CHAPTER 5: Analytic Trigonometry - Kkuniyuk

(Answers for Chapter 5: Analytic Trigonometry) A.5.1CHAPTER 5:Analytic TrigonometrySECTION 5.1:FUNDAMENTAL TRIGONOMETRIC IDENTITIES1)Left Sidecsc ( x )tan ( x )tan ( x )Right Side1sin ( x )1cot ( x )sin ( x )cos ( x )Type of Identity (ID)Reciprocal IDReciprocal IDQuotient ID π tan x 2 cot ( x )Cofunction IDcos ( x ) π sin x 2 Cofunction ID( )cos ( x )tan ( x ) sin ( x )Even / Odd (Negative-Angle) IDcos ( x )Even / Odd (Negative-Angle) ID tan ( x )Even / Odd (Negative-Angle) ID1Pythagorean IDsec 2 ( x )Pythagorean IDcsc 2 ( x )Pythagorean IDsin xsin 2 ( x ) cos 2 ( x )tan 2 ( x ) 11 cot 2 ( x )()2) a) sec ( x ) ; b) sec 2 (θ ) ; c) 1; d) csc 4 x ; e) sin ( t ) ; f) sin (α )3) a) 4 cos (θ ) ; b) 6sec (θ ) ; c) 3tan (θ )SECTION 5.2: VERIFYING TRIGONOMETRIC IDENTITIES1) Solutions will vary.

(Answers for Chapter 5: Analytic Trigonometry) A.5.2SECTION 5.3: SOLVING TRIGONOMETRIC EQUATIONS1) π2π π 2π 2π n n . In [ 0, 2π ) : ,a) x x 2π n or x .33 3 3 3π 2π n n , or, equivalently,b) θ θ 4 3π5π 3π 5π θ θ 2πnorθ 2πnn . In [ 0, 2π ) : , .44 4 4 c) No real solutions; the solution set is . No real solutions in [ 0, 2π ) .(())() 3π 3π 2π n n . Solutions in [ 0, 2π ) : .d) u u 2 2 π π 3π e) u u π n n . Solutions in [ 0, 2π ) : , .2 2 2 7π11π 2π n or u 2π n n , or, equivalently,f) u u 66 7π 11π 7ππ 2π n or u 2π n n . In [ 0, 2π ) : , u u .666 6 πg) x x 2π n n , or, equivalently,3 π 5π π5π 2π n n . In [ 0, 2π ) : , . x x 2π n or x 33 3 3 ()()()(())()h) No real solutions; the solution set is . No real solutions in [ 0, 2π ) . π 7π πi) x x π n n . Solutions in [ 0, 2π ) : , .6 6 6 π π 3π j) θ θ π n n . Solutions in [ 0, 2π ) : , .2 2 2 πk) θ θ π n n , or, equivalently,6 π5π π 5π 7π 11π π n ( n ) . In [ 0,2π ) : , ,, θ θ π n or θ .66 6 6 6 6 ()()()

(Answers for Chapter 5: Analytic Trigonometry) A.5.3 3π π n n , or, equivalently,l) θ θ π n or θ 4 π7π 3π, π, θ θ π n or θ π n n . In [ 0, 2π ) : 0, .444 ()() ππ5π 2π n n .m) x x 2π n or x 2π n or x 626 π π 5π Solutions in [ 0, 2π ) : , , . 6 2 6 π2π nn , by rotational symmetry. Lessn) x x π n or x 23 π2π 2π n n efficiently: x x π n or x 2π n or x 23 (())( π 2π 4π 3π Solutions in [ 0, 2π ) : 0, ,,, .2332 π πn n . The following form mayo) x x 12 2 ( ) . ) π πn5π π n or x n .be more useful for later: x x 12 212 2 π 5π 7π 11π 13π 17π 19π 23π Solutions in [ 0, 2π ) : ,,,,,,, . 12 12 12 12 12 12 12 12 π 2π n π 5π 3π n . In [ 0, 2π ) : ,p) x x , .63 6 6 2 (()) π πnn . The following form may be more useful forq) x x 93 π πn2π π nor x n . Solutions in [ 0, 2π ) :later: x x 9393 π 2π 4π 5π 7π 8π 10π 11π 13π 14π 16π 17π ,,,,,,,,,, , .999999 9 9 9 9 9 9()(2) a){arctan 2,)π arctan 2} ; equivalently, { tan 1 2, π tan 1 2} .b) Approximately: {1.107, 4.249} . (Make sure your calculator is in radian mode.)c){ x x arctan 2 π n ( n )} , or { x x tan 1()}2 π n n .

(Answers for Chapter 5: Analytic Trigonometry) A.5.43) 1 1 a) Solutions in [ 0, 2π ) : arccos , π arccos . Equivalent forms: 5 5 1 1 1 1 1 1 cos , π cos , π arccos , π arccos , and55 55 1 1 arccos , 2π arccos .55 b) Approximately: {1.772, 4.511} . (Make sure your calculator is in radian mode.) 1 c) x x arccos 2π n n , or, equivalently, 5 1 1 x x cos 2πnn , or, equivalently, 5 1 x x arccos 2n 1 π n . 5 ()(() ())SECTIONS 5.4 and 5.5:MORE TRIGONOMETRIC IDENTITIES1)Left SideRight SideType of Identity (ID)sin u v()sin ( u ) cos ( v ) cos ( u ) sin ( v )Sum ID()cos ( u ) cos ( v ) sin ( u ) sin ( 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()cos u vtan u vcos u vtan u v( )sin 2utan ( u ) tan ( v )1 tan ( u ) tan ( v )tan ( u ) tan ( v )1 tan ( u ) tan ( v )2 sin ( u ) cos ( u )Sum IDDifference IDDouble-Angle ID

(Answers for Chapter 5: Analytic Trigonometry) A.5.5Left SideRight Sidecos ( u ) sin ( u ) , 1 2sin ( u ) , and2( )cos 2u22cos ( u ) 122 tan ( u )tan ( 2u )1 cos ( 2u )21 cos 2u( )cos 2 ( u )2 θ sin 2 or1 1 cos ( 2u )2 2or1 1 cos 2u2 2( ) 1 cos (θ ) 1 cos (θ )2(Choose the sign appropriately.) θ cos 2 2(Choose the sign appropriately.) 1 cos (θ )1 cos (θ ) 1 cos (θ )sin (θ )Type of Identity (ID)Double-Angle ID(write all three versions)Double-Angle ID1 tan 2 ( u )sin 2 ( u ) θ tan 2 2 sin (θ )1 cos (θ )(Choose the sign appropriately.)Power-Reducing ID(PRI)Power-Reducing ID(PRI)Half-Angle IDHalf-Angle IDHalf-Angle ID(write all three versions)2)a)2 6; b)43)2 224)2 226 2; c)43 2 (rationalize the denominator in3 3).3 31323; b) ; c); d)22226) cos ( 2θ )5) a)7)tan ( 4x )68)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 numeratorand the denominator by cos ( u ) cos ( v ) .

(Answers for Chapter 5: Analytic Trigonometry) A.5.6.9) π5π π n or x π n n a) All real solutions: x x 1212 π 5π 13π 17π Solutions in [ 0, 2π ) : ,,, 12 12 12 12 ( ) . 2π 2π n or x π 2π n n , or,b) All real solutions: x x 3 2π4π 2π n or x 2π n or x π 2π n n equivalently, x x 33 ()(4π 2πSolutions in [ 0, 2π ) : , π , 3 3 πc) All real solutions: x x π n or x 2π n n , or,3 π5π 2π n n ,equivalently, x x π n or x 2π n or x 33 π 2π nor, equivalently, x x 2π n or x n .33 ()(())5π πSolutions in [ 0, 2π ) : 0, , π , 3 310) 2x 1 x 211) cos 4 ( x ) 38 1cos ( 2x ) 21cos ( 4x )812)a)11 cos ( 2θ ) cos ( 8θ ) , which is simplified from cos ( 2θ ) cos ( 8θ ) ;22b) 2 cos ( 4α ) cos (α ) ; c) 2sin ( 2x ) cos ( x )d)11 sin (19θ ) sin (θ ) , which is simplified from sin (19θ ) sin ( θ ) ;22e)1 cos 3x cos 5x ; f) 2sin 4x sin 3x ; g) 2cos 5α sin 3α ; 2 h)1 sin ( 9α ) sin (α ) 2 ( )( )( ) ( )( ) ( ) ) .

(Answers for Chapter 6: Additional Topics in Trigonometry) A.6.1CHAPTER 6:Additional Topics in TrigonometrySECTION 6.1: THE LAW OF SINES1) a) 35.0 m; b) 22.0 m; c) 372 m 22) a) 180.09 ft; b) 224.86 ft; c) 20,137 ft 2SECTION 6.2: THE LAW OF COSINES1) a) 25.8 ; b) 140.2 ; c) No (that would violate the Triangle Inequality); d) 496 ft 22) 13.8 miSECTION 6.3: VECTORS IN THE PLANE1) 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)1v3v 3vb) 4, 3c)d)e) 5, 5

(Answers for Chapter 6: Additional Topics in Trigonometry) A.6.2.4) 8.0 ft, 8.9 ft5) a) 2 29 5 298 29 20 29,,; b) 111.8 ; c) 292929296) a) 327.53 ; b)22 17014 170, 17177) Yes8) No (they point in opposite directions)9) a) 20.3 mph; b) 18.3 mphSECTION 6.4: VECTORS AND DOT PRODUCTS1) 142) a) scalar; b) vector; c) undefined; d) scalar; e) undefined; f) undefined3) 104) Hint: v w5)v2 w2 (v w) (v w) .26) The Pythagorean Theorem7) 19.7 ; acute8) 167.7 ; obtuse 9) 47.7 . Hint: Find the angle between the vectors BA and BC .10) a) 0 ; b) 180 ; c) 90 11) Yes12) No13) 0 and 1()14) Hint: Use the formula: cos θ 15)14 1717v w.v w

(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 Identity (ID) csc(x) 1 sin(x) Reciprocal ID tan(x) 1