Math141HWAnswers4

Math141HWAnswers4 - (Answers for Chapter 5: Analytic...

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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 Reciprocal ID Reciprocal ID Quotient ID Cofunction ID csc x tan x tan x 1 sin x 1 cot x sin x cos x x cot x tan 2 cos x sin sin 2 sin x x Cofunction ID Even / Odd (Negative-Angle) ID Even / Odd (Negative-Angle) ID Even / Odd (Negative-Angle) ID Pythagorean ID Pythagorean ID Pythagorean ID ( x) ( x) ( x) cos cos x tan x 1 tan sin 2 x + cos 2 x tan 2 x + 1 1 + cot 2 x sec 2 x csc 2 x (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) 3tan 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 b) All real solutions: x x= 3 + 2 n or x = 2 +2 n n 3 ( ) . ): 2 3 3 , = 3 +2 n n 4 5 +2 n n 4 ( ) ) , or, equivalently, . = 3 + 2 n or 4 = ( Solutions in [ 0, 2 ): 3 5 , 4 4 . c) No real solutions; the solution set is No real solutions in [ 0, 2 ) . d) All real solutions: Solutions in [ 0, 2 e) All real solutions: Solutions in [ 0, 2 u u= 3 +2 n n 2 ( ) . ): u 3 2 . u= 2 . + n n ( ) . ): 3 2 2 , (Answers for Chapter 5: Analytic Trigonometry) A.5.3 f) All real solutions: equivalently, Solutions in [ 0, 2 u u u= u= 7 11 + 2 n or u = +2 n n 6 6 6 +2 n n ( ) ) . , or, 7 + 2 n or u = 6 . ( ): x 7 11 , 6 6 g) All real solutions: x= 3 +2 n n ( ) ) , or, equivalently, x x= 3 + 2 n or x = 5 +2 n n 3 ( Solutions in [ 0, 2 ): 5 . 3 3 , . h) No real solutions; the solution set is No real solutions in [ 0, 2 ) . i) All real solutions: Solutions in [ 0, 2 x x= 7 6 6 , 6 . + n n ( ) . ): j) All real solutions: Solutions in [ 0, 2 = 2 . + n n ( ) . ): 3 2 2 , k) All real solutions: = + n or = 6 + n n ( ) ) , or, equivalently, . = 6 Solutions in [ 0, 2 ): 5 + n (n 6 5 7 11 . , , , 6 6 6 6 (Answers for Chapter 5: Analytic Trigonometry) A.5.4 l) All real solutions: = n or = + n n . = 3 + n n 4 ( ) , or, equivalently, = n or Solutions in [ 0, 2 m) All real solutions: 4 ( ) . ): 0, 3 7 , , 4 4 x x= 6 + 2 n or x = 2 + 2 n or x = . 5 +2 n n 6 ( ) . Solutions in [ 0, 2 n) All real solutions: ): x 5 6 2 6 , , x= 2 + n or x = 2 n 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: ): x 0, 2 4 3 , , 2 3 3 2 , x= x + . 12 n n 2 12 + ( ) . The following form may be more useful for later: Solutions in [ 0, 2 p) All real solutions: Solutions in [ 0, 2 x= n n 5 or x = + n 12 2 2 . ( ) . ): x 5 7 11 13 17 19 23 , , , , , , 12 12 12 12 12 12 12 12 , x= + 2 n n 3 . 6 ( ) . ): 5 3 , 6 6 2 , (Answers for Chapter 5: Analytic Trigonometry) A.5.5 q) All real solutions: x x= x 9 + n n 3 9 + ( ) . The following form may be more useful for later: x= n n 2 or x = + n 9 3 3 ( ) . Solutions in [ 0, 2 ) : 2 4 5 7 8 10 11 13 14 16 17 , , , , , , , , , , , 9 9 9 9 9 9 9 9 9 9 9 9 2) a) Solutions in [ 0, 2 . {tan ) : {arctan 2, 1 2, + tan 1 2 . } + arctan 2} , or, equivalently, b) Approximately: {1.107, 4.249} . (Make sure your calculator is in radian mode.) c) {x {x x = arctan 2 + n n x = tan 1 2 + )} , or, equivalently, n (n )} . ( 1 , 5 1 5 + arccos 1 5 3) a) Solutions in [ 0, 2 ): arccos 1 , or, equivalently, cos 1 1 , 5 1 , 5 + cos , or, equivalently, arccos arccos + arccos arccos 1 5 1 5 , or, equivalently, . 1 , 2 5 b) Approximately: {1.772, 4.511} . (Make sure your calculator is in radian mode.) c) x x x x = arccos x = cos 1 1 +2 n n 5 1 +2 n n 5 1 + 2n + 1 5 ( ) , or, equivalently, ( ) , or, equivalently, x = arccos ( ) (n ) . (Answers for Chapter 5: Analytic Trigonometry) A.5.6 SECTIONS 5.4 and 5.5: MORE TRIGONOMETRIC IDENTITIES 1) Left Side Right Side sin u cos v + cosu sin v Type of ID Sum ID Sum ID Sum ID Difference ID Difference ID Difference ID Double-Angle ID Double-Angle ID (write all three versions) Double-Angle ID 1 cos 2u 2 sin u + v ( ) ) ) ) ) ) cos u + v tan u + v sin u v ( cosu cos v sin u sin v tan u + tan v 1 tan u tan v ( ( sin u cos v cosu sin v cosu cos v + sin u sin v tan u tan v 1 + tan u tan v cos u v tan u v sin 2u ( ( ( ) ( ) 2 sin u cosu cos 2 u sin 2 u , 1 2sin 2 u , and 2cos 2 u 1 2 tan u 1 tan 2 u 1 cos 2u 2 1 + cos 2u 2 cos 2u tan ( 2u ) sin 2 u cos 2 u ( ) ( ) or 1 2 ( ) ( ) Power-Reducing ID (PRI) Power-Reducing ID (PRI) or 1 1 + cos 2u 2 2 sin 2 1 cos 2 (Choose the sign appropriately.) 1 + cos 2 (Choose the sign appropriately.) Half-Angle ID cos 2 Half-Angle ID (Answers for Chapter 5: Analytic Trigonometry) A.5.7 2) a) b) c) 2+ 6 4 6 4 3 + 2 . (Remember to rationalize the denominator in 2 3+3 .) 3 3 3) 2 2 2 4) 2+ 2 2 5) a) 1 3 2 3 , b) , c) , d) 2 2 2 2 6) cos ( 2 7) 8) ) tan ( 4x ) 6 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 + n n 12 ( ) . ): 5 13 17 , , 12 12 12 12 , (Answers for Chapter 5: Analytic Trigonometry) A.5.8. b) All real solutions: equivalently, x x= x= 2 + 2 n or x = 3 +2 n n ( ) , or, x 2 4 + 2 n or x = + 2 n or x = 3 3 +2 n n ( ) . Solutions in [ 0, 2 c) All real solutions: equivalently, Solutions in [ 0, 2 10) 2x 1 x 2 11) cos 4 x = 12) a) ): x x 2 4 , , 3 3 x = n or x = x = n or x = +2 n n 3 ( ) , or, 3 + 2 n or x = 5 +2 n n 3 ( ) . ): 0, 3 , , 5 3 3 8 + 1 2 cos ( 2x ) + 1 8 cos ( 4x ) 1 cos ( 2 2 ) + cos ( 8 ) ) cos , which is simplified from 1 cos ( 2 2 ) + cos ( 8 ) b) 2 cos ( 4 c) 2 sin ( 2x ) cos x d) e) f) g) h) 1 sin (19 2 1 cos 3x 2 ) sin cos 5x , which is simplified from 1 sin (19 2 ) + sin ( ) ( ) ( ) ( ) ( ) 2cos (5 ) sin ( 3 ) 2sin 4x sin 3x 1 sin 9 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) 3) a) v 5, 3 or 5 m, 3 m , b) 34 m, c) 210.96 1 v 3 3v b) 4, 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 22 170 , 17 14 170 17 8 29 20 29 , 29 29 6) a) 327.53 , b) 7) Yes 8) No (they point in opposite directions) 9) a) 20.3 mph, b) 18.3 mph (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) 2 = (v + w) (v + w) . v 2 + 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 = vw . v w 15) 14 17 17 ...
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