Name: Majority (about 95%) of the final exam will cover material from quarter 2. This final will cover chapters 6-8 and 10-13

Transcription

1 FINAL REVIEW PACKET Name: Date: Period: FINAL REVIEW TOPICS Majority (about 95%) of the final exam will cover material from quarter. This final will cover chapters 6-8 and Topics to focus on: 1. Oblique Triangles law of sines, law of cosines, ambiguous case of law of sines. Graphing Trig Functions period, amplitude, translations 3. Proving Trig Identities 4. Solving Trig Equations 5. Using the Trig Formulas to find trig values half angle, double angle, and angle sum/difference 6. Systems of Equations & Matrices Solving systems by: elimination, substitution, Gauss-Jordan elimination/rref, Cramer s Rule, and inverse matrices. Matrices: addition, subtraction & multiplication, determinants, finding inverses 7. Conic Sections getting them into their standard forms, finding the center, vertices, co-vertices, directrix, radius, eccentricity, asymptotes, foci 8. Probability combinations, permutations, distinguishable permutations, basic probability, binomial probability, AND probability problems, OR probability problems. 9. Sequences and Series formulas, finding arithmetic & geometric terms, finding arithmetic & geometric series, finding infinite geometric series, Pascal s triangle & Binomial theorem, finding specific terms in a binomial expansion. Complete each problem on a separate sheet of paper. Show all of your work. NO WORK = NO CREDIT! Trig Review Sheet #1 For questions 1-4, graph one period of each function. Fully label your axes!! Use color for your final function! 1. f( x) 3sec4( x π ) =. f( x) = 1+ 4sin ( θ + 45 ) 3. f( x) tan3( θ 30 ) = π f( x) = cot x+ 3 Mrs. Nguyen Final Review Packet page 1

2 For questions 5-8, prove each identity sec x sec xtan x 1 sinx = sec x cos x = sec x + 1+ cos x sec x cos x π π 7. tan x+ + 1 = cos xsec x tanx (1 + tan x) tan x= 1 tanx For questions 9-13, solve each equation in the indicated domain. 9. cot x+ cotx= 0, Domain :[0, π ] 10. 4csc θ + 4cscθ + 1= 0, Domain: [0,360 ) cosθ = sin θ, Domain: [ 180,180 ) 1. tan10 θ + tan 50 3 =, Domain: (0,90 ) 1 tan10θ tan sin ( x) cos( x) sin ( x) =, Domain :[0, π ] cos 3θ cos1 sin 3θ sin1 =, Domain: [ 10,10 ) For questions 15-19, find the specified side or angle. Round your answers to decimal places. 15. In HJK, h = 8, j = 6, m K = 17, find k. N 16. In BAD, a =.897, d = 5.91, m B = 119, find b. Baker W E 17. In PEG, p = 1, e = 0, g = 16, find m E. 18. In ABC, m A = 40, m B = 60, a = 4, find c. 150 mi Boat S 19. In ABC, m A= 35, m B= 15, c= 5, find a. Able 0. Coast Guard Station Able is located 150 miles due south of Station Baker. A ship at sea sends an SOS call that is received by each station. The call to Station Able indicates that the ship is located N55 E; the call to Station Baker indicates that the ship is located S60 E. How far is each station from the ship? 1. An aircraft is spotted by two observers, one on each side of the plane, who are 1000 feet apart. As the airplane passes over the line joining them, each observer takes a sighting of the angle of elevation to the plane. Observer A has an angle of elevation of 40 and observer B has an angle of elevation of 35. How high is the plane? Mrs. Nguyen Final Review Packet page

3 Trig Review Sheet # 1. The 6 inch cd rotates at 3600 rpm. What is the linear velocity of the cd at the edge of the cd?. A windmill used to generate electricity has blades that are 10 feet in length. The propeller is rotating at 4 revolutions per second. Find the linear velocity of the tips of the blades. 3. The radius of a wheel rolling on the ground is 80 cm. If the wheel rotates through an angle of 60, how many centimeters does it move? 4. A Ferris wheel has a radius of 5 feet. The wheel is moving at 0 feet per minute. Find the angular velocity in radians per second. 5. A watch contains gears that are intertwined to make the hands move. The larger gear has a diameter of 30 inches and the smaller gear has a diameter of 10 inches. The smaller gear turns 14 radians per second. Find the angular velocity of the teeth on the large gear. 6. The wheels on a bike are connected by a chain that goes around the front and back sprockets. The front sprocket is 40 cm in diameter and the back sprocket is 90 cm in diameter. The diameter of the back wheel is 180 cm in diameter. The back wheel is moving at 18 kilometers per hour. Find the angular velocity of the front sprocket in radians per hour. 7. A telephone pole is 55 feet tall. A guy wire 80 feet long is attached from the ground to the top of the pole. Find the angle between the wire and the pole to the nearest degree. 8. A plane rises from take-off and flies at an angle of 10 with the horizontal runway. When it has gained 500 feet, find the distance, to the nearest foot, the plane has flown. 9. When a six-foot pole casts a four-foot shadow, what is the angle of elevation of the sun? Round to the nearest whole degree. 10. When the angle of elevation of the sun is 6, a telephone pole that is titled at an angle of 8 directly away from the sun casts a shadow 0 feet long. Determine the length of the pole to the nearest tenth of a foot. 11. Closed to tourists since 1990, the Leaning Tower of Pisa in Italy leans at an angle of about 85.7 with the ground. From a point 171 feet from the base of the tower, the angle of elevation to the top is 50. Find the distance, to the nearest tenth of a foot, from the base to the top of the tower. Assume that you are standing at a point where the tower is leaning toward you. Round your answer to the nearest tenth of a foot. 1. Find the distance across a lake from A to C, to the nearest yard, using the facts that the distance from A to B is 140 yards, the distance from B to C is 160 yards, the measure of angle B is 5. Round your answer to the nearest tenth of a yard. For questions 13-16, solve triangle PQR. Round angle measures to the nearest degree and side lengths to decimal places. 13. P = 46, r = 8 and q = r = 3, p = 9, and q = Q = 40, q = 54, and p = P = 50, Q = 33.5 and r = 76. For questions 17-18, find the area of each triangle. Round your answers to decimal places. 17. c = 4 meters, a = 10 meters, T = m = 6 feet, d = 16 feet, and b = 18 feet. Mrs. Nguyen Final Review Packet page 3

4 Systems of Equations & Matrices Final Exam Review Sheet For questions 1-, solve each system by Gauss Jordan Elimination/RREF. 3x+ 3y+ z = 6 8x+ 9y = x + y + 4z = 6x y = 16 x + y + 6z = For questions 3-4, solve using any method. x+ 3y+ z = x+ 8y 6z = 4 4. x+ 4y 3z = 1 3x 3y+ 6z = 6 x + y z = 5 5x 8y+ 13z = 7 For questions 5-7, solve by elimination: x+ y = 7x y = x + 3y 4z = 6 5x+ 6y = 0 5x 6y + z = x+ y = 1 y 5z = 1 x + y + z = 6 For questions 8-11, simplify each expression, if possible given the following matrices A= 6 4 5, B 8 3, C = = A C 9. CA 10. AB 11. 3C + 4A For questions 1-13, solve each system using inverse matrices. 5x 3y+ 5z = 1 3x+ y = x+ y+ 4z = 11 x + 5y = 18 x y+ 3z = 4 For questions 14-15, solve each system using Cramer s Rule. x+ y z = 4 5x 4y = x + 4y z = 6 6x 5y = 1 x + 3y + z = 3 Mrs. Nguyen Final Review Packet page 4

5 Conic Sections Final Exam Review Sheet Write each equation in standard form. Then, find all the important parts to the conic (i.e. center, vertices, co-vertices, directrix, radius, foci, asymptotes, eccentricity) x 9y 3x 18y+ 19 = 0. x + 5y 8x+ 100y+ 91 = 0 4. x + y 6x y 15= x y = y x + 4y + 3x+ 4y 1 = x + y + 8x 6y = 0 x= y 6y x + 9y + 4x+ 7y+ 144 = 0 5x 4y 40x 16y 36 = 0 y = x 4x 1 3 Write the equation of each conic section described. 11. A circle with the endpoints of the diameter at (4, 3) and (, 7). 1. An ellipse with foci of (4, 0) and ( 4, 0) and co-vertices of (0, ) and (0, ). 13. An ellipse with a center of (, 4) and a co-vertex at (, 1) and a vertex at (7, 4). 14. A parabola with a vertex of (, 3) and directrix of y = A parabola with a focus of (5, 6) and directrix of x = A vertical parabola with a vertex of (6, 1) and contains the point (4, 5). 17. A hyperbola with vertices at (4, 3) and (, 3) and foci at ( 3± 10,3). 18. A hyperbola with vertices at (4, -) and (4, -6) and foci at (4, 1) and (4, -9). For each question, write each conic section in its standard form. For parabolas, find the vertex, focus, and equation of the directrix. For ellipses, find the center, vertices, co-vertices, and foci. For hyperbolas, find the center, vertices, foci, and equations of the asymptotes. For circles, find the center and radius x 8x y+ 19= 0 0. x + y x+ 6y = 3. 9x 4y 4y = 7 3x + 6x+ y 6y = 3 4x + 9y + 16x 54y = 61 Mrs. Nguyen Final Review Packet page 5

6 Sequence and Series Final Review Sheet Find the specified term th term of 3, 7, 11,. 1 th term of 1, 3, 9, 7, 3. 9 th term of 0, 15, 10, 5, th term of 1, 1,1,... 4 Find out which term the given number is in the indicated sequence in the arithmetic sequence with a 1 = 7 and d = in the geometric sequence with a 1 = 178 and r = ½ Find S n for the indicated series. 7. S 16 for S 0 for S 3 for a series with a 1 = 17 and d = S 6 for a series with a 1 = 5 and r = S 13 for a series with a 1 = -3 and r = 1.4 A partial sum of a series is given, along with other information. Find the number of terms in the partial sum. 1. Arithmetic series, S n = , a 1 =.8, t 3 = 3.6. Find n 13. Geometric series, S n = , a 1 = 47, r = Find n. Tell whether the geometric series converges. If so, find the value to which it converges. 14. a 1 = 15, r = ½ 15. a 1 = 78, r = -/3 16. a 1 = 95, r = 1. Find each probability. Probability Final Exam Review Sheet 1. Kate has a collection of 45 CDs- 0 R&B and 5 rock. As she is leaving for a trip, she randomly chooses 13 CDs to take with her. What is the probability that she selects 8 R&B and 5 rock?. When 5 coins are tossed, what is the probability that all five are heads? 3. Stephanie has a collection of 17 books- 6 are fiction and 11 are non-fiction. He randomly chooses 7 books to take with him on vacation. What is the probability that he chooses 5 fiction and nonfiction? 4. Sara has 10 blue pens, 5 black pens, and 3 red pens in her desk. If she selects three pens at random with no replacement, what is the probability that she will first select a blue pen, then a black pen, and then another blue pen? 5. Tony, Susan, Carla, and Dustin are four candidates for student council president. If their names are placed in random order on the ballot, what is the probability that Dustin s name will be first on the ballot followed by Carla s name second? Mrs. Nguyen Final Review Packet page 6

7 6. According to the weather reports, the probability of rain on a certain day is 0% in Yellow Falls and 60% in Copper Creek. What is the probability that it will rain in both cities? 7. A contestant on a game show reaches into a container without looking and picks three paper bills. There are 5 $100 bills, 9 $50 bills, 14 $0 bills, and 18 $10 bills. What is the probability that the contestant draws 3 $100 bills one after the other without replacement? 8. Fifteen percent of a batch of toothpaste is defective. Six tubes of toothpaste are selected. a. P(0 defective) b. P(exactly one defective) c. P(at least 3 defective) d. P(less than 3 defective) 8. Two marbles are simultaneously drawn at random from a bag containing 8 red, blue, and 6 green marbles. a. P(at least one red marble) b. P(at least one green marble) c. P(two marbles of the same color) d. P(two marbles of different colors) 9. There are 8 girls and 8 boys on ASB. Five of the students are seniors. What is the probability that a person selected from ASB is not a senior? 10. There are 400 subscribers to an Internet service provider. Of these, 100 own Brand A computers, 500 own Brand B, and 100 own both A and B. What is the probability that a subscriber selected at random owns either Brand A or Brand B? 11. Diana has a stack of playing cards consisting of 10 hearts, 8 spades, and 7 clubs. If she selects a card at random from this stack, what is the probability that it is a heart or a club? 1. Susan is drawing cards from a standard deck. What is the probability that she ll pull 3 red cards and 4 black cards all at the same time? 13. Carla is arranging items for a garage sale. How many ways can she arrange 5 shirts, 8 pairs of pants, 4 pairs of shoes, and 9 skirts if items if each kind (i.e. purses, scarves, hats, and watches) are considered identical? 14. A soccer coach is trying to figure out how many ways he can arrange his players on the field. The coach has 4 forwards, 6 half backs, 8 full backs, and 3 goalies to choose from. Find the number of ways he can choose forwards, 3 half backs, 3 full backs, and 1 goalie. 15. Sandy is arranging books on a shelf. In how many ways can she arrange the 6 books on her shelf? 16. Mary is picking groups of 4 people to serve on a judging panel. In how many ways can she pick these groups if she has fourteen to choose from? 17. How many ways can the letters in the word MATHEMATICS be arranged if the new word must begin with an A and end with a consonant? Mrs. Nguyen Final Review Packet page 7

8 Chapters 6-8 and Final Review Sheet For #1-4, determine the number of triangles (0, 1, or ) that can be formed from the given information. Then solve the triangles. Round angle measures to the nearest whole number and side lengths to decimal places. A calculator may be used on these! 1. a= 8, b= 4, A= 1. a= 7, b= 30, A= a= 9, b=, A= a= 4.6, b= 9., A= 30 For #5-7, find the exact value of each expression using the sum/difference of angles formulas. NO CALCULATOR! 13π 5. cos( 15 ) 6. tan sin 1 1 π For #8-16, find each value if sin A = with < A < π and 13 NO CALCULATOR! B 8. cos A 9. sin B 10. tan 7 sin B = with 3 π < B < π. 5 A B 11. sin 1. cos 13. tan A 14. sin( A+ B) 15. cos( A B) 16. tan( A B) For #17-0, write each equation in standard form. Then, find all the important parts to the conic (i.e. center, vertices, co-vertices, directrix, radius, foci, asymptotes, eccentricity). NO CALCULATOR! x 9y 3x 18y+ 19 = x + 5y 8x+ 100y+ 91 = 0 0. x + y + 8x 6y = 0 x= y 6y+ 1 3 For #1-4, solve each system by the method stated. A calculator may be used! 3x y = 4 1. Gauss-Jordan Elimination (RREF):. Cramer s Rule: 9x + 3y = 1 x y+ 3z = 19 x + 3y z = 10 3x+ 5y+ z = 3 3. Elimination: x+ y z = 5 x+ y+ z = 8 x+ 3y z = 13 x+ 3y = 1 4. Inverse Matrices: 4x+ 6y = 8 Mrs. Nguyen Final Review Packet page 8