Ch 7 & 8 Exam Review. Note: This is only a sample. Anything covered in class or homework may appear on the exam.

Transcription

1 Ch 7 & 8 Exam Review Note: This is only a sample. Anything covered in class or homework may appear on the exam. Determine whether there is sufficient information for solving a triangle, with the given combination of angles and sides, by the law of sines. 1) A, c, and a 2) a, b, and C Solve the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate. 3) 30 m 4) B = 53.2 C = b = 31.5 Find the area of triangle ABC with the given parts. Round to the nearest tenth when necessary. 5) A = 39.7 b = 11.8 in. c = 6.3 in. Solve the problem. 6) An airplane is sighted at the same time by two ground observers who are 2 miles apart and both directly west of the airplane. They report the angles of elevation as 12 and 21. How high is the airplane? Round to the nearest hundredth of a mile. Find the missing parts of the triangle. 7) 6 12 If necessary, round angles to the nearest degree and give exact values of side lengths. 8) B = 40.1 b = 24.1 in. c = 19.4 in. If necessary, round angles and side lengths to the nearest tenth.

2 9) B = 29.5 b = a = If necessary, round angles to the nearest tenth and side lengths to the nearest hundredth. 10) A = 78 a = 36 yd b = 73 yd If necessary, round angles to the nearest degree and side lengths to the nearest yard. Solve the problem. 11) A ship sailing parallel to shore sights a lighthouse at an angle of 15 from its direction of travel. After traveling 3 miles farther, the angle is 21. At that time, how far is the ship from the lighthouse? If necessary, round to the nearest hundredth of a mile. Assume a triangle ABC has standard labeling. Determine whether SAA, ASA, SSA, SAS, or SSS is given. Then decide whether the law of sines or the law of cosines should be used to begin solving the triangle. 12) a, b, and C Find the indicated angle or side. Give an exact answer. 13) Find the exact length of side a. 3 2 Assume a triangle ABC has standard labeling. Determine whether SAA, ASA, SSA, SAS, or SSS is given. Then decide whether the law of sines or the law of cosines should be used to begin solving the triangle. 14) b, c, and C Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate. 15) C = a = 7.5 m b = 9.6 m Find the area of triangle ABC with the given parts. Round to the nearest tenth when necessary. 16) a = 46 ft b = 54 ft c = 62 ft Solve the problem. 17) Two ships leave a harbor together traveling on courses that have an angle of 129 between them. If they each travel 505 miles, how far apart are they (to the nearest mile)?

3 Draw a sketch to represent the vector. Refer to the vectors pictured here. 18) 3d 19) d - a Sketch the vectors u and w with angle between them and sketch the resultant. 20) u = 5, w = 5, = 45 Two forces act at a point in the plane. The angle between the two forces is given. Find the magnitude of the resultant force. 21) forces of 59 and 81 newtons, forming an angle of 90 (round to the nearest newton) Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the figure. Round to one decimal place. 22) Solve the problem. 23) Two forces of 425 newtons and 267 newtons act at a point. The resultant force is 507 newtons. Find the angle between the forces. 24) A box weighing 64 lb is hanging from the end of a rope. The box is pulled sideways by a horizontal rope with a force of 27 lb. What angle, to the nearest degree, does the first rope make with the vertical? Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in [0,360 ]. 25) -16, 0 26) -4, -3 Vector v has the given magnitude and direction. Find the horizontal or vertical component of v, as indicated, if is the direction angle of v from the horizontal. Round to the nearest tenth when necessary. 27) = 23.7, v = 751; Find the vertical component of v.

4 Write the vector in the form <a, b>. If necessary, round values to the nearest hundredth. 28) Find the indicated vector. 29) Let u = 3i, v = i + j. Find 4u - v. 30) Let u = 5, 2. Find -9u. Find the component form of the indicated vector. 31) Let u = 9, -3, v = -1, 8. Find -6u + v. Write the vector in the form ai + bj. 32) -5, 6 Find the dot product for the pair of vectors. 33) 5, -13, 15, 15 34) i + 3j, 8i - j Find the angle between the pair of vectors to the nearest tenth of a degree. 35) 6i + 3j, -2i + 3j 36) 7i - 5j, 2i - 7j Solve the problem. 37) If u = -5, 7, v = -1, 6, and w = -11, 2, evaluate u (v + w). Determine whether the pair of vectors is orthogonal. 38) 3, 6, -6, 3 39) -8, -1, -8, 32 Write the number as the product of a real number and i. 40)

5 41) -425 Solve the quadratic equation and express all nonreal complex solutions in terms of i. 42) x = 0 43) x = 5x Perform the indicated operations. Simplify the answer. 44) Write the number in standard form a + bi. 45) Find the quotient. Write the answer in standard form. 46) 9 + 2i 2-5i 47) 1 + 3i 5 + 3i Perform the indicated operation. Write the result in standard form. 48) (-4-8i) + (9 + 5i) Find the product. Write the answer in standard form. 49) (3-7i)(2-5i) Perform the indicated operation. Write the result in standard form. 50) (9 + 8i) - (-6 + i) Find the product. Write the answer in standard form. 51) (6-5i) 2 Simplify the power of i. 52) i 73 53) 1 i -9 54) 1 i 35

6 Graph the complex number. 55) -3-6i 56) + i Find sum of the pair of complex numbers and check graphically. 57) + 6i, -4 Write the complex number in rectangular form. 58) 4(cos 5 + i sin 5 ) 59) 6(cos i sin 330 ) 60) 9 cis 240 Write the complex number in trigonometric form r(cos + i sin ), with in the interval [0, 360 ). 61) i 62) i

7 Plot the point. 63) 2, ) -2, -3 4 For the given rectangular equation, give its equivalent polar equation. 65) x - y = 15 66) x 2 + y 2 = 81 Find an equivalent equation in rectangular coordinates. 67) r = cos 68) r(cos - sin ) = 3 69) r = 5 6 sin + 4 cos

8 The graph of a polar equation is given. Select the polar equation for the graph. 70) A) r = 4 B) r = 8 cos C) r sin = 4 D) r = 8 sin Graph the polar equation for in [0, 360 ). 71) r = 5 sin 2 72) r = 8 cos 7 Find the polar coordinates of the point(s) of intersection of the given curves for 0 < 2. 73) r = 7, r = 7 + sin

9 74) r = 3 + cos, r = 3 - sin Considering the given value of t, choose the ordered pair that lies on the graph of the given pair of parametric equations. 75) x = -4t + 8, y = 3t - 3; t = -7 A) (-3, -7) B) (4, 0) C) (20, -18) D) (36, -24) 76) x = t 2-6, y = t 2 + 1; t = 3 A) (15, 8) B) (3, 10) C) (10, 3) D) (, 15) Use a table of values to graph the plane curve defined by the following parametric equations. Find a rectangular equation for the curve. 77) x = 3t, y = t + 1, for t in [-2, 3] 78) x = t, y = 4t + 3, for t in [0, 4] Give two parametric representations for the equation of the parabola. 79) y = (x - 5)

10 Answer Key Testname: CH 7 & 8 EXAM REVIEW 1) Yes 2) No 3) C = 103, a = 13.5 m, b = 23.9 m 4) A = 26.2, a = 17.4, c = ) 23.7 in.2 6) 0.95 mi 7) B = 60, C = 90, b = 6 3 8) A = 108.7, C = 31.2, a= 35.4 in. 9) A1 = 30, C1 = 120.5, c1 = 34.03; A2 = 150, C2 = 0.5, c2 = ) no such triangle 11) 7.43 mi 12) SAS; law of cosines 13) 13 14) SSA; law of sines 15) c = 14.4 m, A = 28.4, B = 38 16) 1206 ft 2 17) 912 mi 18) 19) 20) 21) 100 newtons 22) lb 23) ) 23 25) 16; ) 5; ) ) 4.7, ) 11i - j 30) -45, ) -55, 26 32) -5i + 6j 33) ) 5 35) ) ) ) Yes 39) No 40) -11i

11 Answer Key Testname: CH 7 & 8 EXAM REVIEW 41) 5i 17 42) {±4i 5} 43) 5 2 ± 115 i 2 44) ) 2 + i ) i 47) i 48) 5-3i 49) i 50) i 51) 11-60i 52) i 53) 54) i 55) 56) 57) i 58) i 59) 3 3-3i 60) i ) 10(cos 30 + i sin 30 )

12 Answer Key Testname: CH 7 & 8 EXAM REVIEW 62) 15(cos i sin ) 63) 64) 65) r = 66) r = 9 15 cos - sin 67) x2 + y2 = x 68) x - y = 3 69) 6y + 4x = 5 70) A 71)

13 Answer Key Testname: CH 7 & 8 EXAM REVIEW 72) 73) (7, 0), (7, ) 74) 3-1 2, 3 4, , ) D 76) B 77) 78) y = 1 x + 1, for x in [-6, 9] 3 y = 4x2 + 3, for x in [0, 2] 79) x = t, y = (t - 5) for t in (, ); x = t + 5, y = t for t in (, )