CHAPTER 7 ISM - Herrerasmath.weebly
SECTION 7-1175CHAPTER 7Section 7 12. Answers will vary.4. In the calculator window one can never see the entire domain of most functions. Moreover, the resolutioncan never be enough to distinguish small differences in values. The calculator screen can never prove thatan equation is an identity. However, if the graphs of the left and the right sides functions differ visibly, thenthis shows that the equation is not an identity.6. Verify: cos θ csc θ cot θcos θ csc θ cos θ · 1sin Reciprocal Identitycos sin Algebra cot θQuotient Identity8. Verify: tan θ csc θ cos θ 1tan θ csc θ cos θ sin 1·· cos θcos sin 1Quotient and Reciprocal IdentitiesAlgebra10. Verify: cot( x)tan(x) 1cot( x)tan(x) cos( x) sin x·cos xsin( x)Quotient Identitycos xsin x· sin x cos xIdentities for Negatives 112. Verify: tan α Algebracos sec cot cos cos1 cos sec cos cot sin Quotient and Reciprocal Identities1Algebracos sin sin cos Algebra tan αQuotient Identity14. Verify: tan u 1 sec u(sin u cos u)1(sin u cos u)cos usin ucos u cos ucos usec u(sin u cos u) tan u 1Reciprocal IdentityAlgebraQuotient Identity
176CHAPTER 7 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS16. Verify:cos 2 x sin 2 x cot x tan xsin x cos xcos 2 x sin 2 xcos 2 xsin 2 x sin x cos xsin x cos xsin x cos xcos xsin x sin xcos x cot x tan x18. Verify:Quotient Identitycos t sin t csc tsin tAlgebraPythagorean Identity csc tVerify:Algebra2cos 2 tcos 2 t sin 2 t sin t sin tsin t1 sin t20.Algebrasin u1 cos 2 usin uReciprocal Identity csc u21 cos u sin uPythagorean Identitysin 2 u1 sin uAlgebra csc uReciprocal Identity22. Verify: (1 sin t)(1 sin t) cos2 t(1 sin t)(1 sin t) 1 sin t sin t sin2 t 1 sin2 t cos2 tAlgebraAlgebraPythagorean Identity24. Verify: (sin x cos x)2 1 2 sin x cos x(sin x cos x)2 sin2 x 2 sin x cos x cos2 x sin2 x cos2 x 2 sin x cos x 1 2 sin x cos xAlgebraAlgebraPythagorean Identity26. Verify: (csc t 1)(csc t 1) cot2 t(csc t 1)(csc t 1) csc2 t csc t csc t 1 csc2 t 1 1 cot2 t 1 cot2 t28. Verify: sec2 u tan2 u 1sec2 u tan2 u 1 tan2 u tan2 u 1AlgebraAlgebraPythagorean IdentityAlgebraPythagorean IdentityAlgebra30. Verify: sin m(csc m sin m) cos2 m 1 sin m sin m sin m(csc m sin m) sin m 1 sin2 m cos2 m32.Plug in x 0Reciprocal IdentityAlgebraPythagorean Identity34. Plug in x 5
SECTION 7-1Left side: 5 – 4 1 1Left side: 02 10 0 25 25 5Right side: 0 5 5The equation is not an identity.36.Plug in x Right side: 52 16 9 3The equation is not an identity.38. Plug in x 0Left side: cos (–0) cos 0 1Right side: –cos 0 –1The equation is not an identity. 2Left side: sin2 – cos2 12 – 0 2 122Right side: –1The equation is not an identity.40. The two graphs appear to coincide.The equation 3 – 3cos2 x 3sin2 x appears to be anidentity.42. The two graphs do not appear to coincide.The equation sec x cot x sin x is not an identity.44. The two graphs appear to coincide.The equation cos x – sec x –sin x tan x appears to bean identity.46. Not an identity. If x is negative, the left side is 5.48. Yes. As long as x is positive, the equation is always true, and the domain of both sides is x 0.50. Not an identity. sin 11 cos 1.442252. Yes. π – πcos2 x π(1 – cos2 x) πsin2 x by the Pythagorean Identity.54. Verify:1 cos 2 y tan2 y(1 sin y )(1 sin y )sin 2 y1 cos 2 y (1 sin y )(1 sin y )1 sin 2 y sin 2 ycos 2 y sin y 2 cos y tan2 y56.tan 1Verify: sin θ cos θ sec sin 1tan 1cos cos 1·sec cos cos sin θ cos θPythagorean Identity, AlgebraPythagorean IdentityAlgebraQuotient IdentityQuotient Identity, AlgebraAlgebra177
178CHAPTER 7 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS58. Verify: 1 sin y cos 2 y1 sin y1 sin ycos 2 ycos 2 y ·1 sin y1 sin y 1 sin y Algebra(1 sin 2 y )(1 sin y )Pythagorean Identity(1 sin 2 y ) 1 sin yAlgebra60. Verify: sec2 x csc2 x sec2 x csc2 xsec2 x csc2 x 11 cos 2 xsin 2 xsin 2 x cos 2 xsin 2 x cos 2 x1cos 2 x sin 2 x11cos 2 x·sin 2 xReciprocal IdentityAlgebraPythagorean IdentityAlgebra sec2 x csc2 x62.Verify:Reciprocal Identity1 sec csc θsin tan 1 cos1 1 sec cos ·sin sin tan cos sin cos cos 1sin cos sin cos 1 sin (cos 1) 1sin Quotient and Reciprocal Identities, AlgebraAlgebraAlgebraAlgebra csc θReciprocal Identity64. Verify: ln(cot x) ln(cos x) ln(sin x) cos x sin x ln(cos x) ln(sin x) ln ln(cot x)66.Verify:AlgebraQuotient Identity1 csc ysin y 1 1 csc ysin y 11 sin1 y1 csc ysin y ·1 sin1 y sin y1 csc y sin y 1sin y 168. Verify: sin4 x 2 sin2 x cos2 x cos4 x 1sin4 x 2 sin2 x cos2 x cos4 x (sin2 x cos2 x)2 12 1Reciprocal Identity, AlgebraAlgebraAlgebraPythagorean IdentityAlgebra
SECTION 7-170. Verify: csc n sin n cot n1 cos nsin n1sin ncsc n 1 cos nsin n1 cos n72. Verify:sin 2 t 4sin t 32 1 cos n sin 2 nsin n(1 cos n)Algebra 1 sin 2 n cos nsin n(1 cos n)Algebra cos 2 n cos nsin n(1 cos n)Pythagorean Identity cos n(cos n 1)sin n(1 cos n)Algebra cos nsin nAlgebracos t (sin t 1)(sin t 3)1 sin 2 t(1 sin t )(3 sin t ) (1 sin t )(1 sin t ) 74. Verify:3 sin t1 sin tAlgebra, Pythagorean IdentityAlgebraAlgebracos3 u sin 3 u 1 sin u cos ucos u sin ucos3 u sin 3 u(cos u sin u )(cos 2 u cos u sin u sin 2 u ) (cos u sin u )cos u sin u cos2 u sin2 u cos u sin u 1 sin u cos u76. As the graph shows,defined.Quotient Identity3 sin t1 sin tcos tsin 2 t 4sin t 32Reciprocal Identity cot n179AlgebraAlgebraPythagorean Identitycos( x) 1 is not an identity since the left hand side is 1 for all x for which it issin x cot( x)
180CHAPTER 7 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS78. As the graph shows,cos x 1 appears to be an identity.sin( x) cot( x )cos xcos x x)sin( x) cot( x ) sin x cos(sin( x ) cos xx sin x cossin x 180. As the graphs show,1 tan 2 x1 cot 2 x tan2 x does not appear to be an identity.Y1 LHSY2 RHSThe last screen shows a value of x for which both sides are defined but not equal.82. As the graph shows,tan 2 x 11 cot 2 x tan2 x appears to be an identity.tan 2 x 11 cot 2 x sin 2 x 1cos 2 x21 cos2 xsin xQuotient Identity sin 2 x cos 2 xcos 2 xsin 2 x cos 2 xsin 2 xAlgebra 1cos 2 x·sin 2 x1 tan2 x84.Pythagorean IdentityAlgebra, Quotient Identitycos xcos x 2 sec x appears to be an identity.As the graph shows,1 sin x1 sin xcos xcos x 1 sin x1 sin xcos x (1 sin x) cos x(1 sin x) (1 sin x)(1 sin x ) Algebracos x cos x sin x cos x cos x sin x1 sin 2 x2 cos xcos 2 x2 cos x 2 sec xAlgebraAlgebra, Pythagorean IdentityAlgebraReciprocal Identity
SECTION 7-186.Verify:3cos 2 z 5sin z 53sin z 2 1 sin zcos 2 z23cos z 5sin z 53(1 sin 2 z ) 5(1 sin z )cos 2 z 1 sin 2 z(1 sin z )[3(1 sin z ) 5] (1 sin z )(1 sin z )3(1 sin z ) 51 sin z3 3sin z 5 1 sin z3sin z 2 1 sin z Pythagorean Identity, AlgebraAlgebraAlgebraAlgebraAlgebrasin x cos y cos x sin ytan x tan y cos x cos y sin x sin y1 tan x tan y88. Verify:sin x sin ytan x tan ycos x cos ycos xcos y ·ysinsin x1 tan x tan ycos x cos y 1 cosx cos y sin x cos y cos x sin ycos x cos y sin x sin ycot cot tan tan cot cot 11 tan tan 1 1tan tan cot cot 90. Verify:1 tan tan 1 cot1 cot1 ·cot cot cot cot cot cot cot cot 1cot cot cot cot 192. (A) tan x sin xcos x(B) sin2 x cos2 x 1(C) 96.1 cos 2 x sin x . Both sides 0 x in all quadrants.1 sin xReciprocal Identity, Algebra1 sec xcos x1 sin 2 x cos x;2AlgebraAlgebra94.sin xQuotient Identity, AlgebraAlgebra 98.1 sin 2 x 0 cos x 0 x must be in quadrants I or IV. tan x. 1 sin 2 x 0 identity when sin x and tan x have opposite signs x in quadrantsII or III.100.181a2 u 2 a 2 a 2 cos 2 x a 2 (1 cos 2 x) a sin 2 x and since for 0 x π, a sin xsin 2 x sin x
182102.CHAPTER 7 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONSa2 u 2a 2 a 2 cot 2 x a 2 (1 cot 2 x) a csc2 x and since for 0 x a csc x ,2csc 2 x csc xSection 7 22. Since cos (x – y) cos x cos y sin x sin y, cos x cos cos x sin sin x;22 2 2since cos 0 and sin 1, the right side reduces to sin x.24.It makes no difference how the angles are measured as long as it is done consistently.6.Directly using the sum identity involves tan 2, which is undefined. sin x sin cos x cos sin xcos x2 22However, tan x –cot x. 2 cos x cos cos x sin sin x sin x 2 22 8. Plug in x 2, y 1. 10. Plug in x 2, y .Left side: (2 – 1)3 13 16Right side: 23 – 13 8 – 1 7 2Left side: 2 tan The equation is not an identity.63 Right side: tan 2 3 6 The equation is not an identity. ,y .66 Left side: tan tan 366 14. Plug in x 12. Plug in x Right side: tan ,y .62 3 Left side: sin sin32 2 6 3 112 tan 66333Right side: sin 11 sin 1 6222The equation is not an identity.The equation is not an identity. ,y .44 Left side: sin sin 1442 18. cos(x π) cos x cos π sin x sin π cos xNot an identity. 22 Right side: sin sin224420. cot(x π) 16. Plug in x 2The equation is not an identity.22. sec(2π x) 111 sec xcos(2 x) cos 2 cos( x) sin 2 sin x cos x cos( x )cos x cos sin x sin sin x cos cos x sin sin( x ) cos x cot x sin xYesYes
SECTION 7-2 24. cos x cos x cos sin x sin sin x2 22 2 Not an identity.26. Verify: tan x cot x tan x 2 2 x cos 2 x Quotient Identitycos xsin xCofunction Identitysin cot x Quotient Identity 28. Verify: sec x csc x2 2 sec x 1cosReciprocal Identity 2 x 1sin xCofunction Identity csc xReciprocal Identity 30. Verify: sin(π x) sin xsin(π x) sin π cos x – cos π sin x 0 cos x – (–1) sin x sin xDifference IdentityKnown ValuesAlgebra32. tan(x y) tan[x – (–y)]Algebratan x tan( y ) 1 tan x tan( y ) Difference Identitytan x tan y1 tan x tan yIdentities for Negatives34. sin(x 45 ) sin x cos 45 sin 45 cos x sin x · 1212 1cos x2(sin x cos x) tan tan x Known ValuesAlgebra36. cos(x 180 ) cos x cos 180 sin x sin 180 cos x( 1) sin x(0) cos x438. tan x 1 tan 4 tan x 4 Difference IdentitySum IdentityKnown ValuesAlgebraDifference Identity1 tan x1 tan xKnown Values40. sin 75 sin(45 30 ) sin 45 cos 30 sin 30 cos 45 12·3113 1 · or2222 23 12 2·22 6 24183
184CHAPTER 7 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS42. cos cos coscos sinsin661244 4 6 12·3113 1 · ·2222 222 44. sin 22 cos 38 cos 22 sin 38 sin(22 38 ) sin 60 46.tan110 tan 50 tan(110 50 ) tan 60 1 tan110 tan 50 48. sin x 6 2432321(QII), cos y (QIII).34Angle x: a 32 22 5 cos x 531542 1 15 5 2 5 3sin(x y) sin x cos y sin y cos x 3 4 4312Angle y: b 42 ( 1)2 15 sin y tan x tan(x y) 25, tan y 15 2 15tan x tan y5 2 5 35 · 2 1 tan x tan y1 1555 2 15550. cos x 11(QII), tan y (QIII)32Angle x: b 32 ( 1) 2 Angle y: c ( 2) 2 ( 1) 2 8, tan x 83 2 18 sin x 5 cos y sin(x y) sin x cos y sin y cos x tan(x y) 8 12tan x tan y 1 tan x tan y1 ( 8) 12558 2 1 1 4 2 1· · 33553 5 ·52. Verify: sin 2x sin(x x)sin(x x) sin x cos x cos x sin x 2 sin x cos x54. Verify: cot(x y) , sin y 21 4 2 22 2 2AlgebraSum IdentityAlgebracot x cot y 1cot y cot xcot(x y) cos( x y )sin( x y )Quotient Identity
SECTION 7-256. Verify: cot 2x cos x cos y sin x sin y·sin x cos y sin y cos x cot x cot y 1cot y cot x1sin x sin y1sin x sin yAlgebra, Quotient Identitycot 2 x 12 cot xcot 2x cot(x x)Algebracos( x x) sin( x x) Quotient Identitycos x cos x sin x sin xsin x cos x sin x cos xcos 2 x sin 2 x· 2sin x cos xSum Identity1sin 2 x1sin 2 xAlgebracot 2 x 12 cot xtan u tan vsin(u v) tan u tan vsin(u v)Algebra, Quotient Identity 58. Verify:sin u cos v sin v cos usin(u v) ·sin u cos v sin v cos usin(u v)1cos u cos v1cos u cos vtan u tan vtan u tan vsin( x y )tan x tan y cos x cos y 60. Verify:sin( x y )sin x cos y sin y cos x cos x cos ycos x cos y sin x cos ysin y cos x cos x cos ycos x cos y tan x tan y62.Difference Identities, AlgebraSum and Difference Identities, AlgebraAlgebra, Quotient IdentityDifference IdentityAlgebraAlgebra, Quotient Identitycot x cot yVerify: tan(x y) cot x cot y 1tan(x y) 64. Verify:tan x tan y·1 tan x tan y1tan x tan y1tan x tan ySum Identity, Algebra 1 1tan ytan x1 1tan x tan yAlgebra cot y cot xcot x cot y 1Reciprocal Identity cot x cot ycot x cot y 1Algebrasin( x h) sin x cos h 1 sin h sin x cos x hh h sin( x h) sin xsin x cos h sin h cos x sin x hhSum Identity185
186CHAPTER 7 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONSsin x(cos h 1) sin h cos xhcosh 1 sin h sin x cos x h h 66. x 3AlgebraAlgebra4 3,y cos(x y) cos x cos y – sin x sin y 4 5 1Left Side: cos cos 3Right Side: cos3 24 4 1– sin sin 3332cos3 3 3 3 1 3 1 1 2 – 2 2 4 4 2 sin(x y) sin x cos y cos x sin y3 4 5 Left Side: sin sin 3 Right Side: sin68.x 33 cos2 3 4 4 3 1 1 3 333 cos sin 2 2 2 2 4423335 3 ,y 44cos(x y) cos x cos y – sin x sin y 5 3 Left Side: cos cos 0 4Right Side: cos 4 25 5 3 3 1 1 cos – sinsin –442 2 4 4 1 1 1 1 02 2 2 2 sin(x y) sin x cos y cos x sin y 5 3 Left Side: sin sin 1 4Right Side: sin 4 25 5 3 3 1 1 cos cossin 4442 2 4 1 1 1 1 12 2 2 2 70. x 3.042, y 2.384sin(x y) sin(3.042 2.384) sin(0.658) 0.6115sin(x y) sin x cos y sin y cos x sin 3.042 cos 2.384 sin 2.384 cos 3.042 0.6115tan(x y) tan(3.042 2.384) tan 5.426 1.155tan(x y) tan 3.042 tan 2.384tan x tan y 1.1551 tan x tan y 1 tan 3.042 tan 2.38472. x 128.3 , y 25.62 sin(x y) sin(128.3 25.62 ) sin(102.68 ) 0.9756sin x cos y sin y cos x sin 128.3 cos 25.62 sin 25.62 cos 128.3 0.9756tan(x y) tan(128.3 25.62 ) tan(153.92 ) 0.4895tan128.3 tan 25.62 tan x tan y 0.48951 tan128.3 tan 25.62 1 tan x tan y74. Evaluate each side for a particular set of values of x and y for which each side is defined. If the left side isnot equal to the right side, then the equation is not an identity. For example, for x 2 and y 1, both sidesare defined, but are not equal. 13 sincos x sin x cos x76. y sin x sin x cos33223
SECTION 7-2187y1 sin(x π/3)y2 13sin x cos x22 78. y cos x 5 5 31 5 cos x cos sin x sin cos x sin x 6226 6 y1 cos(x 5π/6)y2 31cos x sin x22tan x tan tan x 1480. y tan x 1 tan x tan 44 1 tan x y1 tan(x π/4)tan x 1y2 1 tan x 3 4 3 4 3 4 82. cos sin 1 cos 1 cos sin 1 cos cos 1 sin sin 1 sin cos 1 555555 4 4 3 3 16 9 1 2525 5 5 5 5 84. cos arccos 3 3 3 1 1 1 arcsin cos arccos cos arcsin sin arccos sin arcsin 2 2222 2 3 3 1 1 31 1 2 22244 86. Angle x: b x, c 1, a 1 x 2Angle y: a y, c 1, b 1 y 2cos(sin 1 x cos 1 y) cos(sin 1 x)cos(cos 1 y) sin(sin 1 x)sin(cos 1 y) 1 x2 · y x · 1 y2 y 1 x2 x 1 y288. Verify: sin(x y z) sin x cos y cos z cos x sin y cos z cos x cos y sin z sin x sin y sin zsin(x y z) sin((x y) z)Algebra sin(x y)cos z sin z cos(x y)Sum Identity cos z[sin x cos y sin y cos x] sin z[cos x cos y sin x sin y]Sum Identity sin x cos y cos z cos x sin y cos z cos x cos y sin z sin x sin y sin z Algebra
188CHAPTER 7 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS90.sin( x y ) sin x cos y cos x sin y cos( x y ) cos x cos y sin x sin y94.sin x cos y cos x sin ysin x cos y cos x sin y cos x cos ycos x cos y cos x cos y cos x cos y sin x sin y cos x cos y sin x sin y cos x cos ycos x cos y cos x cos y92.sin x sin y tan x tan ycos x cos y sin x sin y 1 tan x tan y1 cos x cos y1x 123 12m mtan(θ2 θ1) 2 1 1 m1m21 3 1296. y 3x 1 and y ·6 12 12 32θ2 θ1 45 98. α 43 , M 0.25 inch, N 0.11 inchtan β tan α N0.11sec α tan 43 sec 43 0.33089M0.25β 18 Section 7 32.sin(x – y) sin x cos y – cos x sin y. After replacing y with x, this reduces tosin(x – x) sin x cos x – cos x sin x, or sin 0 0, or 0 0.4.cos(x – y) cos x cos y sin x sin y. After replacing y with x, this reduces tocos(x – x) cos x cos x sin x sin x, or cos 0 cos2 x sin2 x, or cos 0 1, or 1 1.6. Since sinxxand cos are functions, they can take on22only one value for any x. The choice is madexlies.2according to the quadrant in which8.Verify: sin 2x 2 sin x cos x for x 45 sin 2x sin(2 · 45 ) sin 90 12 sin x cos x 2 sin 45 cos 45 22· 1221 cos xx ,x 12. Verify: cos222x x , Quad I: sign of cosis .242 x2 1cos cos 2 cos 22242 2·10. Verify: tan 2x 2 tan x21 tan x tan 2x tan 2 tan 3 6 2 tan x21 tan x 2 tan 61 tan 2 6 14. tan 75 tan150 2 1 231313 6for x 2 63 3 31 cos150 1 cos150 1 231 23 233 3 1 13 3 333 6 3 61 cos x 23 16. tan 15 tan 1 1 30 232321 cos 221 021 2221 cos 30 1 cos 30
SECTION 7-3 2 32 31 1 2 32 32 32 3 2 3 2 6 18. cos cos 212 2 3 2 32 34 3 1892 32 3 2 331 cos 62327 7 4 sin 20. sin 82 2 2 34 2 32 1 7 1 cos 4 22222 24 2 2222. Verify: sin 2x (tan x)(1 cos 2x)sin x(1 2 cos2 x 1)cos xsin x (2 cos2 x)cos x(tan x)(1 cos 2x) 2 sin x cos x sin 2x1(cos 2x 1)211(cos 2x 1) (2 cos2 x 1 1)221 (2 cos2 x)2Quotient Identity, Double-angle IdentityAlgebraAlgebraDouble-angle Identity24. Verify: cos2 x cos2 x26. Verify: 1 sin 2t (sin t cos t)21 sin 2t 1 2 sin t cos t sin2 t cos2 t 2 sin t cos t sin2 t 2 sin t cos t cos2 t (sin t cos t)2x1 cos x 22 1 cos x 2xcos2 22 1 cos x 2cot x tan xVerify: cot 2x 2111 tan 2 xcot 2x 2 tan x tan 2 x2 tan x2Double-angle IdentityAlgebraAlgebraDouble-angle IdentityPythagorean IdentityAlgebraAlgebra28. Verify: cos230.1 tan xHalf-angle IdentityAlgebra 1 cos sin 21 cot tan 2232. Verify: cot1sin 1 cos 1 cos sin
190CHAPTER 7 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS 34. Verify:1tan x cot x tan x 2 tan x22cos 2u1 tan u 1 sin 2u1 tan ucos 2ucos 2 u sin 2 u 1 sin 2u1 2sin u cos u(cos u sin u )(cos u sin u ) cos 2 u 2sin u cos u sin 2 u(cos u sin u )(cos u sin u )(cos u sin u ) 2 cos u sin u·cos u sin u 1 tan u1 tan u1cos u1cos u 12 cos 2 x 1·1cos 2 x1cos 2 xDouble-angle Identity, Algebra1cos 2 x2 12cos xAlgebrasec 2 xReciprocal Identity2 sec 2 xcot tan cot tan cos sin cos sin 2sin cos sin cos sin cos sin cos cos sin 2 cos 2 sin 2 cos 2 1 cos 2 α .6 AlgebraReciprocal Identitycot tan cot tan Plug in x AlgebraAlgebra, Quotient Identity2 sec 2 x1sec 2x cos 2 x40.Algebra, Pythagorean Identitysec 2 x36. Verify: sec 2x 38. Verify: cos 2α Double-angle IdentityQuotient Identity, AlgebraAlgebraDouble-angle Identity, Pythagorean IdentityAlgebra .3 31Left side: tan tan 62342. Plug in x 1Left side: cos 2 cos 32 6
SECTION 7-3Right side: 2 cos 63Right side:The equation is not an identity.7 .37 7 31Left side: sin sin 6221 cos 73 2 1 122 46. Plug in x .6 Left side: tan 2 tan3 6 12Right side:The equation is not an identity. .2 2Left side: sin sin 42 11Right side: sin ·1 222 13tan 322The equation is not an identity.44. Plug in x Right side:1912 cot 61 cot 2 6 32( 3)1 ( 3) 2 3The equation is not an identity.48. Plug in x 50. csc 2x 111 csc x sec xsin 2 x 2sin x cos x 2Not an identity.1212The equation is not an identity.52. Plug in x : tan 4 tan π 04 4 4 tanNot an identity.4556. cos x ,54. Plug in x 442 tan 6 cot 6Not an identity. x π2a 4, r 5, b 33 4 24· 5 5 25sin 2x 2 sin x cos x 2 · 3 2187cos 2x 1 2 sin2 x 1 2 1 2525 5 tan 2x 58. cot x : tan 2 tan 63 6 2 tan x21 tan x 64 16 2421 169 16 16 91 34 2 34 5, x 0212sin 2x 2 sin x cos x 2 a 5, b 12, r 13 12 5120 13 13169247213 33 21 33 3
192CHAPTER 7 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS288 119 12 2 1 169 169 13 cos 2x 1 2 sin2 x 1 2 tan 2x 2 tan x21 tan x 245 25 12021 14425 25 1441 125 252 125 1460. cos x , π x sin3 a 1, r 4, b 1521 411 cos xx 222 120119 x3 (QII)2244 152 82 2 21041 411 cos x4 1 3 2 6x 228242 2 2 11 151 cos x4 1 15 5 15x4tan 15sin x1532 15 154cos62. tan x 3, π x 24a 4, b 3, r 5, x (QIV)224sin1 54 3 101 cos x5 4310x 221010210 10cos1 541 cos xx 222tan1 cos x 1 54 5 4 9x 3sin x 53 3 32105 4110 101010 1054 50 θ 90 0 2 θ 180 and since sec 2 θ 0, 2 θ is in QII.4 5 4 cos 2 θ sec 2θ 5464. Find the exact values of sin θ and cos θ, given sec 2 θ , 0 θ 90 .(A)(B) 4 2sin 2θ 1 cos 2 2 1 5 (C)sin θ 1 cos 2 2cos θ 1 cos 2 2 1 16 2593 255(D) & (E) θ is a quadrant I angle, sosin θ 1 cos 2 21 54cos θ 1 cos 2 21 5422 3 105 4310 101010 10 105 4110 101010 1066. x 72.358 (A) tan 2x tan(2·72.358 ) tan 144.716 0.70762tan 2x 2 tan x21 tan x 2 tan 72.358 21 tan 72.358 0.7076268. x 4(A) tan 2x tan(2·4) tan 8 6.7997tan 2x 2 tan x21 tan x 2 tan 41 tan 2 4 6.7997
SECTION 7-3x72.358 cos cos 36.179 0.80718221 cos x1 cos 72.358 xcos 0.80718222193x4 cos cos 2 0.41615221 cos x1 cos 4xcos 0.41615222(B) cos(B) cos(Quadrant III)70.Y1Y1 and Y2 are identities on the intervals [ 2π, π] and [π, 2π].Y272.Y1Y1 and Y2 are identities on the interval [0, 2π].74. Verify: sin 3x 3 sin x 4 sin3 xsin 3x sin(2x x) sin 2x cos x sin x cos 2x 2 sin x cos x cos x sin x(1 2 sin2 x) 2 sin x cos2 x sin x 2 sin3 x 2 sin x(1 sin2 x) sin x 2 sin3 x 2 sin x 2 sin3 x sin x 2 sin3 x 3 sin x 4 sin3 xSum IdentityDouble-angle IdentityAlgebraPythagorean IdentityAlgebraAlgebra76. Verify: sin 4x (cos x)(4 sin x 8 sin3 x)sin 4x sin(2x 2x) sin 2x cos 2x sin 2x cos 2x 2 sin 2x cos 2x 2(2 sin x cos x)(1 2 sin2 x) cos x(4 sin x)(1 2 sin2 x) (cos x)(4 sin x 8 sin3 x)AlgebraSum IdentityAlgebraDouble-angle IdentityAlgebraAlgebraY278. Use sin 2θ 2 sin θ cos θ 3 3 3 4324· sin 2 cos 1 2 sin cos 1 cos cos 1 2 ·5 5 255 5 5 80. Use tan 2θ 2 tan 1 tan 2 12 34 16 3 2424 2 1 34 16 16 9 72 tan tan 4 3 tan 2 tan 1 4 1 tan 2 tan 1 34
194CHAPTER 7 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS82. Use sin1 cos , QIV22 1 cos tan 1 43 1 1 4 sin tan 2 3 284.1 53 2cos(2x) 2 cos2x – 11 cos 2x 2 cos2x 5 3210 2 55 1010510 1086. tan 2x 1 cos 2 x cos 2 x21 cos 2 xcos 2 x 288. If 0 x π, sin x 0, then 0 Quadrant I. 2 tan x1 tan 2 x 2 tan x cot 2 x(1 tan 2 x) cot 2 x2 tan x cot x cot x222cot x tan x cot x 2 cot xcot 2 x 1 xxx , tan 0. This applies if x is in Quadrants I or II, with in2222 xxx π, tan 0. This applies if x is in Quadrants III or IV, with in2222 xxQuadrant II. ( If x π, and tan is undefined.)222xThe truth of the statement that tan and sin x always have the same sign follows for all other values of x2 x 2k since sin(x 2kπ) sin x and tan tan(x kπ) tan x. 2 If π x 2π, sin x 0, then90.tanx1 cos x 21 cos x (A) Half-angle identity1 cos x 1 cos x 1 cos x 1 cos xMultiplied by 1(1 cos x) 2Algebra1 cos 2 x(1 cos x) 2sin 2 x(B) Pythagorean identity(1 cos x) 2a b2sin x1 cos x sin x (C)absin 2 x sin x becausea 2 a for any real number a;(1 cos x) 2 1 cos x because 1 – cos x is never negative. 1 cos xsin xxand sin x always have the same sign, and since 1 – cos x2xis never negative, tan and (1 – cos x)/sin x always have the same sign2(D) Since tanfor any x.
SECTION 7-492.tan 2θ 2 tan 2, tan θ 1 tan 2 2x6x2 22xx1 2x19526, tan 2θ xx 64x 2xx 46x2 24 4x2x2 12x 2 3 3.464 ft.tan θ 221 x2 33θ 30.000 Ms/22s/25s 25 s s s2 42 2 AM 94.s/2s NAsin 2 MN2AM 2 s45 s2 22 5 2MN 2 s s 2 2 s2 21 cos 21 cos 21 20 1022 11 cos θ 10 54cos θ 5Section 7 42. A sum-product identity expresses a sum of two trigonometric functions as a product of two othertrigonometric functions.4. The identity for cos(x – y).6. The double-angle identities for sine and cosine.1[cos(x y) cos(x y)]21111cos 7A cos 5A [cos(7A 5A) cos(7A 5A)] (cos 12A cos 2A) cos 12A cos 2A22228. cos x cos y 1[sin(x y) sin(x y)]2111cos 2θ sin 3θ [sin(2θ 3θ) sin(2θ 3θ)] [sin 5θ sin( θ)] [sin 5θ sin θ]22211 sin 5θ sin θ22x yx ycos x cos y 2 coscos227 5 7 5 cos 7θ cos 5θ 2 coscos 2 cos 6θ cos θ2210. cos x sin y 12.2 s2
196CHAPTER 7 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONSx yx ysin22u 5uu 5usin u sin 5u 2 cossin 2 cos 3u sin( 2u) 2 cos 3u sin 2u2214. sin x sin y 2 cos1[sin(x y) sin(x y)]211cos 75 sin 15 [sin(75 15 ) sin(75 15 )] [sin 90 sin 60 ]221 3 1 2 3 2 3 1 2 2 2 2 41sin x sin y [cos(x y) cos(x y)]211sin 112.5 sin 22.5 [cos(112.5 22.5 ) cos(112.5 22.5 )] [cos 90 cos 135 ]221 2 2 0 2 2 41sin x cos y [sin(x y) sin(x y)]211sin 262.5 cos 52.5 [sin(262.5 52.5 ) sin(262.5 52.5 )] [sin 315 sin 210 ]221 2 1 ( 2 1) 2 22 416. cos x sin y 18.20.22. cos x cos y cos1[cos(x y) cos(x y)]23 7 1 3 7 cos cos 8882 8 5 1 2 2 3 7 1 cos 8 8 2 cos 2 cos 4 2 0 2 4 1[sin(x y) sin(x y)]27 5 1 7 5 1 1 1 7 5 1 cossin sin sin 2 sin sin 6 2 0 2 4121212122 12 12 24. cos x sin y 26. sin x sin y sin1[cos(x y) cos(x y)]217 1sin 24 224 17 2 3 1 1 2 1 2 17 1 cos cos cos cos 434 2 2 2 24 24 2 24 24 x yx ycos22105 15 105 15 1 22cos 105 cos 15 2 coscos 2 cos 60 cos 45 2 2 222228. cos x cos y 2 cos
SECTION 7-4x yx ycos22165 105 165 105 2 36sin 165 sin 105 2 sin cos 2 sin 135 cos 30 2 2 222230. sin x sin y 2 sin32. cos x cos y 2 sinx yx ysin2213 13 5 cos cos 2 sin121234. sin x sin y 2 cos12 5 13 1212sin2 5 122 2 sin 3 2 36sin 2 32 242x yx ysin2211 7 sin sin 2 cos121211 127 12211 sin12 7 122 2 cos 3 2 12sin 2 62 2421[cos(x y) cos(x y)]211[cos(x y) cos(x y)] [cos x cos y sin x sin y (cos x cos y sin x sin y)]221 [2 sin x sin y]236. Verify: sin x sin y sin x sin y38. Start with the product-sum identitycos u cos v Let x u vy u v1[cos(u v) cos(u v)]2x yx y,v . Substituting into the product-sum identity,22x yx y1x yx ycoscos [cos x cos y] or cos x cos y 2 cossin22222Solving this system gives u 40. Verify:cos t cos 3t tan tsin t sin 3t 2sin t 23t sin t 23tcos t cos 3t 2sin t 23t cos t 23tsin t sin 3t sin 2t sin( t )sin 2t cos( t )sin 2t sin tsin 2t cos tsin t cos t tan tSum-product IdentitiesAlgebraIdentities for negativesAlgebraQuotient Identity197
198CHAPTER 7 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS42. Verify:sin x sin yx y tan2cos x cos y2sin x 2 y cos x 2 ysin x sin y cos x cos y2 cos x 2 y cos x 2 y x y2cos x 2 ysin44. Verify:Algebrax y2 tanQuotient Identitycos x cos yx y tan2sin x sin y 2sin x 2 y sin x 2 ycos x cos y sin x sin y2sin x 2 y cos x 2 y sincos tan46. Verify:x y2x y2Sum-product IdentitiesAlgebrax y2Quotient Identitytan 12 ( x y ) sin x sin y sin x sin ytan 12 ( x y ) 2 sin x 2 y cos x 2 ysin x sin y sin x sin y2 cos x 2 y sin x 2 y x ycos2x ycos 2 sinsinx y2x y2x yx ycot22x y1 tan2 tan x y2 tan 48.Sum-product IdentitiestantanLet x y 0.Left side: cos 0 sin 0 0Right side: cos 0 – sin 0 1The equation is not an identity.x y2x y2Sum-product IdentitiesAlgebraQuotient IdentitiesReciprocal IdentityAlgebra and y .66 3Left side: cos cos 664 1 Right side: cos cos 32 6 6 50. Let x The equation is not an identity.
SECTION 7-452. Let x 0 and y .2 and y 0.2 Left side: cos cos 0 12 2 0 2 0 Right side: 2 sinsin 2 sin2 142254. Let x 12 Right side: sin 0 sin 02Left side: sin 0 sinThe equation is not an identity.The equation is n
Yes. π – πcos2 x π(1 – cos2 x) πsin2 x by the Pythagorean Identity. 54. Verify: 1cos2 (1 sin )(1 sin ) y y y tan2 y 1cos2 (1 sin )(1 sin ) y y y 2 2 sin 1sin y y Pythagorean Identity, Algebra 2 2 sin cos y y Pythagorean Identity sin cos y y 2 Algebra tan2 y Quotient
Part One: Heir of Ash Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 .
TO KILL A MOCKINGBIRD. Contents Dedication Epigraph Part One Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Part Two Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18. Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26
DEDICATION PART ONE Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 PART TWO Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 .
A. The ISM Code The ISM Code was borne when on 1 July 1998; the 1994 amendments to the International Convention for the Safety of Life at Sea (SOLAS), 1974 came into force. With the introduction of the chapter IX (Management for the safe operation of ships), the ISM Code was made mandatory.
Dear Supply Management and Business Leaders ISM has a proud history of establishing and advancing global standards for the supply management profession. In an important continuation of our tradition of professional leadership, the ISM Board of Directors approved important updates to the ISM Principles
ISR 2911 ISR 2911-ISM ISR 2901 ISR 2901-ISM ISR 1941 ISR 1941-ISM ISR 1921 ISR 1921-ISM ISR 892FSP Licensing The IWAN Application is a component of the APIC-EM. APIC-EM can be purchased a-la-carte or with the Cisco ONE Software. With the a-la-carte option, Cisco provides a
Automated storage systems ISM UltraFlex 3600 ISM UltraFlex 3900 ISM 2000 ISM 1100 Max. number of storage places 3,622 3,840 2,059 1,100 Storage/retrieval quantity 54 — 27 27 Weight (including cases, w/o materials) 1,500 kg approx. 1,500 kg 1,650 kg 458 kg 2,450 mm 2,450 mm 2,400 mm 2,200
in pile foundations for Level 1 earthquake situation. The proposed load factors in the study are a function of the chosen soil investigation/testing and piling method, which is applied to the bending moment in piles. Therefore, better choices of soil investigation/testing and high quality piling method will result in more reasonable design results. Introduction Reliability-based design .
