Practice quizzes for this test can be found at:

- The Circle Model Practice Test
- Trigonometric Functions Practice Test
- Graphs and Inverses Practice Test
- Hyperbolic Functions Practice Test
- Polar Coordinates Practice Test
- Three-Space and Vectors Practice Test

**What is Trigonometry Practice Test**

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

1. With respect to the circular functions, an angle whose measure is equal to –45° is the same as an angle whose measure is

(a) 45°

(b) 135°

(c) 225°

(d) 315°

(e) undefined

2. Suppose there is a triangle whose sides are 5, 12, and 13 units, respectively. What is the sine of the angle opposite the side that measures 13 units, accurate to three decimal places?

(a) 0.385

(b) 0.417

(c) 0.923

(d) 1.000

(e) It cannot be determined without more information

3. What is the arcsine of 3?

(a) 30°

(b) 60°

(c) 90°

(d) 180°

(e) It is not defined

4. In Fig. Test 1-1, the solid curve represents the hyperbolic cosine function ( *y* = cosh *x* ), and the dashed curve represents the inverse of the hyperbolic cosine function ( *y* = arccosh *x* ). From this, it appears that the domain of *f* ( *x* ) = arccosh *x*

(a) includes all real numbers

(b) includes all non-negative real numbers

(c) includes all real numbers greater than 1

(d) includes all real numbers greater than or equal to 1

(e) does not include any real numbers

5. Based on the information shown in Fig. Test 1-1 and given in Question 4, it appears that the hyperbolic cosine of 0 is

(a) equal to 1

(b) equal to 0

(c) non-negative

(d) greater than or equal to 1

(e) not defined

6. Based on the information shown in Fig. Test 1-1 and given in Question 4, it appears that the hyperbolic arccosine of 0 is

(a) equal to 1

(b) equal to 0

(c) non-negative

(d) greater than or equal to 1

(e) not defined

7. Consider a system of Cartesian coordinates where *x* represents the independent variable and *y* represents the dependent variable. A function in this system is

(a) a relation in which every *x* value corresponds to at least one *y* value

(b) a relation in which every *y* value corresponds to at least one *x* value

(c) a relation in which every *x* value corresponds to at most one *y* value

(d) a relation in which every *y* value corresponds to at most one *x* value

(e) not described by any of the above

8. Suppose a balloon, hovering high in the atmosphere, is located in a position with respect to an observer defined by the following: azimuth 45°, elevation 60°, radius (also called distance or range) 25 kilometers. This is an expression of the balloon’s position in

(a) Cartesian coordinates

(b) polar coordinates

(c) cylindrical coordinates

(d) spherical coordinates

(e) rectangular coordinates

9. In navigator’s polar coordinates, the azimuth angle is measured

(a) counterclockwise around the horizon, relative to a ray pointing east

(b) clockwise around the horizon, relative to a ray pointing north

(c) downward relative to a ray pointing toward the zenith

(d) upward relative to a ray pointing toward the horizon

(e) upward relative to a ray pointing toward the nadir

10. Suppose **v** is a vector in three-space. Let the magnitude of this vector be denoted *v* . What is **v** · **v** (the dot product of vector **v** with itself)?

(a) *2v*

(b) v ^{2}

(c) 0

(d) 1

(e) It is impossible to determine without more information

11. For which of the following angles is the value of the tangent function not defined?

(a) 0 rad

(b) π/6 rad

(c) π/4 rad

(d) *it/2* rad

(e) It is defined for all of the above values

12. In Fig. Test 1-2, which of the following ratios represents csc *θ* ?

(a) *e/f*

(b) *d/f*

(c) *d/e*

(d) *e/d*

(e) None of the above

13. In Fig. Test 1-2, which of the following ratios represents cos *θ* ?

(a) *e/f*

(b) *d/f*

(c) *d/e*

(d) *e/d*

(e) None of the above

14. In Fig. Test 1-2, which of the following ratios represents cot *θ* ?

(a) *e/f*

(b) *d/f*

(c) *d/e*

(d) *e/d*

(e) None of the above

15. As long as the measure of an angle is not equal to any integer multiple of 180°, its sine is equal to the reciprocal of its

(a) cosecant

(b) cosine

(c) secant

(d) tangent

(e) cotangent

16. Given that csch *x* = 2/( *e* ^{x} – *e* ^{–x} ), what can be said about csch 0?

(a) It is equal to zero

(b) It is a positive real number less than 1

(c) It is equal to 1

(d) It is a positive real number greater than 1

(e) It is undefined

17. If a vector is multiplied by 2, what happens to its orientation?

(a) It does not change

(b) It is shifted counterclockwise by 90°

(c) It is shifted clockwise by 90°

(d) It is shifted by 180°

(e) It doubles

18. Suppose we are told that the measure of an angle *θ* lies somewhere between (but not including) 90° and 270°. We can be certain that the value of sin ^{2} *θ* + cos ^{2} *θ* is

(a) greater than 0 but less than 1

(b) greater than –1 but less than 0

(c) greater than –1 but less than 1

(d) equal to 1

(e) equal to 0

19. If a vector is multiplied by 2, what happens to its magnitude?

(a) It does not change

(b) It doubles

(c) It is cut to ½ its previous value

(d) It quadruples

(e) It is cut to ¼ its previous value

20. The range of the function *y* = sin *x* encompasses

(a) all real numbers between but not including –1 and 1

(b) all real numbers between but not including 0 and 1

(c) all real numbers between and including –1 and 1

(d) all real numbers between and including 0 and 1

(e) all real numbers

21. Suppose a broadcast tower is constructed in a perfectly square field that measures 100 meters on each side. The tower is in the center of the field and is 50 meters high. It is guyed from the middle and from the top. The guy wires run to the corners of the field. At what angle, to the nearest degree and relative to the horizontal, do the top guy wires slant?

(a) 35°

(b) 45°

(c) 55°

(d) 65°

(e) It cannot be determined without more information

22. In the graph of Fig. Test 1-3, suppose that *x* _{0} = 2.91 and *y* _{0} = 3.58. What is the value of *r* _{0} , rounded to two decimal places?

(a) 3.25

(b) 4.61

(c) 6.49

(d) 21.28

(e) It cannot be determined without more information

23. In the graph of Fig. Test 1-3, suppose that *x* _{0} = 2.91 and *y* _{0} = 3.58. What is the value of *θ* _{0} , rounded to the nearest degree?

(a) 36°

(b) 39°

(c) 51°

(d) 54°

(e) It cannot be determined without more information

24. In the graph of Fig. Test 1-3, suppose that *θ* _{0} = 45° and *r* _{0} = 5.35. What is the Cartesian coordinate ( *x* _{0} , *y* _{0} ) of point *P* ? Express both values rounded to two decimal places.

(a) (3.78,3.78)

(b) (5.35,5.35)

(c) (2.31,2.31)

(d) (2.68,2.68)

(e) It cannot be determined without more information

25. What is the arcsine of 0.5?

(a) 0°

(b) 30°

(c) 60°

(d) 90°

(e) It is not defined

26. Suppose you are standing on a flat, empty playing field and it is a sunny day. You measure the length of your shadow and discover that it is exactly twice your height. What is the angle of the sun above the horizon (if that angle can be determined) to the nearest degree?

(a) 27°

(b) 30°

(c) 60°

(d) 63°

(e) It depends on how tall you are

27. Suppose you are standing on a flat, empty playing field and it is a sunny day. You measure the length of your shadow and discover that it is exactly 1 meter greater than your height. What is the angle of the sun above the horizon (if the angle can be determined) to the nearest degree?

(a) 27°

(b) 30°

(c) 60°

(d) 63°

(e) It depends on how tall you are

28. Suppose a point is located at ( *x* _{0} , *y* _{0} ) in a Cartesian coordinate system. What is the radius, *r* , in mathematician’s polar coordinates?

29. Suppose there are two vectors, **a** and **b** , and that vector a points straight west while vector **b** points straight north. In what direction does vector **a** × **b** point?

(a) Southeast

(b) Northwest

(c) Straight up

(d) Straight down

(e) This question has no answer, because **a** × **b** is a scalar, not a vector

30. A radian is the equivalent of

(a) 2π angular degrees

(b) 1/(2π) of the angle comprising a full circle

(c) the circumference of a unit circle

(d) ¼ of the angle comprising a full circle

(e) ½ of the circumference of a circle

31. Suppose we restrict the domain of *y* = sin *x* to allow only values of *x* between, but not including, –30° and 30°. What is the range of the resulting function?

(a) 0 < *y* < 0.5

(b) –0.5 < *y* < 0

(c) –0.5 < *y* < 0.5

(d) The entire set of real numbers

(e) It is undefined

32. Given that arccsch *x* = ln [x ^{–1} + ( *x* ^{–2} + 1) ^{1/2} ], what can be said about arccsch 0?

(a) It is equal to zero

(b) It is a positive real number less than 1

(c) It is equal to 1

(d) It is a positive real number greater than 1

(e) It is undefined

33. In mathematician’s polar coordinates, an angle of –90° is equivalent to an angle of

(a) π/4 rad

(b) π/2 rad

(c) 3π/4 rad

(d) 5π/4 rad

(e) None of the above

34. Suppose an antenna tower is 250 meters high and stands in a perfectly flat field. The highest set of guy wires comes down from the top of the tower at a 45° angle relative to the tower itself. How long is each of these guy wires? Express your answer (if an answer exists) to the nearest meter.

(a) 250 meters

(b) 354 meters

(c) 375 meters

(d) 400 meters

(e) It is impossible to tell without more information

35. What does the graph of the equation *r* = – *θ* /(20π) look like in mathematician’s polar coordinates, when *θ* is expressed in radians?

(a) A large circle

(b) A 20-leafed clover

(c) A large cardioid

(d) A tightly wound spiral

(e) Nothing, because the radius must always be negative, and such a condition is not defined

36. What is the vector sum **a** + **b** in Fig. Test 1-4?

(a) (7.4,0.4)

(b) (3.9,6.1)

(c) (1.7,6.1)

(d) (6.1,1.7)

(e) (0.4,7.4)

37. What is the angle *θ* _{a} in Fig. Test 1-4 to the nearest degree?

(a) 28°

(b) 31°

(c) 32°

(d) 35°

(e) It cannot be determined without more information

38. What is the angle *θ* _{b} in Fig. Test 1-4?

(a) arctan [4.6/(–1.1)]

(b) π + arctan [(4.6/(–1.1)]

(c) arctan (–1.1/4.6)

(d) π + arctan (–1.1/4.6)

(e) It cannot be determined without more information

39. Suppose, in a set of mathematician’s polar coordinates, an object has a radius coordinate *r* = 7 units. What is the angle coordinate *θ* of the object in radians?

(a) *θ* = π/2

(b) *θ* = π

(c) *θ* = 3π/2

(d) *θ* = 2π

(e) It cannot be determined without more information

40. In Fig. Test 1-5, the *x* value of point *Q* is equal to

(a) sin *θ*

(b) cos *θ*

(c) tan *θ*

(d) arctan *θ*

(e) arccos *θ*

41. In Fig. Test 1-5, the *y* value of point *P* is equal to

(a) sin *θ*

(b) cos *θ*

(c) tan *θ*

(d) arctan *θ*

(e) arccos *θ*

42. In Fig. Test 1-5, let *q* represent the *x* value of point *Q,* and let *p* represent the *y* value of point *P.* Which of the following statements is false?

(a) arcsin *p* = *θ*

(b) arccos *q* – *θ*

(c) *q* ^{2} + *p* ^{2} = 1

(d) *p/q* = tan *θ*

(e) *1/p* = cot *θ*

43. Suppose an object is located 30 kilometers south and 30 kilometers west of the origin in a set of navigator’s polar coordinates. What is the azimuth of this object?

(a) 45°

(b) 135°

(c) 225°

(d) 315°

(e) It is not defined

44. In Fig. Test 1-6, it is apparent that if the curve represents a trigonometric relation, then

(a) the *x* axis is graduated in radians

(b) the *y* axis is graduated in radians

(c) both the *x* and the *y* axes are graduated in radians

(d) neither the *x* axis nor the *y* axis is graduated in radians

(e) the system of coordinates is sinusoidal

45. By inspecting Fig. Test 1-6, we can conclude that

(a) the wave-shaped curve represents a relation between *x* and *y,* but not a function of *x*

(b) the wave-shaped curve represents a function of *x,* but not a relation between *x* and *y*

(c) the wave-shaped curve represents a function of *x*

(d) the wave-shaped curve represents a function of *y*

(e) None of the above

46. Suppose we are told that the curve in Fig. Test 1-6 has a sinusoidal shape. Also suppose that the maximum *y* value attained by the curve is 2, and the minimum y value is *–2* (as shown by the dashed line and the labeled points). From this it is apparent that

(a) the curve represents the graph of *y* = (sin *x)/2*

(b) the curve represents the graph of *y* = (cos *x)/2*

(c) the curve represents the graph of *y* = 2 sin *x*

(d) the curve represents the graph of *y* = 2 cos *x*

(e) the curve represents the graph of *y* = sin *2x*

47. What is the value of cosh [arccosh ( *k* + 1)], where *k* is a positive real number?

(a) *k* + 1

(b) *k* – 1

(c) *e* ^{k} + *e*

(d) ln ( *k* – 1)

(e) It cannot be determined without more information

48. The sum of the measures of the interior angles of a triangle is

(a) *π* /2 rad

(b) *π* rad

(c) 3π/2 rad

(d) *2π* rad

(e) dependent on the relative lengths of the sides

49. The hyperbolic functions arise from a curve that has a certain equation in the rectangular *xy* -plane. What is that equation?

(a) *y* = *x* ^{2} + 2 *x* + 1

(b) *y* = *x* ^{2}

(c) *x* = *y* ^{2}

(d) *x* ^{2} + *y* ^{2} = 1

(e) *x* ^{2} – *y* ^{2} – 1

50. One second of arc is equal to

(a) π/60 radians

(b) π/3600 radians

(c) 1/60 of an angular degree

(d) 1/3600 of an angular degree

(e) a meaningless expression

**Answers:**

1. d

2. d

3. e

4. d

5. a

6. e

7. c

8. d

9. b

10. b

11. d

12. e

13. b

14. c

15. a

16. e

17. a

18. d

19. b

20. c

21. a

22. b

23. c

24. a

25. b

26. a

27. e

28. e

29. d

30. b

31. c

32. e

33. e

34. b

35. d

36. c

37. a

38. b

39. e

40. b

41. a

42. e

43. c

44. a

45. c

46. d

47. a

48. b

49. e

50. d

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