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A) 0.9372 B) A) B) -0.9372 4 4 3 5 A) B) C) 3 10) sin-1 sin 4 4 4 5 21) cos-1 cos 3 10 2) cos-1 22 3 -1Test Practice 2 A) B) 5) sinfor (0.5) 5 A) B) 10 5 C) - 11 10 10 7 7 A) B) A) Find the exact value of the expression A) B)6 C) 6 6 6 3 a. b. 4 11) tan-1 tan 22) sin-1 sin 5 3 Secant, Cosecant, and Cotangent Functio 2 Define the Inverse 5 3 -1 3) cos 6) tan-1 4 3 A) - 2 B) A) B)CHOICE. 5 5 - 5 MULTIPLE Choose the oneC)alternative that best comp 5 55 7 c. d. A) B) A) B) C) 66 6 3 Find the exact 6value of the expression. 3 12) -1-1-115)sec cotsin 23) tan-1 tan 2 8 Value Define Inverse Secant, Cosecant, Cotangent 22 Find anthe Approximate of an Inverse Cosine, or Tangent Function -1 (1)Sine, 4)and tan 3 Functions A) B) 3or answers 4 4 A) 2 B) 1 MULTIPLE CHOICE. Choose the one alternative that best completes the statement the question. A) B) C) 8 e. MULTIPLE CHOICE. Choose the onef. alternative that best completes the or answers th A) 8 B) statement 8 4 4 Find the exact value of the expression. Use a calculator to find the value of 16) the csc expression -1 - 2 rounded to two decimal places. -1 -1 4 Find 15) thecot Inverse Function of a Trigonometric 3 -1Function 7) sin-13(-0.3) 13) sec-1tan 5) sin (0.5) 2 3 B) f A) C) D) B) -1 of the function B) FindA) the inverse function f A) B) C) 1.88 -0.30 -17.46 4 4 7 4 or answers 3the q MULTIPLE CHOICE. Choose the one alternative that 4 best completes3the statement A) 2 3 B) a. f(x) =2 sin x β 5 A) 6 B) 2 6 b.cos f(x) =7 tan 8x f-1 of the function f. 3 -1 -1 Find the16) inverse function 8) (0.3) csc - 2 c. f(x) = cos(x-6) β 7 17) sec -1 1 24) f(x) A) 21.27 sin x - 5 = B) 72.54 C) 0.30 A) B) C) D) 3 C) Find the domain of f and f-1x + 5 6) tan4-1 4 3 6 x 5 + A) 0 B) 2 B) f-1 (x) = cos A) f-1 (x) = sin-1 -1 3 14) tan sin -1=(2.3) 2 2 f(x) -5cos(9x) 9)a.tan 2 -1 b. f(x) = 5sin(8x-1) 17) secA) 1.16 1 C) A)B) 66.50 B) c. C) f(x)f-1 = tan(x-3) + 6-1 x + 2 -10.41 -1 - 5 (x) = sin D) f 6 -1 A) 1 (-2) B) 3(x) 2 = 2 sin x D) 18) 5 A) 0 B) sec C) 6 3 D) Find the exact solution of the equation 2 B) A) ! -1(4x) = 2 Find an Approximate Value of an Inverse Sine, a. βsin 3 Cosine, or Tang 3 25) f(x) 7 cos x 6 = + ! 18)b.sec2-1 (-2) -1 cos x = π Page 602 x-6 x-6 -1 2 2 -1(x) c. A) -4 tan x == π cos-1MULTIPLE CHOICE. Choose the one4 alternative B) f-1 (x) =that sin best complete f B) C) D) A) 7 7 3 Use a Calculator to Evaluate sec^-1 x, csc^-1 x, andPage cot^-1 3 3 3x 3 610 E) x +a6calculator to find the value of the expression rounded -1 x + 6to two C) f-1 (x) = cos-1Use f-1 (x) = 7 cos MULTIPLE CHOICE. Choose the oneD)alternative that best comp 3 Use a Calculator to Evaluate sec^-1 x, csc^-1 x, and cot^-1 x 7 Copyright © 2011 Pearson Education, Inc. 7) sin-1 (-0.3) 4) tan-1 (1) MULTIPLE CHOICE. Choose the oneaalternative that completes the statement or answers the question.m A) -0.30 -17.46 Use calculator tobest find the value of theB)expression in radian Copyright © 2011 Pearson E 26) f(x) = 7 tan(8x) 7 -1 in radian measure rounded to two decimal Use a calculator to find the of the expression places. csc 1 value -1 x 8)19) -1 (x) = 1 tan-1 x -1 tan B) f A) f-1 (x) = 4 cos (0.3) 7 7 8 7 8 19) csc-1 4 1 -1 (x) = C) A) f0.61 7 tan(8x) 20) sec -1 - 8 27) f(x)A) = -8 1.70cos(3x) x 1 A) f-1 (x) = cos-1 5 8 3 21) cot-1 - A)A) 1.27 0.61 B) 34.85 -1-1 (2.3) 9)20) tansec -8 A)A) 1.16 1.70 B) 97.18 5 21) cot-1 14 14 x 1 -1 -1 C) (x) = - cos A) f-1.23 B) -70.35 A) -1.23 8 3 B) 72.54 B) 34.85 C) 0.96D) f-1 (x) = 7 tan-1 (8x) D) 55.15 C) -0.13 B) 66.50 B) 97.18 D) -7.18 x 1 B) f-1 (x) = cos-1 3 8 D) f-1 (x) = -8 cos-1Page (3x) 602 C) -0.34 D) -19.65 B) -70.35 sindepth sinis rin below the surface of of the andfrom S is theexample, total horizontal i = n r(also where M thekilometers) horizontal movement (in meters) atcenter arepresented distance d earthquake, (in earthquake, D is the 75)niThe seasonal variation in the length of daylight canthe be by akilometers) sine function. For the daily Solving for , we obtain kilometers from an displacement in meters) at the line. is theofhorizontal movement r (also in (also depth kilometers) below thefault surface ofWhat the center the earthquake, the 41and5 S5 is 2 xtotal horizontal number of hours of daylight in a certain city in the U.S. can be given by h = + sin , where x is the n earthquake centered 3 kilometers below the surface with a total horizontal4displacement meters? an Round displacement 3 5 kilometers 365 of 4 from -1 i sin i (also in meters) at the fault line. What is the horizontal movement r = sin nr the answer to the nearest 0.01 meter. earthquake centered 3 kilometers below the surface with a total horizontal displacement of 4 meters? Round number of days after March 21 ( disregarding leap year). On what day(s) will there be about 10 hours of the answer to the nearest 0.01 meter. Find daylight? r for crown glass (n i = 1.52), water(n r = 1.33), and i = 38°. 75) The seasonal variation in the length of daylight can be represented by a sine function. For example, the daily 6.4 74) The ground movement of an earthquake near a fault line is modeled by the equation 75) The seasonal variation in the length of daylight can be represented by a sine function. the daily 41 5 For 2 example, x 2M , where x is the number hours d = D tanof Identities 1 - of daylight in a certain city in the U.S. can be given by h = + sin Trigonometric 41 5 2 x 2 S 4 3 365 , where x is the number of hours of daylight in a certain city in the U.S. can be given by h = + sin 4 3 365 where M is the horizontal movement (in meters) at a distance d (in kilometers) from the earthquake, D is the number of daysTrigonometric after March 21Expressions ( disregarding leap year). On what day(s) will there be about 10 hours of 1 Use Algebra to Simplify depth (also in kilometers) below the surface of the center of the earthquake, and S is the total horizontal number of days after March 21 ( disregarding leap year). On what day(s) will there be about 10 hours of daylight? displacement (also in meters) at the fault line. What is the horizontal movement 5 kilometers from an daylight? SHORT ANSWER. Write the 3word or phrase completes each statement or answers question. earthquake centered kilometers belowthat the best surface with a total horizontal displacement ofthe 4 meters? Round the answer to the nearest 0.01 meter. 6.4 Trigonometric Identities Simplify the trigonometricIdentities expression by following the indicated direction. 6.4 Trigonometric 75) The seasonal variation in the length of daylight canxbe represented by a sine function. For example, the daily 1) Rewrite of sine and cosine: tan x · cot 1 Use Algebrain toterms Simplify Trigonometric Expressions 41 5 2 x 1 Use number Algebraofto Simplify Trigonometric Expressions in the U.S. can be given by h = sin , where x is the hours of daylight in a certain city + 4 3 365 SHORTnumber ANSWER. Write the or phrase thatleap bestyear). completes each statement or answers the question. 1word cos(or sin + 21 of days afterthe March disregarding On what day(s) will there about 10 hours of SHORT ANSWER. Write word phrase that best completes statement orbeanswers question. 2) Multiply 49) When light travels from oneby medium to another from air to water, foreach instance it changes direction.the (This is daylight? 1 - cos 1 cos + F) aWrite trigonometric expression as an in u: of incidence r is the angle in why pencil, partially submerged in water, asalgebraic though isexpression bent.) The angle Simplify the the trigonometric expression bylooks following theitindicated direction. Simplify thesin trigonometric expression by following the indicated direction. -1u) a. (cot the first medium; the angle of refraction is the second medium. (See illustration.) Each medium has an index 1) Rewrite inIdentities terms of sine and cosine: tan x · cot x 6.4 Trigonometric r 1) Rewrite in-1u) terms of sine and cosine: tan1x · cot x 1 b. cos (csc of refraction nover , respectively which can be found in i and-1an rcommon Rewrite denominator: + tables. Snell s law relates these quantities in the 1 Use 3) Algebra to Simplify Expressions c. tan (sec u) Trigonometric 1 - sin 1 + sin formula 1 cos sin + 1 cos sin + by 2) Multiply SHORT ANSWER. wordby or phrase that best completes each statement or answers the question. sinMultiply sin 1 the i 2) i = nWrite rthe 11 ++ cos cos using the directions provided: G)nSimplify following 1 r-- cos cos expressions Solving for , we obtain (tanby following + 1)(tan the + 1)indicated - sec 2 direction. Simplify the trigonometric expression r 4) Multiply and simplify: 1) Rewrite innterms of sine and cosine: tan x cot x tan· i 11 11 sin = sin-1 r3) 3) Rewrite Rewrite over iaa common common denominator: denominator: 1 - sin ++ 1 + sin n r over 1 - sin 1 + sin 1 + cos sin by 2) Multiply Find r for air (n 1.0003), methylene 16+sin cos2 +iodide 1 i-=cos 7 sin (n+r =1 1.74), and i = 14.7°. 5) Factor and simplify: 2 -++ 11)(tan ++ 1) -- sec22 (tan sin 1 1 4) Multiply and simplify: 3) Rewrite over a common denominator: + 1 - tan sin 1 + sin MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 2 (tan6 sin 1) - sec + 1)(tan 2 + 7 +sin +1 4) Multiply and simplify: 5) expression. Factor and simplify: tan Simplify the sin2 - 1 cos 6) + tan 1 + sinand simplify: 6 sin2 + 7 sin + 1 5) Factor alternative that best completes the statement or sin2one -1 MULTIPLE CHOICE. Choose the or answers answers the the question. question. A) sec the expression: B) cos + sin C) 1 D) sin2 H) Simplify Simplify the expression. expression. MULTIPLE Choose the one alternative that best completes the statement or answers the question. Simplify the a. CHOICE. cos cos 7) (1 cot )(1 + tan ) - csc2 6) expression.-++ cot Simplify the tan 6) 1 ++ sin sin 1 cos 2 A) -2 cot B) 0 C) 2 D) 2 cot2 6) + tan 1 + sinA) sec 2 B) cos sin C) 1 D) + A) sec B) cos + sin C) 1 D) sin sin2 2 A) sec B) cos + sin C) 1 D) sin b. 2 Establish Identities 7) (1 + cot )(1 - cot ) - csc2 2 7)+(1 cot + cot (1 cot )(1 cot) - csc)2- csc - )(1 SHORT 7) ANSWER. Write that each statement the question. 2the word or phrase B) A) cot 0 best completes C) C) 2 or answersD) D) 2 cot22 -2 2 cot2 A) -2 cot B) 0 B) 0 2 2 cot2 A) C) 2 D) 2 cot -2 G) Establish the following identities Establish the identity. Establish Identities 2 2Establish Identities 2 Establish Identities 8) (sin x)(tan x cos x - cot x cos x) = 1 - 2 cos 2 x Page 608 SHORT ANSWER. Write the word or phrase that best completes each statement or answersor theanswers question.the question. SHORT ANSWER. Write the word or phrase that best completes each statement SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Establish the identity. Establish identity. 3the 2x cos 2 xx)-=cos 28) x cos x = xsin x (cos 1 -42x) cos 2 x 8) sin (sinthe x)(tan x cos - cot Page 621 Establish identity. Inc. 1 - Pearson 2 cos 2Education, x 8) (sin x)(tan x cos x - cot Copyright x cos x)©=2011 8) (sin x)(tan x cos x - cot x cos x) = 1 - 2 cos 2 x 29) csc3 x tan2 x = csc x (1 + tan2 x) Page 621 H) Complete the follwing identities (multiple choice). Page 621 Copyright © 2011 Pearson Education, Inc. 1. 30) cot x sec 4 x = cot x + 2 tan x + tan3 x Page 621 Copyright © 2011 Pearson Education, Inc. sin x sin x 31) + = 2 tan 2 x csc x - 1 csc x + 1 2. 32) cos x 2 cos x cos x = sec x - 1 sec x + 1 tan2 x 33) 1 - 2 sec x - 3 sec 2 x 1 - 3 sec x = 2 1 - sec x Copyright © 2011 Pearson Education, Inc. Copyright © 2011 Pearson Education, Inc. 3. 4. 5. 6. I) Use sum and difference formulas to fin exact values of the following expressions: a. cos 285° !! b. tan c. d. !" !!! sinβ !" !!!"# !"°!"#!"° !"# !"°!!"#$!° J) Find the exact values under the specified conditions a. b. c. K) Establish the following identities using sum and difference formulae. 5 4 59) cos tan-1 - cos-1 12 5 A) 63 65 B) 13 24 C) 7 13 D) 52 65 Write the trigonometric expression as an algebraic expression containing u and v. 60) cos (sin-1 u - cos-1 v) L) Complete the following 2 identities using 2 sum and difference formulae. A) v 1 - 5u + u 1 -4v 59) cos tan-1 - cos-1 C) uv - ( 121 - u2 )( 15 - v2 ) 63 -1 -1 61) cosA) (tan 65 u + tan v) B) 13 24 B) v 1 - u2 - u 1 - v2 D) uv + ( 1 - u2 )( 1 - v2 ) C) 7 13 D) 5 4 1 + uv u2 + 1 · v2 + 1 59) cosA)tan-1 1 - -uvcos-1 B) C) D) 12 5 1 uv 2 2 2 2 v 1 v 1 u 1 u 1 · + · + + + Write the trigonometric expression as an algebraic expression containing u and v. 63 13 7 B) C) 60) cosA)(sin-1 u - cos-1 v) 65-1 24 13 62) sinA) (tan 2 + -1 v 1 u- +utan u v)1 - v2 B) v 1 - u2 - u 1 - v2 1 + uv containing u and v. u2 + 1 · v22+ 1 u + vexpressions M) Write the trigonometric as2algebraic expressions C) uv - ( 1 expression 1 -v.u )( 1 - v2 )D) - u2 )( 1 - v C)D) uv +u (and A) Write the trigonometric as)anB)algebraic expression containing u +1· v +1 u +1· v +1 60) cos (sin-1 u - cos-1 v) 1. -1 u + tan-1 v) 2.61) cos (tan 2 u 1 - v2 A) v -11u--utan+-1 63) sin (tan v) 3. 1 + uv 1 - uv 2 A) uv - ( u 1- -v u )( 1 - v2 ) B) C) 1 - uv N) Solve theA) following 0u2 β€ΞΈ<2π B) v2 + 1 on the interval u2 +trig 1 ·equations + 1 · v2 + 1 2 2 u2 + 1 · v2 + 1 u2 + 1 · v2 + 1 ! ,0 < π < ! u B) v 1 - u2 - u 1 - v2 u2 + 1 · v2 +2 1 C)D)uuv ) 2 + +1 (· 1v2- +u1 )( 1 - v2D) 1 - uv C) D) 1 - uv -1=u0+ tan-1 v) 1. πππ π 61) π πππ cos +(tan 2.62)π πππ = πππ π -1 u + tan-1 v) sin (tan-1 -1 v) 1 + uv u2 + 1 · v2 + 1 uv 64) cos (sin u 1+= cos 3. 3π πππ β πππ π β1 B) C) 2 A) v2 + 12 u +211· - uv 2+1 2 1+ +1uv 2-+uu12+·-vuv21 +- 1v2 A) v 1 B) v 1 u u 1-v + v u u · B) C) A) O) Use the information given about the angle to π, 0 β€ π β€ 2π, to find the exact value of the 1 - uv 2 2 2 2 + 1functions. v 1 v 1 u 1 u · + · + + indicated trigonometric C) uv - ( 1 - u2 )( 1 - v2 ) D) uv + ( 1 - u2 )( 1 - v2 ) 1. π πππ = D 1 - uv 2 2 u u D D) , find cos 2ΞΈ. 62) sin (tan-1 !u!!+ tan-1 v) 2. π‘πππ , π < π < , Equations find-1 sin 2ΞΈ. Linear in Sine and Cosine 4 Solve Trigonometric !" (tan-1 u! - tan 63)= sin v) 1 + uv u2 + 1 · v2 + 1 u+v B) C) A) 2 P) Solve the problem v2 + 1 the questioD 1 -best uv completes the statement u + 1or u - v the one alternative that · -answers MULTIPLE CHOICE. Choose 1 uv 2 2 B) C) D) A)! u2 + 1 · v2 + 1 u find + 1sin· 2ΞΈ.v + 1 1. If sin ΞΈ = β , and2ΞΈ terminates in Quadrant IV, then 1 uv 2 2 2 ! v +1 0 u +1· v +1 uon+ the 1 · interval ! Solve the equation <2 . !" !" 2. If tan ΞΈ = β !" , and ΞΈ terminates in Quadrant III, then find cos 2ΞΈ. 0 -1 + sin -1 u -= tan 63) sin (tan v) -1 7u + cos-1 v) 3 64) cosA)(sin , u-v 4 2 4 A) 65) cos A) v 1 - u - u 1 - v2 u2 + 1 · v2 + 1 C) uv - ( 1 - u2 )( 1 - v2 ) 66) cos = sin 5 3 B)4 B) 1 - uv u2 + 1 · 7 v2 + 1 3 2 C) u2 + 1 · v2 + 1 D) - 6 D B) v 1 - u2 + u 1 - v2 1 - uv D) uv + ( 1 - u2 )( 1 - v2 ) C) 3 5 3 , , C) , D) -1 -1 v) LinearB)in Sine 64)Trigonometric cosA)(sin 4 Solve Equations 4 4u + cos 4 4 and Cosine 4 4 4 2 2 2 2 A) v 1 - u - u 1 - v B) v 1 - u + u 1 - v MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the ques 20) If cos 2 = A) 24 , and < 2 < , then find sin . 25 2 7 2 10 B) - 7 2 10 C) 7 5 D) - 7 5 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Q) 21) The path of a projectile fired at an inclination (in degrees) to the horizontal with an initial speed v0 is a parabola. The range R of the projectile, that is, the horizontal distance that the projectile travels, is found by using the formula 2 v0 R= sin (2 ) g where g is the acceleration due to gravity. The maximum height H of the projectile is 2 v0 H= 4g (1 - cos (2 )) Find the range R and the maximum height H in terms of g if the projectile is fired with an initial speed of 200 meters per second at an angle of 15° and then at an angle of 22.5 °. Do not use a calculator, but simplify the answers. R) Use the information given about the angle to π, 0 β€ π β€ 2π, to find the exact value of the 2 Use Double-angle Formulas to Establish Identities indicated trigonometric functions (use half angle formulae). SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. ! 1. π πππ = , 0 < π < ! ! , find sin . ! ! ! Establish the identity. !! ! , π2u) β€ π , find 2 u (1=+3cos 22) 2. tanπ‘πππ 2u tan . = 1β€- cos ! ! S) The polar coordinates of a point are given. Find the rectangular coordinates of the point. 1 23) a. cos4 x = 8 ( 3 + 4 cos 2x + cos 4x) 2 21) -3, The letters r and represent polar coordinates. Write the equation using rectangular coordinates (x, y). 24) cos 3x 3 = cos3 x - 3 sin2 x cos x 46) r = cos 3 -3 3 3 3 3 3 3 3 -3 b. (-3, -135 °) 3x 232,+sin A) y 22u= cos B) x-2 + ,y 2 = y C) (x ,+ y)2 = x D) (x A) C) D) - + ,y)2 = y 25) c. sin(-3, 4u x 2u = 45°) 2 2 2 2 2 2 2 2 47)T) r =The 1 +=rectangular 2(4sin 26) sin 4x sin x cos x)(1 - 2 sin2 x) of a point are given. Find polar coordinates for the point. coordinates 32 (8, 22)a.-5, A) 0) x + y2 = x2 + y2 + 2y B) x2 + y2 = x2 + y2 + 2x 4 b. ( 3, β1) 1 + y2 The letters represent coordinates. Write the equation coordinates (x, y). y26x) 2y D) x2 + y2using y2 + 2x = x2-+cos +polar = x2 + rectangular 27) sin3C)r3xand 3x)(1 =x2 (sin 5 2 2 -5 2 5 2 5 2 -5 2 5 2 -5 2 -5 2 46) rA)= cos , B) , C) , D) , U) The letters x and y represent rectangular coordinates. Write the2equation 2 2 2 using polar 2 2 48) r = 10 sin2 2 2 2 2 2 2 A) x (r, y =x B) x + y = y C) (x + y) = x D) (x + y) + ΞΈ). coordinates 2 + y 2 = 10x 2 + y2 = 10y 2 + y2 = 10x 2 x2 +2y 2 = 10y A) B) x C) x D) x a. x3 + y - 4x = 0 Page 634 23) 3, 2 47) r= 14y sin4 + 22 = b. x + 4 49) r = 2(sin - cos ) c. xy =1 22-=2x x2 + y2 B) 22++2y2y 2-3 3 2 2 -3 2 B) 2 -3 2 2 2+ 2x x2 + y2 =D) + 2y A) A) x2-3+xy222+=3y2y C) x32 +2y, 23= 2x D) x2x - 2y B) 2x + ,2y = y - x C) , =x-y d. yA) =5 2 , 2 2 2 2 2 2 2 C) x2 + y2 = x2 + y2 Copyright D) x2 + y2 = x2 + y2 + 2x + 2y © 2011 Pearson Education, Inc. 50)V) r =The 5 letters r and ΞΈ represent polar coordinates. Write the equation using rectangular 4x2 + y 2(x, D) x + y = 5 25 B) x2 - y 2 = 25 C) x + y = 25 = y). coordinates 24) 5,A) 48) r-= 310 sin a. 2 2 2 2 5 5x25+ 3y 2 = 10y D) x2 + 5 B) 5 x 3 + y = 10x 5 3 5 C) x + y = 10y 5 3 5 51) r =A) A) , B) , C) , D) , 1 + cos 2 2 2 2 2 2 2 2 2 49) r 2(sin - cos ) B) y2 = 10x - 25 C) x2 = 25 - 10y D) x2 = 10y - 25 b. A)=y = 25 - 10x 25) (-7, 120°) A) x2 + y 2 = 2y - 2x B) 2x2 + 2y2 = y - x C) x2 + y 2 = 2x - 2y 52) r sin 7= 10 -7 3 A) y =, 10 A) 2 2 50) r = 5 7 -7 3 B) x-= 10 , B) 2 2 53) r(1 - 2A) cosx2)+= y12 = 25 26) (-3, -135°) A) x2 + y2 = 1 + 2x 3 2 3 2 A) 5, 2 51) r = 2 B) x2 + y 2 = 1 + 2x 3 2 -3 2 B) , 2 2 27) (5, 270°) A) y2 = 25 - 10x A) (0, -5) B) (-5, 0) 1 + cos 7 7 3 C) C) y2=, 10x 2 C) x + y = 25 B) x2 - y 2 = 25 B) y2 = 10x - 25 C) x2 + y2 = 2 + x -3 2 -3 2 C) , 2 2 C) (0, 5) D) 2x2 + 7 7 3 D) , D) x- =2 10y 2 D) x + y = D) x2 + y 2 = 2 + x -3 2 3 2 D) , 2 2 C) x2 = 25 - 10y D) (5, 0) D) x2 = 1