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# How is Trigonometry Used Practice Test

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By McGraw-Hill Professional
Updated on Oct 3, 2011

Practice quizzes for this test can be found at:

## How is Trigonometry Used Practice Test

You may draw diagrams or use a calculator if necessary. A good score is at least 38 correct.

1. Fill in the blank: “A kilometer is ______ orders of magnitude larger than a millimeter.”

(a) 0

(b) 2

(c) 4

(d) 6

(e) 8

2. Suppose an aircraft is detected on radar at azimuth (or bearing) 90°. It flies on a heading directly north, and continues on that heading. As we watch the aircraft on the screen

(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

3. The period of a sine wave contains

(a) 90° of phase

(b) 180° of phase

(c) 270° of phase

(d) 360° of phase

(e) none of the above

4. The angular diameter of a distant object (in degrees or radians) can be used to determine the distance to that object, if the actual diameter of the object (in linear units such as kilometers) is known. This technique is called

(a) parallax

(b) line sighting

(c) angle sighting

(d) surveying

5. Assume that waves X and Y shown in Fig. Test 2-1 are both sinusoidal. Also assume that the time and amplitude scales are linear. The peak amplitude of wave Y is

(a) four times the peak amplitude of wave X

(b) twice the peak amplitude of wave X

(c) half the peak amplitude of wave X

(d) a quarter of the peak amplitude of wave X

(e) the same as the peak amplitude of wave X

Fig. Test 2-1 . Illustration for Questions 5, 6, and 7 in the test for Part Two.

6. Assume that waves X and F shown in Fig. Test 2-1 are both sinusoidal. Also assume that the time and amplitude scales are linear. The frequency of wave Y is

(a) four times the frequency of wave X

(b) twice the frequency of wave X

(c) half the frequency of wave X

(d) a quarter of the frequency of wave X

(e) the same as the frequency of wave X

7. Assume that waves X and F shown in Fig. Test 2-1 are both sinusoidal. Also assume that the time and amplitude scales are linear. Wave Y

(a) lags wave X by π/2 radians of phase

(c) is in phase coincidence with wave X

(d) is in phase opposition relative to wave X

(e) bears no phase relationship to wave X

8. The number 5.33 × 10 –4 , written out in full, is

(a) 5330000

(b) 53300

(c) 5.33

(d) 0.0533

(e) 0.000533

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

y = sin x

y = –4 cos x

(a) 0°

(b) 45°

(c) 90°

(d) 180°

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

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

y = sin x

y = cos (–4x)

(a) 0°

(b) 45°

(c) 90°

(d) 180°

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

11. What is the square root of 29, truncated (not rounded) to two significant figures?

(a) 5.3

(b) 5.38

(c) 5.4

(d) 5.39

(e) None of the above

12. Imagine two alternating-current, sinusoidal waves X and Y that have the same frequency. Suppose wave X leads wave Y by 300° of phase. The more common way of saying this is

(a) wave X leads wave Y by 60°

(b) wave Y leads wave X by 60°

(c) wave Y lags wave X by 300°

(d) the two waves are in phase coincidence

(e) nothing! The described situation is impossible

13. Suppose a surveyor uses triangulation to measure the distances to various objects. As the distance to an object increases, assuming all other factors remain constant, the absolute error, expressed in meters, of the distance measurement

(a) diminishes

(b) stays the same

(c) increases

(d) cannot be determined

(e) is a meaningless expression

14. What is the product of 5.66 × 10 5 and 1.56999 × 10 –3 , rounded to the justifiable number of significant figures?

(a) 8.88

(b) 888

(c) 8.89

(d) 889

(e) None of the above

15. In some transparent materials, the index of refraction depends on the color of the light. This effect is called

(a) refraction

(b) total internal reflection

(c) declination

(d) dispersion

(e) distortion

16. The critical angle for light rays that strike a boundary between two transparent substances depends on

(a) the ratio of the indices of refraction of the substances

(b) the ratio of the declinations of the substances

(c) the ratio of the distortions of the substances

(d) the ratio of the dispersions of the substances

(e) none of the above

17. Fill in the blank: “The lengths of the sides of any triangle are in a constant ratio relative to the ______ of the angles opposite those sides.”

(a) tangents

(b) sines

(c) cosines

(d) secants

(e) cotangents

18. Figure Test 2-2 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 = 35° and ø = 60°, 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

19. 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. Test 2-2. 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

Fig. Test 2-2 . Illustration for Questions 18, 19, and 20 in the test for Part Two.

Suppose we are told, in addition to all of the above information, that ø = 60° 00′, θ = 35° 00′, and r = 1.880. From this, we can determine that

(a) s = 0.532

(b) s = 1.000

(c) s = 1.245

(d) s = 2.134

(e) none of the above

20. 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. Test 2-2. 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 less than π/2 rad

(b) The angle θ must be less than π/3 rad

(c) The angle θ must be less than 1 rad

(d) The angle θ must be less than π /4 rad

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

21. In the equation θ 1 + θ 2 = arctan x, the variables θ 1 and θ 2 represent

(a) circular functions

(b) tangents

(c) angles

(d) logarithms

(e) hyperbolic functions

22. 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 )

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

(a) All of the incident rays pass through

(b) None of the incident rays pass through

(c) Only the incident rays striking at less than 30° relative to the normal pass through

(d) Only the incident rays striking at more than 30° relative to the normal pass through

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

23. The shortest distance between two points on the surface of a sphere, as determined over the surface (not cutting through the sphere), is known as

(a) an arc of a great circle

(b) a spherical line segment

(c) a linear sphere segment

(d) a spherical angle

(e) a surface route

24. Imagine a point P on a sphere where two different great circles C 1 and C 2 intersect. Now imagine some point Q (other than P ) on great circle C 1 , and some point R (other than P ) on great circle C 2 . The angle QPR, as expressed on the surface of the sphere, is an example of

(a) an acute angle

(b) a triangular angle

(c) an obtuse angle

(d) a circumferential angle

(e) a spherical angle

25. As you drive along a highway, the compass bearings of nearby objects change more rapidly than the compass bearings of distant objects. This is because of

(b) parsec effect

(c) parallax

(d) direction finding

(e) angular error

26. Suppose a ray of light, traveling at first through the air, strikes a flat pane of crown glass having uniform thickness at an angle of 30° relative to the normal. The index of refraction of the glass is 1.33. The ray goes through the glass and emerges into the air again. At what angle relative to the normal does the ray emerge?

(a) 30°

(b) 22°

(c) 42°

(d) The ray does not emerge, but is totally reflected within the glass

27. Suppose a ray of light, traveling at first through the air, strikes a flat pane of flint glass having uniform thickness at an angle of 30° relative to the normal. The index of refraction of the glass is 1.52. The ray goes through the glass and emerges into the air again. At what angle relative to the normal does the ray emerge?

(a) 30°

(b) 19°

(c) 50°

(d) The ray does not emerge, but is totally reflected within the glass

28. The upper equation in Fig. Test 2-3 expresses the spherical law of sines. This equation is useful if

(a) we know the angular lengths of two of the sides of the spherical triangle QRS and the measure of the spherical angle between them, and we want to find the angular length of the third side

(b) we know the measures of all three spherical angles and the angular length of one of the sides of the spherical triangle QRS, and we want to find the angular lengths of the other two sides

(c) we know the angular lengths of all three sides of the spherical triangle QRS, and we want to find the radius of the sphere in linear units

Fig. Test 2-3 . Illustration for Questions 28, 29, and 30 in the test for Part Two.

(d) we know the angular length of one of the sides of the spherical triangle QRS, and we want to find the angular lengths of the other two sides

(e) we know the measure of one of the spherical angles of the spherical triangle QRS, and we want to find the measures of the other two spherical angles

29. The lower equation in Fig. Test 2-3 expresses the spherical law of cosines. This equation is useful if

(a) we know the angular lengths of two of the sides of the spherical triangle QRS and the measure of the spherical angle between them, and we want to find the angular length of the third side

(b) we know the measures of all three spherical angles and the angular length of one of the sides of the spherical triangle QRS, and we want to find the angular lengths of the other two sides

(c) we know the angular lengths of all three sides of the spherical triangle QRS, and we want to find the radius of the sphere in linear units

(d) we know the angular length of one of the sides of the spherical triangle QRS, and we want to find the angular lengths of the other two sides

(e) we know the measure of one of the spherical angles of the spherical triangle QRS, and we want to find the measures of the other two spherical angles

30. The two equations shown in Fig. Test 2-3 approach the laws of sines and cosines for triangles in a flat plane as the points Q, R, and S

(a) become farther and farther apart relative to the size of the sphere

(b) become more and more nearly the vertices of an equilateral spherical triangle

(c) become more and more nearly the vertices of a right spherical triangle

(d) become more and more nearly the vertices of an isosceles spherical triangle

(e) become closer and closer together relative to the size of the sphere

31. Suppose a certain angle is stated as being equal to 10°, plus or minus a measurement error of up to 1.00 minute of arc. What is this error figure, expressed as a percentage?

(a) ±10.0%

(b) ±1.67%

(c) ±1.00%

(d) ±0.167%

32. The absolute accuracy (in fixed units such as meters) with which the distance to an object can be measured using parallax depends on all of the following factors except:

(a) the distance to the object

(b) the length of the observation base line

(c) the size of the object

(d) the precision of the angle-measuring equipment

(e) the distance between the two observation points

33. Imagine a sphere of shiny metal. Imagine a light source is so distant that the rays can be considered parallel. When reflected from the sphere, the light rays

(a) diverge

(b) converge

(c) remain parallel

(d) come to a focus

(e) none of the above

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

(a) 360°

(b) 540°

(c) 630°

(d) 720°

(e) 810°

35. Imagine a spherical triangle with vertices Q, R, and S. Point Q is at the south geographic pole. Point R is on the equator at 30° east longitude. Point S is on the equator at 20° west longitude. What is the measure of ∠ sph RQS ?

(a) 20°

(b) 30°

(c) 50°

(d) 90°

36. Imagine a spherical triangle with vertices Q, R, and S. Point Q is at the south geographic pole. Point R is on the equator at 30° east longitude. Point S is on the equator at 20° west longitude. What is the sum of the measures of the interior spherical angles of Δ sph QRS

(a) 180°

(b) 200°

(c) 210°

(d) 230°

37. 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 )

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

(a) All incident rays pass through

(b) No incident rays pass through

(c) All incident rays pass through, except those striking normal to the boundary

(d) No incident rays pass through, except those striking normal to the boundary

(e) This formula tells us nothing at all if r = s

38. Imagine a spherical quadrilateral with vertices P, Q, R, and S. Point P is at the north geographic pole. Point Q is on the equator at 30° east longitude. Point R is at the south geographic pole. Point S is on the equator at 20° west longitude. What is the sum of the measures of the interior spherical angles of this spherical quadrilateral?

(a) 360°

(b) 410°

(c) 460°

(d) 540°

39. Which of the following must be a great circle on the surface of the earth?

(a) Any circle on the surface that is centered at one of the poles

(b) Any circle on the surface that is centered at a point on the Greenwich meridian

(c) Any circle on the surface that is centered at a point on the equator

(d) Any circle on the surface whose center coincides with the center of the earth

(e) Any circle on the surface that passes through one of the poles

40. Suppose we observe a target on radar, and we see that its range is increasing while its azimuth (or bearing) is not changing. From this, we can conclude that

(a) the target is heading north

(b) the target is heading south

(c) the target is heading east

(d) the target is heading west

(e) none of the above

41. What is the value of 2 3 × 5 + 2 × 3?

(a) 16

(b) 46

(c) 126

(d) 168

(e) This expression is ambiguous

42. Refer to the wave vector diagram of Fig. Test 2-4. Assume waves X and Y have identical frequency, and that the radial scale is linear. Which of the following is apparent?

(a) Waves X and Y differ in peak amplitude by 50°

(b) Waves X and Y are in phase opposition

(c) Wave X lags wave Y by 50° of phase

(d) Wave X leads wave Y by 50° of phase

(e) The phase of wave Y is 4/5 of the phase of wave X

Fig. Test 2-4 . Illustration for Questions 42 and 43 in the test for Part Two .

43. Refer to the wave vector diagram of Fig. Test 2-4. Assume waves X and Y have identical frequency, and that the radial scale is linear. Which of the following is apparent?

(a) The peak amplitude of wave Y is 4/5 of the peak amplitude of wave X

(b) The peak amplitude of wave X is 110°, and the peak amplitude of wave Y is 160°

(c) Wave X travels in a different direction from wave Y

(d) The dot product of X and Y is equal to 0

(e) The cross product of X and Y is the zero vector

44. Imagine two alternating-current, sinusoidal waves X and Y that have the same frequency. Suppose wave X leads wave Y by 2π radians of phase. The more common way of saying this is

(c) wave Y lags wave X by π/2 radians

(d) the two waves are in phase coincidence

(e) nothing! The described situation is impossible

45. Suppose a surveyor measures the distance to an object using triangulation. As the length of the base line increases, assuming all other factors remain constant, the absolute accuracy (as expressed in terms of the maximum possible error in meters) of the distance measurement

(a) improves

(b) stays the same

(c) gets worse

(d) depends on factors not mentioned here

(e) cannot be determined

46. The expression 3.457e–5 is another way of writing

(a) 3457

(b) 3.457

(c) 3.457 × 10 5

(d) 3.457 × 10 –5

(e) none of the above

47. Refer to Fig. Test 2-5. A celestial object, which lies in the plane of the earth’s orbit around the sun, is observed at two intervals three months apart, as shown. The angle θ is measured as 1.000°. Recall that an astronomical unit (AU) is the mean distance of the earth from the sun. The distance to the celestial object, accurate to three significant figures, is

(a) 1.00 AU

(b) 57.3 AU

(c) 100 AU

(d) 360 AU

Fig. Test 2-5 . Illustration for Questions 47, 48, and 49 in the test for Part Two.

48. Suppose a celestial object is observed as in Fig. Test 2-5, and the angle θ for it is determined to be precisely 0° 00′ 01″. The distance to this object, accurate to four significant figures, is

(a) 1000 AU

(b) 100.0 AU

(c) 10.00 AU

(d) 1.000 AU

(e) none of the above

49. Consider the scenario of Fig. Test 2-5 in general, for celestial objects that are many AU away from the earth. The size of the angle θ varies

(a) in direct proportion to the square of the distance to an object

(b) in direct proportion to the distance to an object

(c) inversely as to the distance to the object

(d) inversely as to the square of the distance to the object

(e) none of the above

50. Suppose the measure of a certain angle is stated as (4.66 × 10 6 )°. From this, we can surmise that

(a) the angle represents a tiny fraction of one revolution

(b) the angle represents many revolutions

(c) the sine of the angle is greater than 1

(d) the sine of the angle is less than – 1

(e) the expression contains a typo, because angles cannot be expressed in power-of-10 notation

1. d

2. c

3. d

4. e

5. b

6. e

7. a

8. e

9. c

10. e

11. a

12. b

13. c

14. d

15. d

16. a

17. b

18. a

19. c

20. a

21. c

22. c

23. a

24. e

25. c

26. a

27. a

28. b

29. a

30. e

31. d

32. c

33. a

34. a

35. c

36. d

37. a

38. c

39. d

40. e

41. b

42. c

43. a

44. d

45. a

46. d

47. b

48. e

49. c

50. b

150 Characters allowed