics in Trigonometry) A.6.1 CHAPTER 6: Additional Topics in Trigonometry SECTION 6.1: THE LAW OF SINES 1) a) 35.0 m, b) 22.0 m, c) 372 m 2 2) a) 180.09 ft, b) 224.86 ft, c) 20,137 ft 2 SECTION 6.2: THE LAW OF COSINES 1) a) 25.8 , b) 140.2 , c) No (that would violate the Triangle Inequality), d) 496 ft 2 2) 13.8 mi (Answers for Chapter 6: Additional Topics in Trigonometry) A.6.2 SECTION 6.3: VECTORS IN THE PLANE 1) a) 2, 3 or 2 m, 3 m , b) 13 m, c) 56.3 2) a) 5, 3 or 5 m, 3 m , b) 34 m, c) 210.96 3) a) v 3v b) 4, 3 1 v 3 (Answers for Chapter 6: Additional Topics in Trigonometry) A.6.3 c) d) e) 5, 5 4) 8.0 ft, 8.9 ft 5) a) 2 29 5 29 , , b) 111.8 , c) 29 29 6) a) 327.53 , b) 22 170 , 17 14 170 17 7) Yes 8) No (they point in opposite directions) 9) a) 20.3 mph, b) 18.3 mph 8 29 20 29 , 29 29 (Answers for Chapter 6: Additional Topics in Trigonometry) A.6.4 SECTION 6.4: VECTORS AND DOT PRODUCTS 1) 14 2) a) scalar, b) vector, c) undefined, d) scalar, e) undefined, f) undefined 3) 10 4) Hint: v + w 5) v 2 +w 2 = (v + w) • (v + w) . 2 6) The Pythagorean Theorem 7) 19.7 ; acute 8) 167.7 ; obtuse 9) 47.7 . Hint: Find the angle between the vectors BA and BC . 10) a) 0 , b) 180 , c) 90 11) Yes 12) No 13) 0 and 1 14) Hint: Use the formula: cos = 15) 14 17 17 v•w . vw ...
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.

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