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# Trigonometry Demystified Practice Test

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By McGraw-Hill Professional
Updated on Oct 3, 2011

## Trigonometry Demystified Practice Test

You may draw diagrams or use a calculator if necessary. A good score is at least 75 correct.

1. Suppose y = csc x . The domain of this function includes all real numbers except

(a) integral multiples of π /4 rad

(b) integral multiples of π /2 rad

(c) integral multiples of π rad

(d) integral multiples of 2 π rad

(e) integral multiples of 4 π rad

2. The expression sin –1 ( x ) is equivalent to the expression

(a) 1/(sin x )

(b) arcsin x

(c) sin x – 1

(d) sin ( x – 1)

(e) –sin x

3. In Fig. Exam-1, which of the graphs represent functions of x ?

(a) L only

Fig. Exam-1. Illustration for Questions 3, 4, and 5 in the final exam.

(b) L and N

(c) M and N

(d) M only

(e) none of them

4. The coordinate scheme in Fig. Exam-1 is an example of

(a) a polar system

(b) an equilateral system

(c) a logarithmic system

(d) a rectangular system

(e) Euclidean three-space

5. In a coordinate system such as that shown in Fig. Exam-1, how far is the point (3, –4) from the origin?

(a) 3 units

(b) 4 units

(c) –3 units

(d) –4 units

(e) none of the above

6. The cosine of the negative of an angle is equal to the cosine of the angle. The following formula holds for any angle θ :

cos – θ = cos θ

Based on this trigonometric identity, we can conclude that cos 320° is the same as

(a) –cos 320°

(b) –cos 40°

(c) cos 40°

(d) sin 40°

(e) –sin 40°

7. Refer to Fig. Exam-2. The bold, solid curves represent the function y = coth x . The bold, dashed curves represent the inverse of this function, which can be denoted as

(a) y = (coth x ) –1

(b) y = coth –1 x

(c) y = 1/(coth x )

(d) x = arc coth y

(e) none of the above

Fig. Exam-2. Illustration for Questions 7, 8, and 9 in the final exam.

8. Refer to Fig. Exam-2. Which of the following quantities (a), (b), or (c) is in the domain of the function shown by the bold, dashed curves?

(a) x = –1

(b) x = 0

(c) x = 1

(d) All of the above quantities (a), (b), and (c) are in the domain

(e) None of the above quantities (a), (b), or (c) are in the domain

9. Refer to Fig. Exam-2. Which of the following quantities (a), (b), or (c) lies outside the domain of the function shown by the bold, solid curves?

(a) x = –1

(b) x = 0

(c) x = 1

(d) All of the above quantities (a), (b), and (c) are outside the domain

(e) None of the above quantities (a), (b), or (c) are outside the domain

10. The arctangent function, y = arctan x , is defined for

(a) all values of x

(b) x > 0 only

(c) x < 0 only

(d) –2 π < x < 2 π only

(e) –1 < x < 1 only

11. Triangulation using parallax involves the measurement of distance by observing an object from

(a) a single reference point

(b) two reference points that lie on a ray pointing in the direction of the distant object

(c) two reference points that lie on a line perpendicular to a ray pointing in the direction of the distant object

(d) three reference points that lie on a ray pointing in the direction of the distant object

(e) three reference points that lie on a line perpendicular to a ray pointing in the direction of the distant object

12. Suppose you know the lengths of two sides p and q of a triangle, and the measure of the angle θ r between them. Then the length of the third side r is:

r = ( p 2 + q 2 – 2 pq cos θ r ) 1/2

Recall this as the law of cosines. Knowing this, suppose you are at the intersection of two roads. One road runs exactly east/west, and the other runs exactly northeast/southwest. You see a car on one road 500 meters to your southwest, and a car on the other road 700 meters to your east. How far from each other are the cars, as measured along a straight line?

(a) 900 meters

(b) 1.11 kilometers

(c) 1.20 kilometers

(d) 1.41 kilometers

(e) It is impossible to calculate this without more information

13. Refer to Fig. Exam-3. How many orders of magnitude does the horizontal scale encompass?

(a) 3

(b) 7

(c) 10

(d) 21

(e) Infinitely many

Fig. Exam-3. Illustration for Questions 13, 14, and 15 in the final exam.

14. Refer to Fig. Exam-3. How many orders of magnitude does the vertical scale encompass?

(a) 0

(b) 1

(c) 10

(d) 100

(e) Infinitely many

15. In Fig. Exam-3, by how many orders of magnitude (approximately) do the x and y coordinates of point P differ?

(a) 1

(b) 3

(c) 8

(d) Infinitely many

(e) More information is needed to tell

16. In exponential terms, the hyperbolic sine (sinh) function of a variable x can be expressed in the form:

sinh x = (e xe –x )/2

According to this formula, as x becomes larger and larger negatively without limit, what happens to the value of sinh x ?

(a) It becomes larger and larger positively, without limit

(b) It approaches 0 from the positive direction

(c) It becomes larger and larger negatively, without limit

(d) It approaches 0 from the negative direction

(e) It alternates endlessly between negative and positive values

17. In scientific notation, an exponent takes the form of

(a) a subscript

(b) an italicized numeral

(c) a superscript

(d) a boldface quantity

(e) an underlined quantity

18. Suppose we are confronted with the following combination of products and a quotient after having conducted a scientific experiment involving measurements:

(3.55 × 290.992)/(64.24 × 796.66)

How many significant figures can we claim when we calculate the result?

(a) 2

(b) 3

(c) 4

(d) 5

(e) 6

19. If the frequency of a wave is 1000 Hz, then the period of the wave is

(a) 0.001000 second

(b) 0.00628 second

(c) 0.360 second

(d) 1.000 second

(e) impossible to determine without more information

20. The hyperbolic functions are based on certain characteristics of a curve with the equation

(a) x + y = 1

(b) xy = 1

(c) x 2 + y 2 = 1

(d) x 2y 2 = 1

(e) y = x 2 + 2 x + 1

21. Suppose the coordinates of a point in the mathematician’s polar plane are specified as ( θ , r ) = (–π/4,–2). This is equivalent to the coordinates

(a) ( π /4,2)

(b) (3 π /4,2)

(c) (5 π /4,2)

(d) (7 π /4,2)

(e) none of the above

22. Figure Exam-4 illustrates an example of distance measurement by means of

(a) angular deduction

(b) triangulation

(c) the law of sines

(e) parallax comparison

Fig. Exam-4. Illustration for Questions 22, 23, and 24 in the final exam.

23. Approximately what is the distance d in the scenario of Fig. Exam-4?

(a) 8.47 meters

(b) 516 meters

(c) 859 meters

(d) 30.9 kilometers

(e) It is impossible to determine without more information

24. In the scenario of Fig. Exam-4, suppose the distance d doubles, while the human’s height and orientation do not change. Approximately what will be the angular height (or diameter) of the human, as seen from the same point of observation?

(a) 0° 48′ 00″

(b) 0° 24′ 00″

(c) 0° 12′ 00″

(d) 0° 06′ 00″

(e) 0° 03′ 00″

25. Snell’s law is a principle that involves

(a) the behavior of refracted light rays

(b) hyperbolic functions

(c) cylindrical-to-spherical coordinate conversion

(d) Cartesian-to-polar coordinate conversion

(e) wave amplitude versus frequency

26. Fill in the blank to make the following statement the most correct and precise: “In optics, the angle of incidence is usually expressed with respect to a line _______ the surface at the point where reflection takes place.”

(a) parallel to

(b) passing through

(c) normal to

(d) tangent to

(e) that does not intersect

27. Suppose a prism is made out of glass that has an index of refraction of 1.45 at all visible wavelengths. If this prism is placed in a liquid that also has an index of refraction of 1.45 at all visible wavelengths, then

(a) rays of light encountering the prism will behave just as they do when the prism is surrounded by any other transparent substance

(b) rays of light encountering the prism will all be reflected back into the liquid

(c) rays of light encountering the prism will pass straight through it as if it were not there

(d) some of the light entering the prism will be trapped inside by total internal reflection

(e) all of the light entering the prism will be trapped inside by total internal reflection

28. On a radar display, a target appears at azimuth 280°. This is

(a) 10° east of south

(b) 10° west of south

(c) 10° south of west

(d) 10° west of north

(e) none of the above

29. Suppose a pair of tiny, dim stars in mutual orbit, never before seen because we didn’t have powerful enough telescopes, is discovered at a distance of 1 parsec from our Solar System. When the stars are at their maximum angular separation as observed by our telescopes, they are ½ second of arc apart. What is the actual distance between these stars, in astronomical units (AU), when we see them at their maximum angular separation? Remember that an astronomical unit is defined as the mean distance of the earth from the sun.

(b) ¼ AU

(c) ½ AU

(d) 1 AU

(e) 2 AU

30. Suppose two vectors are oriented at a 60° angle relative to each other. The length of vector a is exactly 6 units, and the length of vector b is exactly 2 units. What is the dot product a · b , accurate to three significant figures?

(a) 0.00

(b) 6.00

(c) 10.4

(d) 12.0

31. On a sunny day, your shadow is half as great as your height when the sun is

(a) 15° from the zenith

(b) 45° from the zenith

(c) 60° from the zenith

(d) 75° from the zenith

(e) none of the above

32. When a light ray passes through a boundary from a medium having an index of refraction r into a medium having an index of refraction s , the critical angle, θ c , is given by the formula:

θ = arcsin ( s/r )

What does this formula tell us about rays striking a boundary where r = s /2?

(a) Only those rays striking at an angle of incidence less than 60° pass through

(b) Only those rays striking at an angle of incidence greater than 60° pass through

(c) Only those rays striking at an angle of incidence less than 30° pass through

(d) Only those rays striking at an angle of incidence greater than 30° pass through

(e) The critical angle is not defined if r = s /2

33. A geodesic that circumnavigates a sphere is also called

(a) a spherical circle

(b) a parallel

(c) a meridian

(d) a great circle

(e) a spherical arc

34. The sum of the measures of the interior angles of a spherical pentagon (a five-sided polygon on the surface of a sphere, all of whose sides are geodesic arcs) is always greater than

(a) 540°

(b) 630°

(c) 720°

(d) 810°

(e) 900°

35. What is the shortest possible height for a flat wall mirror that allows a man 180 centimeters tall to see his full reflection?

(a) 180 centimeters

(b) 135 centimeters

(c) 127 centimeters

(d) 90 centimeters

(e) It depends on the distance between the man and the mirror

36. Imagine four distinct points on the earth’s surface. Two of the points are on the Greenwich meridian (longitude 0°) and two of them are at longitude 180°. Suppose each adjacent pair of points is connected by an arc representing the shortest possible path over the earth’s surface. What is the sum of the measures of the interior angles of the resulting spherical quadrilateral?

(a) 360°

(b) 540°

(c) 720°

(e) It cannot be defined

37. Imagine four distinct points on the earth’s surface, all of which lie on the equator. Suppose each adjacent pair of points is connected by an arc representing the shortest possible path over the earth’s surface. What is peculiar about the resulting spherical quadrilateral?

(a) The inside of the quadrilateral can just as well be called the outside, and the outside can just as well be called the inside

(b) All four sides have the same angular length, but all four interior spherical angles have different measures

(c) No two sides can have the same angular length

(d) The interior area of the quadrilateral is greater than the surface area of the earth

(e) The interior area of the quadrilateral cannot be calculated

38. The cotangent of an angle is equal to

(a) the sine divided by the cosine, provided the cosine is not equal to zero

(b) the cosine divided by the sine, provided the sine is not equal to zero

(c) 1 minus the tangent

(d) 90° minus the tangent

(e) the sum of the squares of the sine and the cosine

39. The hyperbolic secant of a quantity x , symbolized sech x , can be defined according to the following formula:

sech x = 2/( e x + e –x )

For which, if any, of the following values of x is this function undefined?

(a) –1 < x < 1

(b) 0 < x < 1

(c) –1 < x < 0

(d) x < 0

(e) None of the above; the function is defined for all real-number values of x

40. Written in scientific notation, the number 255,308 is

(a) 255308

(b) 0.255308 × 10 5

(c) 2.55308 × 10 5

(d) 0.255308 × 10 –5

(e) 2.55308 × 10 –5

41. Figure Exam-5 shows the path of a light ray R , which becomes ray S as it crosses a flat boundary B between media having two different indexes of refraction r and s . Suppose that line N is normal to plane B . Also suppose that line N , ray R , and ray S all intersect plane B at point P . If θ = 55° and ø = 30°, we can conclude that

(a) r > s

(b) r = s

(c) r < s

(d) the illustrated situation is impossible

(e) rays R and S cannot lie in the same plane

Fig. Exam-5. Illustration for Questions 41, 42, and 53 in the final exam.

42. Imagine a light ray R , which becomes ray S as it crosses a flat boundary B between media having two different indexes of refraction r and s , as shown in Fig. Exam-5. Suppose that line N is normal to plane B . Also suppose that line N , ray R , and ray S all intersect plane B at point P . We are given the following equation relating various parameters in this situation:

s sin ø = r sin θ

Suppose we are told, in addition to all of the above information, that θ = 55° 00′, ø = 30° 00′, and r = 1.000. From this, we can determine that

(a) s = 1.638

(b) s = 0.410

(c) s = 1.833

(d) s = 1.000

(e) none of the above

43. Imagine a light ray R , which encounters a flat boundary B between media having two different indexes of refraction r and s , as shown in Fig. Exam-5. Suppose that line N is normal to plane B . Also suppose that line N and ray R intersect plane B at point P . Suppose we are told that r < s . What can we conclude about the angle of incidence θ at which ray R undergoes total internal reflection at the boundary plane B ?

(a) The angle θ must be greater than 0°

(b) The angle θ must be greater than 45°

(c) The angle θ must be less than 90°

(d) The angle θ must be less than 45°

(e) There is no such angle π , because no ray R that strikes B as shown can undergo total internal reflection if r < s

44. Suppose we set off on a bearing of 315° in the navigator’s polar coordinate system. We stay on a straight course. If the starting point is considered the origin, what is the graph of our path in Cartesian coordinates?

(a) y = –x, where x ≤ 0

(b) y = 0, where x ≥ 0

(c) x = 0, where y ≥ 0

(d) y = – x , where x ≥ 0

(e) None of the above

45. What is the angular length of an arc representing the shortest possible distance over the earth’s surface connecting the south geographic pole with the equator?

(a) 0°

(b) 45°

(c) 90°

(d) 135°

(e) It is impossible to answer this without knowing the longitude of the point where the arc intersects the equator

46. Minneapolis, Minnesota is at latitude +45°. What is the angular length of an arc representing the shortest possible distance over the earth’s surface connecting Minneapolis with the south geographic pole?

(a) 0°

(b) 45°

(c) 90°

(d) 135°

(e) It is impossible to answer this without knowing the longitude of Minneapolis

47. When a light ray passes through a boundary from a medium having an index of refraction r into a medium having an index of refraction s , the critical angle, θ c , is given by the formula:

θ c = arcsin ( s/r )

Suppose θ c = 1 rad, and s = 1.225. What is r ?

(a) 0.687

(b) 1.031

(c) 1.456

(e) It is undefined; such a medium cannot exist

48. The equal-angle axes in the mathematician’s polar coordinate system are

(a) rays

(b) spirals

(c) circles

(d) ellipses

(e) hyperbolas

49. The dot product of two vectors that point in opposite directions is

(a) a vector with zero magnitude

(b) a negative real number

(c) a positive real number

(d) a vector perpendicular to the line defined by the two original vectors

(e) a vector parallel to the line defined by the two original vectors

50. The cross product of two vectors that point in opposite directions is

(a) a vector with zero magnitude

(b) a negative real number

(c) a positive real number

(d) a vector perpendicular to the line defined by the two original vectors

(e) a vector parallel to the line defined by the two original vectors

51. What is the phase difference, in radians, between the two waves defined by the following functions:

y = –2 sin x y = 3 sin x

(a) 0

(b) π /4

(c) π /2

(d) π

(e) It is undefined, because the two waves do not have the same frequency

52. What is the phase difference, in radians, between the two waves defined by the following functions:

y = –3 sin x y = 5 cos x

(a) 0

(b) π /4

(c) π /2

(d) π

(e) It is undefined, because the two waves do not have the same frequency

53. What is the phase difference, in radians, between the two waves defined by the following functions:

y = –4 cos x y = –6 sin x

(a) 0

(b) π /4

(c) π /2

(d) π

(e) It is undefined, because the two waves do not have the same frequency

54. Suppose there are two sine waves X and Y . The frequency of wave X is 350 Hz, and the frequency of wave Y is 360 Hz. From this, we know that

(a) wave X leads wave Y by 10° of phase

(b) wave X lags wave Y by 10° of phase

(c) the amplitudes of the waves differ by 10 Hz

(d) the phases of the waves differ by 10 Hz

(e) none of the above

55. Suppose a distant celestial object is observed, and its angular diameter is said to be 0° 0′ 0.5000′ ± 10%. This indicates that the angular diameter is somewhere between

(a) 0° 0′ 0.4000″ and 0° 0′ 0.6000″

(b) 0° 0′ 0.4500″ and 0° 0′ 0.5500″

(c) 0° 0′ 0.4900″ and 0° 0′ 0.5100″

(d) 0° 0′ 0.4950″ and 0° 0′ 0.5050″

(e) 0° 0′ 0.4995″ and 0° 0′ 0.5005″

56. Suppose there are two sine waves X and Y having identical frequency. Suppose that in a vector diagram, the vector for wave X is 80° clockwise from the vector representing wave Y . This means that

(a) wave X leads wave Y by 80°

(b) wave X leads wave Y by 110°

(c) wave X lags wave Y by 80°

(d) wave X lags wave Y by 110°

(e) none of the above

57. In navigator’s polar coordinates, it is important to specify whether 0° refers to magnetic north or geographic north. At a given location on the earth, the difference, as measured in degrees of the compass, between magnetic north and geographic north is called

(a) azimuth imperfection

(b) polar deviation

(c) equatorial inclination

(d) right ascension

(e) declination

58. Refer to Fig. Exam-6. Given that the size of the sphere is constant, the length of arc QR approaches the length of line segment QR as

(a) points Q and R become closer and closer to point P

(b) points Q and R become closer and closer to each other

(c) points Q and R become farther and farther from point P

(d) points Q and R become farther and farther from each other

(e) none of the above

59. Refer to Fig. Exam-6. What is the greatest possible length of line segment QR ?

(a) Half the circumference of the sphere

(b) The circumference of the sphere

(c) Twice the radius of the sphere

(d) The radius of the sphere

(e) None of the above

Fig. Exam-6. Illustration for Questions 58, 59, and 60 in the final exam.

60. Suppose, in the scenario shown by Fig. Exam-6, point Q remains stationary while point R revolves around the great circle, causing the length of arc QR to increase without limit (we allow the arc to represent more than one complete trip around the sphere). As this happens, the length of line segment QR

(a) oscillates between zero and a certain maximum, over and over

(b) increases without limit

(c) reaches a certain maximum and then stays there

(d) becomes impossible to define

(e) none of the above

61. Suppose that the measure of angle θ in Fig. Exam-7 is 27°. Then the measure of ∠ QRP is

(a) 18°

(b) 27°

(c) 63°

(d) 153°

(e) impossible to determine without more information

Fig. Exam-7. Illustration for Questions 61 through 64 in the final exam.

62. In Fig. Exam-7, the ratio e/f represents

(a) cos ø

(b) cos θ

(c) tan ø

(d) tan θ

(e) sec θ

63. In Fig. Exam-7, csc ø is represented by the ratio

(a) d/f

(b) d/e

(c) e/f

(d) f/e

(e) f/d

64. In Fig. Exam-7, which of the following is true?

(a) sin 2 θ + cos 2 θ = 1

(b) sin 2 θ + cos 2 ø = 1

(c) sin 2 θ + cos 2 ø = 0

(d) θø = π /2 rad

(e) None of the above

65. What is the value of arctan (–1) in radians? Consider the range of the arctangent function to be limited to values between, but not including, –π/2 rad and π /2 rad. Do not use a calculator to determine the answer.

(a) –π/3

(b) –π/4

(c) 0

(d) π /4

(e) π /3

66. Suppose a target is detected 10 kilometers east and 13 kilometers north of our position. The azimuth of this target is approximately

(a) 38°

(b) 52°

(c) 128°

(d) 142°

(e) impossible to calculate without more information

67. Suppose a target is detected 20 kilometers west and 48 kilometers south of our position. The distance to this target is approximately

(a) 68 kilometers

(b) 60 kilometers

(c) 56 kilometers

(d) 52 kilometers

(e) impossible to calculate without more information

68. Suppose an airborne target appears on a navigator’s-polar-coordinate radar display at azimuth 270°. The target flies on a heading directly north, and continues on that heading. As we watch the target on the radar display

(a) its azimuth and range both increase

(b) its azimuth increases and its range decreases

(c) its azimuth decreases and its range increases

(d) its azimuth and range both decrease

(e) its azimuth and range both remain constant

69. In 5/8 of an alternating-current wave cycle, there are

(a) 45° of phase

(b) 90° of phase

(c) 135° of phase

(d) 180° of phase

(e) 225° of phase

70. In cylindrical coordinates, the position of a point is specified by

(a) two angles and a distance

(b) two distances and an angle

(c) three distances

(d) three angles

(e) none of the above

71. The expression 3 cos 60° + 2 tan 45°/sin 30° is

(a) ambiguous

(b) equal to 5.5

(c) equal to 7

(d) equal to 27

(e) undefined

72. The sine of an angle can be at most equal to

(a) 1

(b) π

(c) 2 π

(d) 180°

(e) anything! There is no limit to how large the sine of an angle can be

73. Suppose you see a balloon hovering in the sky over a calm ocean. You are told that it is 10 kilometers north of your position, 10 kilometers east of your position, and 10 kilometers above the surface of the ocean.

This information is an example of the position of the balloon expressed in a form of

(a) Cartesian coordinates

(b) cylindrical coordinates

(c) spherical coordinates

(d) celestial coordinates

(e) none of the above

74. In Fig. Exam-8, the frequencies of waves X and Y appear to

(a) differ by a factor of about 2

(b) be about the same

(c) differ by about 180°

(d) differ by about π /2 radians

(e) none of the above

Fig. Exam-8. Illustration for Questions 74, 75, and 76 in the final exam.

75. In Fig. Exam-8, the phases of waves X and Y appear to

(a) differ by a factor of about 2

(b) be about the same

(c) differ by about 180°

(d) differ by about π /2 radians

(e) none of the above

76. In Fig. Exam-8, the amplitudes of waves X and Y appear to

(a) differ by a factor of about 2

(b) be about the same

(c) differ by about 180°

(d) differ by about π /2 radians

(e) none of the above

77. Which, if any, of the following expressions (a), (b), (c), or (d) is undefined?

(a) sin 0°

(b) sin 90°

(c) cos π rad

(d) cos 2 π rad

(e) All of the above expressions are defined

78. As x → 0 + (that is, x approaches 0 from the positive direction), what happens to the value of ln x (the natural logarithm of x )?

(a) It becomes larger and larger positively, without limit

(b) It approaches 0 from the positive direction

(c) It becomes larger and larger negatively, without limit

(d) It approaches 0 from the negative direction

(e) It alternates endlessly between negative and positive values

79. Suppose the measure of a certain angle in mathematician’s polar coordinates is stated as –9.8988 × 10 –75 rad. From this, we can surmise that

(a) the angle is extremely large, and is expressed in a clockwise direction

(b) the angle is extremely large, and is expressed in a counterclockwise direction

(c) the angle is extremely small, and is expressed in a clockwise direction

(d) the angle is extremely small, and is expressed in a counterclockwise direction

(e) the expression contains a typo, because angles cannot be negative

80. The hyperbolic cosine of a quantity x , symbolized cosh x , can be defined according to the following formula:

cosh x = ( e x + e –x )/2

Based on this, what is the value of cosh 0? You should not need a calculator to figure this out.

(a) 0

(b) 1

(c) 2

(d)–1

(e)–2

81. An abscissa is

(a) a coordinate representing a variable

(b) the shortest path between two points

(c) a vector perpendicular to a specified plane

(d) the origin of a coordinate system

(e) the boundary of a coordinate system

82. Suppose you are standing at the north geographic pole. Suppose you fire two guns, call them A and B , simultaneously in horizontal directions, gun A along the Prime Meridian (0° longitude) and gun B along the meridian representing +90° (90° east longitude). Suppose the bullets from both guns travel at 5000 meters per second. Let a be the vector representing the velocity of the bullet from gun A ; let b be the vector representing the velocity of the bullet from gun B . What is the direction of vector a × b the instant after the guns are fired?

(a) +45° (45° east longitude)

(b) Straight up

(c) Straight down

(d) Undefined, because the magnitude of a × b is zero

83. In Fig. Exam-9, θ x, θ y , and θ z,

(a) represent variables in spherical coordinates

(b) represent azimuth, elevation, and declination

(c) are always expressed in a clockwise rotational sense

Fig. Exam-9. Illustration for Questions 83, 84, and 85 in the final exam.

(d) uniquely define the direction of vector a

(e) uniquely define the magnitude of vector a

84. Refer to Fig. Exam-9. Suppose the direction (or orientation) in which vector a points is exactly reversed. What happens to θ x, , θ y , and θ s ?

(a) Their measures all change by 180°

(b) Their measures all remain the same

(c) Their measures are all multiplied by –1

(d) Their measures all increase by π /2 rad

(e) It is impossible to say without more information

85. Refer to Fig. Exam-9. Suppose the values of x a , y a , and z a are all doubled. What happens to θ x, , θ y , and θ s ?

(a) Their measures are all doubled

(b) Their measures all remain the same

(c) Their measures are all quadrupled

(d) Their measures are all divided by 2

(e) It is impossible to say without more information

86. Consider the circle represented by the equation x 2 + y 2 = 9 on the Cartesian plane. Imagine a ray running outward from the origin through a point on the circle where x = y . Consider the angle between the ray and the positive x axis, measured counterclockwise. The tangent of this angle is equal to

(a) x /3

(b) y /3

(c) 1

(d) 0

(e) 3

87. Suppose a computer display has an aspect ratio of 4:3. This means that the width is 4/3 times the height. A diagonal line on this display is slanted at approximately

(a) 30° relative to horizontal

(b) 37° relative to horizontal

(c) 45° relative to horizontal

(d) 53° relative to horizontal

(e) 60° relative to horizontal

88. Suppose a geometric object in the polar coordinate plane is represented by the equation r = –3. The object is

(a) a circle

(b) a hyperbola

(c) a parabola

(d) a straight line

(e) a spiral

89. Suppose a geometric object in the polar coordinate plane is represented by the equation θ = 3π/4. The object is

(a) a circle

(b) a hyperbola

(c) a parabola

(d) a straight line

(e) a spiral

90. The hyperbolic functions are

(a) inverses of the circular functions

(b) negatives of the circular functions

(c) reciprocals of the circular functions

(d) identical with the circular functions

(e) none of the above

91. Refer to Fig. Exam-10. What are the coordinates of point P ? Assume that the curves intersect there.

(a) (–5 π /4,2 1/2 )

(b) (–5 π /4,2 –1/2 )

(c) (–7 π /4,2 1/2 )

(d) (–7 π /4,2 –1/2 )

(e) They cannot be determined without more information

Fig. Exam-10. Illustration for Questions 91, 92, and 93 in the final exam.

92. Refer to Fig. Exam-10. What are the coordinates of point Q ? Assume that the curves intersect there.

(a) (5 π /4,–2 1/2 )

(b) (5 π /4,–2 –1/2 )

(c) (7 π /4,–2 1/2 )

(d) (7 π /4,–2 –1/2 )

(e) They cannot be determined without more information

93. Refer to Fig. Exam-10. By what extent is the cosine wave displaced along the x axis relative to the sine wave?

(a) 180° negatively

(b) 135° negatively

(c) 90° negatively

(d) 45° negatively

94. How many radians are there in an angle representing three-quarters of a circle?

(a) 0.25 π

(b) 0.75 π

(c) π

(d) 1.5 π

(e) This question is meaningless, because the radian is not a unit of angular measure

95. Which of the following functions has a graph that is not sinusoidal?

(a) f ( x ) = 3 sin x

(b) f ( x ) = –2 cos 2 x

(c) f ( x ) = 4 csc 4 x

(d) f ( x ) = 4 cos (–3 x )

(e) f ( x ) = –cos ( πx )

96. Suppose f ( x ) = 3 x + 1. Which of the following statements (a), (b), (c), or (d), if any, is true?

(a) f –1 ( x ) = ( x – 1)/3

(b) f –1 ( x ) = x /3 + 1/3

(c) f –1 ( x ) = –3 x – 1

(d) f –1 ( x ) does not exist; that is, the function f ( x ) = 3 x + 1 has no inverse

(e) None of the above statements (a), (b), (c), or (d) is true

97. Which of the following expressions is undefined?

(a) csc 0°

(b) sec 0°

(c) tan 45°

(d) sin 180°

(e) cot 135°

98. Refer to Fig. Exam-11. If the rectangular coordinates x 0 and y 0 of point P are both doubled, what happens to the value of r 0 ?

(a) It increases by a factor of the square root of 2

(b) It doubles

(d) It does not change

Fig. Exam-11. Illustration for Questions 98, 99, and 100 in the final exam.

99. Refer to Fig. Exam-11. If the rectangular coordinates x 0 and y 0 of point P are both doubled, what happens to the value of θ 0 ?

(a) It increases by a factor of the square root of 2

(b) It doubles

(c) It is multiplied by –1

(d) It does not change

(e) It increases by π rad

100. Refer to Fig. Exam-11. If the rectangular coordinates x 0 and y 0 of point P are both multiplied by –1, what happens to the value of θ 0

(a) It increases by a factor of the square root of 2

(b) It doubles

(c) It is multiplied by –1

(d) It does not change

(e) It increases by π rad

1. c

2. b

3. a

4. d

5. e

6. c

7. b

8. e

9. b

10. a

11. c

12. b

13. c

14. e

15. b

16. c

17. c

18. b

19. a

20. d

21. b

22. d

23. b

24. d

25. a

26. c

27. c

28. e

29. c

30. b

31. e

32. e

33. d

34. a

35. d

36. c

37. a

38. b

39. e

40. c

41. c

42. a

43. e

44. a

45. c

46. d

47. c

48. a

49. b

50. a

51. d

52. c

53. a

54. e

55. b

56. c

57. e

58. b

59. c

60. a

61. c

62. a

63. e

64. a

65. b

66. a

67. d

68. a

69. e

70. b

71. b

72. a

73. a

74. b

75. c

76. a

77. e

78. c

79. c

80. b

81. a

82. b

83. d

84. a

85. b

86. c

87. b

88. a

89. d

90. e

91. d

92. b

93. c

94. d

95. c

96. a

97. a

98. b

99. d

100. e

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