1.1 Angles, Degrees, and Arcs

Transcription

1 MA140 Trig 2015 Homework p. 1 Name: 1.1 Angles, Degrees, and Arcs Find the fraction of a counterclockwise revolution that will form an angle with the indicated number of degrees. 3(a). 45 3(b) (c). 270 Identify the following angles as acute, right, obtuse, or straight. If the angle is none of these, say so Change to decimal degrees accurate to 3 decimal places Change to Degree-Minute-Second (DMS) form:

2 MA140 Trig 2015 Homework p. 2 Name: Find C, θ, or s as indicated on a circle with circumference C, arc length s, and central angle θ. Use the following proportion: s C = θ C = 1, 000 cm, θ = 36, s =? 45. s = 25 km, θ = 20, C =? 49. θ = , s = 50.2 cm, C =? 63. Find the distance between the cities to the nearest mile, ignoring any small difference in longitude. Use r = 3960 mi for the radius of Earth. Dallas at N and Lincoln at N latitude.

3 MA140 Trig 2015 Homework p. 3 Name: 1.2 Similar Triangles Find the indicated lengths, given the triangles abc and a b c are similar. 5. a = 5, b = 15, a = 7. Find b. 7. a = 12, c = 18, a = 2.4. Find c. 15. a = 23.4 m, a = 0.47, b = 1.0, c = 1.1. Find b and c.

4 MA140 Trig 2015 Homework p. 4 Name: 23. Given a right triangle ABC with a point P on AB such that CP AB, identify corresponding angles to explain why triangles ACP and BCP are similar. A P C B 29. Hercomer placed a mirror lying flat on the ground with the center of the mirror 24 feet from the trunk of a tree. He then stood on the opposite side of the mirror and adjusted how close he was to the mirror until he could see the top of the tree in the center of the mirror. He measured his distance to the center of the mirror to be 2.1 feet. If his eyes were 5.5 ft above the ground, find the height of a tree. 31. A 20-ft flagpole casts a 32-ft shadow at the same time as another flagpole casts a 44-ft shadow. How tall is the second flagpole?

5 MA140 Trig 2015 Homework p. 5 Name: 1.3 Right Triangle Trig Functions Supplemental. Fill in the definitions for the six right triangle trig functions: Right Triangle Trig Funcs sin θ cos θ tan θ csc θ sec θ cot θ Find each value to three significant digits using a calculator. 17. cos cot csc sin tan sec Use the inverse function on a calculator: 29. Find θ to the nearest degree if cos θ = Find θ to two decimal places if sin θ =

6 MA140 Trig 2015 Homework p. 6 Name: 35. Find θ to the nearest minute if tan θ = Solve each standard right triangle to two decimal places. α β γ a b c 15.0 mm α β γ a b c 12.8 in α β γ a b c 108 mi 132 mi

7 MA140 Trig 2015 Homework p. 7 Name: 51. When Hercomer was solving a right triangle, he knew that side a should be shorter than side b, but the calculator said the opposite. He doublechecked what he entered, and it looked OK. What is one more thing he should check out? 63. Use the right triangle definitions of the trig functions and similar triangles to verify the geometrical interpretations of sine, tangent, and cosecant on the circle with radius 1 in the figure below: csc cot tan g sin h b o θ a cos t s sec 71. Find the area of the triangle in the figure. Dimensions are in inches. x 12 x 5

8 MA140 Trig 2015 Homework p. 8 Name: 1.4 Right Triangle Applications 1. A ladder 8.0 m long is placed against a building. The angle between the ladder and the level ground is 61. How high will the top of the ladder reach up the building? 3. When the angle of elevation of the sun is 58, the shadow cast by a tree is 28 ft long. How tall is the tree? 9. From a point 21 m away from a tree, the angle of elevation to the top of the tree is 75. How tall is the tree?

9 MA140 Trig 2015 Homework p. 9 Name: 13. From the top of a 70-meter vertical cliff overlooking a lake, the angle of depression to a boat is Find the distance between the boat and the base of the cliff. 17. A glider is flying at an altitude of 8,240 m. The angle of depression to the control tower at an airport is What is the horizontal distance between the glider and the control tower (in km)?

10 MA140 Trig 2015 Homework p. 10 Name: 21. A house is being designed so that the overhang keeps the southern wall of the house shaded from the summer solstice sun (elevation angle 75 ), but lets in plenty of sunshine from the winter solstice sun (elevation angle 27 ). If the bottom of the overhang is 19 ft above the ground, answer the following: How far out from the wall must the overhang extend to keep the wall shaded in at the summer solstice? If the house is built according to the answer for part (a), how much of the top of the wall will be shaded at the winter solstice? 25. A parking space needs to contain a rectangle 8 ft wide and 18 ft long in order to accomodate most vehicles. For diagonal parking, it is made oblique, and it must have an additional right triangle on each end, thus making a parallelogram. If the acute angles of the parallelogram measure 72, how long are the sides of the parallelogram?

11 MA140 Trig 2015 Homework p. 11 Name: 37. Mergatroid was doing some amateur surveying, and found that the angle of elevation to the top of Mt. Peak was 25. He walked 1.0 km straight towards Mt. Peak on level ground, and then measured the angle of elevation again. This time it was 42, but the horizontal distance to Mt. Peak was still unknown (x). Find the height (y) of Mt. Peak above Mergatroid s plains. Mergatroid determined these two equations were valid: tan 42 = y x and tan 25 = y x From the roof of an apartment building, Hercomer finds that the angle of elevation to the top of his office building is Furthermore, the angle of depression to the bottom of the same office building is Knowing that the office building is 847 ft high, determine the distance to the office building and also the height of the apartment building he measured from.

12 MA140 Trig 2015 Homework p. 12 Name: 2.1 Degrees & Radians Mentally estimate the conversions between radians and degrees: to the nearest radian radians to the nearest π 2 radians to the nearest 10. Calculate the exact radian measure for each angle without a calculator: 21(a) (b) (c). 90 Calculate the exact radian measure for each angle without a calculator: 23(a) (b) (c). 315

13 MA140 Trig 2015 Homework p. 13 Name: 25. Calculate the exact degree measure for each angle without a calculator: (a) π 2 (b) π (c) 3π Calculate the exact degree measure for each angle without a calculator: (a) 2π 3 (b) 4π 3 (c) 2π Find the two nearest angles (one positive and one negative) that are coterminal with the given angle: π 6. Convert degrees to radians or radians to degrees. Find the exact value and also rounded to four significant digits π 7

14 MA140 Trig 2015 Homework p. 14 Name: 49. For a circle with radius 5.0 m, find the measure of a central angle subtended by the given arc length. Write the answer in both radians and degrees. (a) 2 m (b) 6 m (c) 12.5 m (d) 20 m 53. For a circle of radius 12.0 cm, find the area of the sector with the given central angle. (a) 2.00 radians (b) 25.0 (c) radians (d) 105

15 MA140 Trig 2015 Homework p. 15 Name: 57. In which quadrant does the terminal side of the angle 17π/6 radians lie? 67. An angle of 3 is inscribed in a circle of radius 85 km. Find the length of the arc it subtends. 73. A clock has a pendulum 22 cm long. If it swings through an angle of 32, how far does the end of the bob travel in one swing? 83. If a spy satellite that can distinguish objects that subtend angles as small as rad, how small of an object can it distinguish from a low earth orbit (an altitude of 400 km)? (Answer in meters.)

16 MA140 Trig 2015 Homework p. 16 Name: 2.2 Linear and Angular Velocity 5. Find the velocity V of a point on the rim of a wheel if r = 12 cm, ω = 0.7 rad/sec. 7. Find the velocity V of a point on the rim of a wheel if r = 1.2 ft, ω = 200 rad/min. 9. Find the angular velocity ω if r = 250 mi and V = 1, 950 mi/hr. 13. Find the angular velocity of a wheel turning 3π radians in hr.

17 MA140 Trig 2015 Homework p. 17 Name: 21. A satellite orbits Earth in a circular orbit at 20,000 mi/hr. If the radius of the orbit is 4,300 mi, find the angular velocity in radians per hour (to three significant digits). 23. The velocity of sound in air is approximately m/s. If the propeller of an airplane has a diameter of m, at what angular velocity will it reach the speed of sound? 25. Earth orbits the sun almost circularly with a radius of mi. What is its angular velocity in radians per hour, and what is its linear velocity to the nearest hundred miles per hour?

18 MA140 Trig 2015 Homework p. 18 Name: 2.3 Trig Functions on Unit Circle Supplemental. Fill in the definitions for the six trig functions on the unit circle: Unit Circle Trig Functions sin θ cos θ tan θ csc θ sec θ cot θ Verify that the point Q lies on the unit circle, then give the exact values of all six trig functions for an angle θ whose terminal side contains point Q. 3. Q = ( 3 5, 4 ) 5 sin θ cos θ tan θ csc θ sec θ cot θ 7. Q = ( 1 2, 3 2 ) sin θ cos θ tan θ csc θ sec θ cot θ

19 MA140 Trig 2015 Homework p. 19 Name: Fill in the definitions for the six trig functions on an arbi- Supplemental. trary circle: Arb. Circle Trig Functions sin θ cos θ tan θ csc θ sec θ cot θ Find the exact values of all six trig functions for an angle θ whose terminal side contains point P. 11. P = (8, 6) sin θ cos θ tan θ csc θ sec θ cot θ 13. P = ( 7, 24) sin θ cos θ tan θ csc θ sec θ cot θ

20 MA140 Trig 2015 Homework p. 20 Name: Find the exact values of the other five trig functions for an angle θ given the following information. 19. cos θ = 13 5, θ is in QIV. sin θ cos θ tan θ csc θ sec θ cot θ 21. tan θ = 3 2, θ is in QIII. sin θ cos θ tan θ csc θ sec θ cot θ 27. csc θ = 25 24, θ is in QIII. sin θ cos θ tan θ csc θ sec θ cot θ

21 MA140 Trig 2015 Homework p. 21 Name: Use a calculator to find the value of the following trig functions to four significant digits. Make sure your calculator is in the correct mode! 31. cos ( 85 ) 33. csc cot Find the exact values of all six trig functions for an angle θ whose terminal side contains the point 3, 1 ( ). sin θ cos θ tan θ csc θ sec θ cot θ

22 MA140 Trig 2015 Homework p. 22 Name: 57. Find the exact values of the other five trig functions for an angle θ given sin θ = 1 2 and tan θ < 0. sin θ cos θ tan θ csc θ sec θ cot θ 81. Light intensity on a flat surface changes with the angle of incidence (how straight the light shines on it) according to I (θ) = k cos θ, where θ is measured as the angle away from perpendicular to the surface, and k is a constant that depends on a given setup. Find the relative intensity in terms of k for these values of θ to 2 decimal places:

23 MA140 Trig 2015 Homework p. 23 Name: 2.5 Exact Values & Props of Trig Functions 9. Sketch θ and find the reference angle ˆθ if θ = π Sketch θ and find the reference angle ˆθ if θ = Find the exact value of sin π 2 without using a calculator. 31. Find the exact value of cos 3π 4 without using a calculator.

24 MA140 Trig 2015 Homework p. 24 Name: 3.1 Basic Graphs 1 & 2. What is the period of each of the six unmodified trigonometric functions? sin θ cos θ tan θ csc θ sec θ cot θ 7. How far does the graph of each of the six unmodified trigonometric functions deviate from the x-axis? sin θ cos θ tan θ csc θ sec θ cot θ Find all the x-intercepts for each function. f (x) All roots k Z on [ 2π, 2π] sin x cos x tan x cot x sec x csc x

25 MA140 Trig 2015 Homework p. 25 Name: 15. For what values of x is each function undefined? f (x) All bads k Z on [ 2π, 2π] sin x cos x tan x cot x sec x csc x 27. Find the max and min (if they exist) of f (x) = 5 cos x. 29. Find the max and min (if they exist) of f (x) = 9 + sin x.

26 MA140 Trig 2015 Homework p. 26 Name: 3.2 Scaling and Frequency State the amplitude A and period T for each equation, then graph one period of it. If the graph is flipped vertically, state the amplitude as a negative value. Label the maximum and minimum values on the graph as well as the left and right boundaries of the period. In this section, ϕ and V are always 0. Here is the setup: A: T: ϕ: V: A 0 A 0 T 7. y = 2 sin x A: T: ϕ: V: 9. y = 1 2 sin x A: T: ϕ: V:

27 MA140 Trig 2015 Homework p. 27 Name: 13. y = sin x 4 A: T: ϕ: V: 15. y = 2 sin 4x A: T: ϕ: V: 19. y = 1 4 sin x 2 A: T: ϕ: V:

28 MA140 Trig 2015 Homework p. 28 Name: 29. y = sin x 2 A: T: ϕ: V: Find the equation of the form y = A sin 2π T x that produces the given graph

29 MA140 Trig 2015 Homework p. 29 Name: 3.3 Phase Shift In this section, we will use the whole setup with ϕ and V not necessarily equal to 0. The full setup is like this: A: T: V + A V ϕ: V: V A ϕ ϕ + T Find the amplitude, period, phase shift, and vertical shift of the function, then graph the function over one period. 7. y = 5 sin (2x + 7π) A: T: ϕ: V: 9. y = 2 sin (3x + 18) + 1 A: T: ϕ: V:

30 MA140 Trig 2015 Homework p. 30 Name: 13. y = 8 10 sin (2πx + π) A: T: ϕ: V: ( 39. y = 2 sin 3x π ) 2 A: T: ϕ: V: 47. Using a graphing calculator and decimal approximations as needed, find the equivalent function in the form y = A sin [ 2π T (x φ) ] + V for the sum of functions y = sin x + 3 cos x.

31 MA140 Trig 2015 Homework p. 31 Name: 4.1 Fundamental Identities Using fundamental identities, find the remaining exact values of the remaining trig functions using the given information: 5. sin x = 2 5 and cos x = 1 5 sin x cos x tan x csc x sec x cot x 7. cos x = 1 10 and csc x = 10 3 sin x cos x tan x csc x sec x cot x 9. tan x = 1 15 and sec x = 4 15 sin x cos x tan x csc x sec x cot x

32 MA140 Trig 2015 Homework p. 32 Name: Simplify using fundamental identities 11. tan u cot u 13. tan x csc x 17. sin 2 θ cos θ + cos θ

33 MA140 Trig 2015 Homework p. 33 Name: sin 2 β Show that the equation is not an identity by finding one counterexample (any single value of x for which the equation is not true). cos (πx) = π cos x 33. Using fundamental identities, find the remaining exact values of the remaining trig functions given tan x = 1 and sin x > 0. 2 sin x cos x tan x csc x sec x cot x 41. Use the given value and properties of sine and cosine functions to answer the following without using a calculator. (a) Given sin (x) = , find sin ( x) without finding x or using a calculator. (b) Given cos (x) = , find cos ( x) without finding x or using a calculator.

34 MA140 Trig 2015 Homework p. 34 Name: Express the following expressions in terms of sines and cosines and simplify. 47. cot ( θ) csc θ + cos θ 51. sec w csc w sec w sin w

35 MA140 Trig 2015 Homework p. 35 Name: 4.2 Verifying Trig Identities 5. Verify the identity tan x sec x = sin x 9. Verify the identity csc x cot x = sec x 13. Show that the equation is not an identity by finding one counterexample (any single value of x for which the equation is not true). csc x sec x = tan x 19. Verify the identity sin α cos α tan α = 1

36 MA140 Trig 2015 Homework p. 36 Name: 21. Verify the identity cos β sec β tan β = cot β 31. Verify the identity ( sin 2 x ) ( 1 + cot 2 x ) = Verify the identity 1 (cos θ sin θ)2 cos θ = 2 sin θ

37 MA140 Trig 2015 Homework p. 37 Name: 57. Verify the identity 2 cos2 θ sin θ = csc θ + sin θ 73. Verify the identity cos2 n 3 cos n + 2 sin 2 n = 2 cos n 1 + cos n 93. Verify the identity cot β csc β + 1 = csc β 1 cot β

38 MA140 Trig 2015 Homework p. 38 Name: 4.3 Sum, Difference, and Cofunction Identities 13. Use difference of angle identities to verify the cofunction identity sin (π/2 x) = cos x Use sum or difference of angle identities to find exact values. Be careful about the quadrant! It is very easy to mess up the algebraic sign. 25. tan cos 102 cos 12 + sin 102 sin 12

39 MA140 Trig 2015 Homework p. 39 Name: 35. tan (π/24) + tan (5π/24) 1 tan (π/24) tan (5π/24) 57. Verify the identity cot (x y) = cot x cot y + 1 cot y cot x

40 MA140 Trig 2015 Homework p. 40 Name: 61. Verify the identity tan α + tan β tan α tan β = sin (α + β) sin (α β) 79. Verify the identity 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 z

41 MA140 Trig 2015 Homework p. 41 Name: 4.4 Double-Angle and Half-Angle Identities 1. sin 2x 2 sin x = which one of the six trig functions? 7. Use a half-angle identity to find the exact value of sin 105. No calculator, no decimal approximations. 15. Verify the identity: sin 2x = (tan x) (1 + cos 2x)

42 MA140 Trig 2015 Homework p. 42 Name: 23. Verify the identity: cot α 2 = 1 + cos α sin α 25. Verify the identity: cos 2t 1 sin 2t = 1 + tan t 1 tan t

43 MA140 Trig 2015 Homework p. 43 Name: 45. Find the exact values of sin 2x, cos 2x, and tan 2x given that sin x = 7 25 and π 2 < x < π. sin 2x cos 2x tan 2x 63. Find the exact values of sin x, cos x, and tan x given that sin 2x = and 0 < x < π 4. sin x cos x tan x

44 MA140 Trig 2015 Homework p. 44 Name: 4.5 Product-Sum and Sum-Product Identities Write the product as a sum or difference involving sines and cosines. 1. cos 4w cos w 5. sin 2B cos 5B Write the sum or difference as a product involving sines and cosines. 9. cos 5θ + cos 3θ 13. sin 3B + sin 5B

45 MA140 Trig 2015 Homework p. 45 Name: 31. Verify the identity sin x + sin y cos x + cos y = tan x + y Verify the identity sin x + sin y sin x sin y = tan 2 1 (x + y) tan 2 1 (x y)

46 MA140 Trig 2015 Homework p. 46 Name: 5.1 Inverse Sin, Cos, and Tan Find exact real values without using a calculator: 11. sin cos tan cos Evaluate to four significant digits: tan Find the exact real number values without using a calculator: ( 27. sin 1 1 ) tan 1 ( 1) ) sin ( 1 2 ( ) 33. sin sin 1 ( 0.6) 51. Find the exact degree measure without using a calculator: θ = sin 1 (tan 45 ) θ = sin 1 (tan 60 )

47 MA140 Trig 2015 Homework p. 47 Name: 57. Find the degree measure to four significant digits: θ = sin 1 ( ) 65. Use [ identities to find exact real number values without using a calculator: sin cos 1 1 ] 2 + sin 1 ( 1) Write each expression free of trig or inverse trig functions: 75. sin ( cos 1 x ), x [ 1, 1] ( ) 77. tan sin 1 x, x [ 1, 1]

48 MA140 Trig 2015 Homework p. 48 Name: 5.3 Trig Equations: An Algebraic Approach Find exact solutions in radians on the interval indicated. 5. cos x = 3/2, x R 9. sin x = 1, x R cos x + 1 = 0, x [0, 2π)

49 MA140 Trig 2015 Homework p. 49 Name: sin θ 1 = 0, x [0, 2π) cos x 6 = 0, x R Find exact solutions if possible, or to four decimal places otherwise. Restrict answers to the interval indicated. Give answers in radians cos x 2 = 0, x [0, 2π)

50 MA140 Trig 2015 Homework p. 50 Name: 31. tan 2 x 1 = 0, [0, 2π) cos x 2 = cos x + 1, all x R 67. A ferris wheel with a radius of 60 ft rotates counterclockwise at 5 revolutions per minute. At its lowest point, each seat is 3 ft from the ground. The height of a seat above the ground (starting at the lowest point at t = 0) after t minutes is given by y = cos (10πt). Find all times in the first half minute when the seat is 80 ft above the ground. Round to the nearest tenth of a second.

51 MA140 Trig 2015 Homework p. 51 Name: 5.4 Trig Equations: A Graphing Calculator Approach 5. Solve to four decimal places for all x: 2x = cos x 13. Solve to four decimal places for x in [0, 2π): cos 2x + 10 cos x = Solve to four significant digits for x (0.006, 0.007): 2 cos 1 x = 950x 4

52 MA140 Trig 2015 Homework p. 52 Name: 6.1 Law of Sines Determine whether the law of sines can be used to solve the given triangle. Do not solve. α β γ a b c 5 in 7 in α β γ 7. a b c 9 mm 10 mm 11 mm Solve each triangle with the given measurements (to 2 decimal places). α β γ a b c 92 cm

53 MA140 Trig 2015 Homework p. 53 Name: α β γ a b c 18.3 cm Determine if the given information allows you to solve for 0, 1, or 2 triangles. Do not solve completely. α β γ a b c 5 ft 4 ft α β γ a b c 3 in 8 in

54 MA140 Trig 2015 Homework p. 54 Name: α β γ a b c 5 cm 6 cm α β γ a b c 3 mm 2 mm α β γ a b c 2 in 3 in

55 MA140 Trig 2015 Homework p. 55 Name: Given the measurements in each table, solve the oblique triangle to 2 decimal places. If there are two solutions, give both of them. If there is only one solution, cross out the table marked Second solution. If there are no solutions, cross out both tables. 39. First solution (if it exists) α β γ a b c Second solution (if it exists) α β γ a b c 57 m B β 57 m c a A α b γ C 43. First solution (if it exists) α β γ 57 a b c 47 ft 62 ft B β Second solution (if it exists) α β γ 57 a b c 47 ft 62 ft c a A α b γ C

56 MA140 Trig 2015 Homework p. 56 Name: Apply the law of sines to answer the following two questions. 59. A forest fire is spotted from two lookout stations, A and B, which are 10.3 miles apart. Using the line connecting A and B as the reference for the initial side, the angle to the fire from A is 25.3, and from B is How far away is the fire from station A? How far from station B? B β c a A α b γ C 65. Hercomer wants to find the height of a mountain peak above a level plain. He starts a point A far away from the peak and finds the angle of elevation at point A is He then walks 1,850 feet straight toward the peak and measures the angle of elevation at point B, which turns out to be How high is the peak above the plain?

57 MA140 Trig 2015 Homework p. 57 Name: 6.2 Law of Cosines Determine which of the law of sines or the law of cosines should be used to solve the given triangle. Do not solve. α β γ 5. a b c 5 ft 6 ft 7 ft α β γ a b c 8 in 11 in α β γ a b c 9 ft 12 ft

58 MA140 Trig 2015 Homework p. 58 Name: Solve each triangle to two decimal places using the law of cosines. α β γ a b c 7.03 mm 7.00 mm α β γ 21. a b c 9 yd 6 yd 10 yd

59 MA140 Trig 2015 Homework p. 59 Name: Solve each triangle using the law of sines or the law of cosines as appropriate. 25. First solution (if it exists) α β γ a b c 67.6 ft Second solution (if it exists) α β γ a b c 67.6 ft 29. First solution (if it exists) α β γ a b c 12.3 cm Second solution (if it exists) α β γ a b c 12.3 cm

60 MA140 Trig 2015 Homework p. 60 Name: 33. First solution (if it exists) α β γ 80.3 a b c Second solution (if it exists) α β γ 80.3 a b c 14.5 mm 10 mm 14.5 mm 10 mm First solution (if it exists) α β γ Second solution (if it exists) α β γ 37. a b c a b c

61 MA140 Trig 2015 Homework p. 61 Name: 6.3 Areas of Triangles 5. Find the area of the standard triangle ABC to 2 decimal places using the given information. h = 12.0 m, c = 17.0 m (h is an altitude perpendicular to c.) B β c h a A α b γ C α β γ a b c 6 cm 8 cm B β c a A α b γ C

62 MA140 Trig 2015 Homework p. 62 Name: α β γ a b c B β 4.5ft c a A α b γ C α β γ a b c 3.2 cm 4.5 cm B β c a A α b γ C

63 MA140 Trig 2015 Homework p. 63 Name: 31. Find the area (to two decimal places) of a regular pentagon with a perimeter of 35 ft.

64 MA140 Trig 2015 Homework p. 64 Name: 6.4 Vectors Represent the vector AB in the form a, b. 9. A = (3, 4); B = (6, 12) 11. A = ( 5, 6); B = ( 10, 2) Find the magnitude of the vector a, b , , For the vectors u = 1, 4, v = 3, 2, and w = 0, 4, find the resultant for each vector operation: a) u + v b) u v c) 2 u 3 v d) 3 u v + 2 w

65 MA140 Trig 2015 Homework p. 65 Name: 25. Form a unit vector u in the same direction as v = 3, Form a unit vector u in the opposite direction as v = 3, 4 Express vector v as a linear combination of unit vectors i and j. 35. v = 4, v = AB; A = (1, 6); B = ( 2, 13) 77. Two tugboat pilots are trying to free a barge that is stuck on a shoal. Tugboat 1 is pulling in one direction with a force of magnitude 1,500 lb. Tugboat 2 is pulling 32 to the right of tugboat 1 s direction with a force of magnitude 1,100 lb. Find the magnitude of the resultant force, and its angle relative to the direction of tugboat 1.

66 MA140 Trig 2015 Homework p. 66 Name: 83. Miss Ambulator is a tightrope walker who weighs 112 lbs, and when she is standing at a certain point between the support poles for the rope, the part of the rope behind her is deflected 4.2 from horizontal. The part of the rope in front of her is deflected 5.3 from horizontal. Ignoring the weight of the rope itself, find the tension in each part of the rope to the nearest pound.

67 MA140 Trig 2015 Homework p. 67 Name: 6.5 Dot Products Find the dot product: 5. 5, 3 2, i 3 j Find the angle (to 0.1 ) between the vectors. 17. u = 0, 1, v = 5, u = 2, 9, v = 2, Polar and Rectangular Coordinates 5. Plot in a polar coordinate system: A = (8, 0 ), B = (5, 90 ), and C = (6, 30 ).

68 MA140 Trig 2015 Homework p. 68 Name: 9. Plot in a polar coordinate system: A = (5, 30 ), B = (4, 45 ), and C = (9, 90 ). Convert the given polar coordinates to exact rectangular coordinates: 23. (5, π) 25. ( 3 2, π ) ( 6, 2π 3 )

69 MA140 Trig 2015 Homework p. 69 Name: Convert the given rectangular coordinates to exact polar coordinates: 31. (25, 0) 35. ( 2, ) 2 Change the given equation to polar form: 53. 6x x 2 = y x = 3 Change the given equation to rectangular form: 67. r (2 cos θ + sin θ) = r 2 cos 2θ = 4

70 MA140 Trig 2015 Homework p. 70 Name: 7.2 Sketching Polar Equations Sketch using rapid polar sketching: 19. r = 4 cos θ 23. r = 6 sin 3θ 27. r = cos θ

71 MA140 Trig 2015 Homework p. 71 Name: 7.3 The Complex Plane Plot each set of complex numbers in a complex plane: 9. A = 4 + i; B = 2 + 3i; C = A = 5e (270 )i ; B = 4e (60 )i ; C = 8e (150 )i Convert from rectangular form to exact polar form with r 0 and π < θ π i i 3

72 MA140 Trig 2015 Homework p. 72 Name: Convert from polar form to exact rectangular form e πi e ( 2π/3)i Find z 1 z 2 and z 1 z 2. Leave answers in polar form. 55. z 1 = 4e (25 )i ; z 2 = 8e (12 )i 61. z 1 = 4e ( π/3)i ; z 2 = 5e ( π/4)i 65. Find the powers algebraically and write the answer in rectangular form. Then convert the number to polar form and find the powers again in polar form with θ in degrees. n n 2 n 3 Rectangular Form 1 + i Polar Form