Stewart Precalc 6e Section 6.3: Trigonometric Functions of Angles

1 Concept - Trigonometric Function Definitions · Level 1

If the angle \(\theta\) is in standard position and \(P(x, y)\) is a point on the terminal side of \(\theta\), and \(r\) is the distance from the origin to \(P\), then \(\sin \theta = \) _____, \(\cos \theta = \) _____, \(\tan \theta = \) _____.

2 Concept - Signs of Trigonometric Functions · Level 1

The sign of a trigonometric function of \(\theta\) depends on the _____ in which the terminal side of the angle \(\theta\) lies. In Quadrant II, \(\sin \theta\) is _____ (positive / negative). In Quadrant III, \(\cos \theta\) is _____ (positive / negative). In Quadrant IV, \(\sin \theta\) is _____ (positive / negative).

3 Reference Angles · Level 1

Find the reference angle for the given angle. (a) \(150^{\circ}\) (b) \(330^{\circ}\) (c) \(780^{\circ}\)

4 Reference Angles · Level 1

Find the reference angle for the given angle. (a) \(120^{\circ}\) (b) \(-210^{\circ}\) (c) \(-105^{\circ}\)

5 Reference Angles · Level 1

Find the reference angle for the given angle. (a) \(225^{\circ}\) (b) \(810^{\circ}\) (c) \(-30^{\circ}\)

6 Reference Angles · Level 1

Find the reference angle for the given angle. (a) \(99^{\circ}\) (b) \(-199^{\circ}\) (c) \(359^{\circ}\)

7 Reference Angles · Level 1

Find the reference angle for the given angle. (a) \(\dfrac{11 \pi}{4}\) (b) \(-\dfrac{11 \pi}{6}\) (c) \(\dfrac{11 \pi}{3}\)

8 Reference Angles · Level 1

Find the reference angle for the given angle. (a) \(\dfrac{4 \pi}{3}\) (b) \(\dfrac{33 \pi}{4}\) (c) \(-\dfrac{23 \pi}{6}\)

9 Reference Angles · Level 2

Find the reference angle for the given angle. (a) \(\dfrac{5 \pi}{7}\) (b) \(-1.4 \pi\) (c) \(1.4\)

10 Reference Angles · Level 2

Find the reference angle for the given angle. (a) \(2.3 \pi\) (b) \(2.3\) (c) \(-10 \pi\)

11 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\sin 150^{\circ}\)

12 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\sin 225^{\circ}\)

13 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\cos 210^{\circ}\)

14 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\cos(-60^{\circ})\)

15 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\tan(-60^{\circ})\)

16 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\sec 300^{\circ}\)

17 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\csc(-630^{\circ})\)

18 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\cot 210^{\circ}\)

19 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\cos 570^{\circ}\)

20 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\sec 120^{\circ}\)

21 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\tan 750^{\circ}\)

22 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\cos 660^{\circ}\)

23 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\sin \dfrac{2 \pi}{3}\)

24 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\sin \dfrac{5 \pi}{3}\)

25 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\sin \dfrac{3 \pi}{2}\)

26 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\cos \dfrac{7 \pi}{3}\)

27 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\cos\left(-\dfrac{7 \pi}{3}\right)\)

28 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\tan \dfrac{5 \pi}{6}\)

29 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\sec \dfrac{17 \pi}{3}\)

30 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\csc \dfrac{5 \pi}{4}\)

31 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\cot\left(-\dfrac{\pi}{4}\right)\)

32 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\cos \dfrac{7 \pi}{4}\)

33 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\tan \dfrac{5 \pi}{2}\)

34 Exact Values · Level 2

Find the exact value of the trigonometric function: \(\sin \dfrac{11 \pi}{6}\)

35 Find the Quadrant · Level 2

Find the quadrant in which \(\theta\) lies from the information given: \(\sin \theta < 0\) and \(\cos \theta < 0\).

36 Find the Quadrant · Level 2

Find the quadrant in which \(\theta\) lies from the information given: \(\tan \theta < 0\) and \(\sin \theta < 0\).

37 Find the Quadrant · Level 2

Find the quadrant in which \(\theta\) lies from the information given: \(\sec \theta > 0\) and \(\tan \theta < 0\).

38 Find the Quadrant · Level 2

Find the quadrant in which \(\theta\) lies from the information given: \(\csc \theta > 0\) and \(\cos \theta < 0\).

39 Express in Terms of Another Function · Level 3

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant: \(\tan \theta\), \(\cos \theta\); \(\theta\) in Quadrant III.

40 Express in Terms of Another Function · Level 3

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant: \(\cot \theta\), \(\sin \theta\); \(\theta\) in Quadrant II.

41 Express in Terms of Another Function · Level 3

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant: \(\cos \theta\), \(\sin \theta\); \(\theta\) in Quadrant IV.

42 Express in Terms of Another Function · Level 3

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant: \(\sec \theta\), \(\sin \theta\); \(\theta\) in Quadrant I.

43 Express in Terms of Another Function · Level 3

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant: \(\sec \theta\), \(\tan \theta\); \(\theta\) in Quadrant II.

44 Express in Terms of Another Function · Level 3

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant: \(\csc \theta\), \(\cot \theta\); \(\theta\) in Quadrant III.

45 All Trigonometric Function Values · Level 3

Find the values of the trigonometric functions of \(\theta\) from the information given: \(\sin \theta = \dfrac{3}{5}\), \(\theta\) in Quadrant II.

46 All Trigonometric Function Values · Level 3

Find the values of the trigonometric functions of \(\theta\) from the information given: \(\cos \theta = -\dfrac{7}{12}\), \(\theta\) in Quadrant III.

47 All Trigonometric Function Values · Level 3

Find the values of the trigonometric functions of \(\theta\) from the information given: \(\tan \theta = -\dfrac{3}{4}\), \(\cos \theta > 0\).

48 All Trigonometric Function Values · Level 3

Find the values of the trigonometric functions of \(\theta\) from the information given: \(\sec \theta = 5\), \(\sin \theta < 0\).

49 All Trigonometric Function Values · Level 3

Find the values of the trigonometric functions of \(\theta\) from the information given: \(\csc \theta = 2\), \(\theta\) in Quadrant I.

50 All Trigonometric Function Values · Level 3

Find the values of the trigonometric functions of \(\theta\) from the information given: \(\cot \theta = \dfrac{1}{4}\), \(\sin \theta < 0\).

51 All Trigonometric Function Values · Level 3

Find the values of the trigonometric functions of \(\theta\) from the information given: \(\cos \theta = -\dfrac{2}{7}\), \(\tan \theta < 0\).

52 All Trigonometric Function Values · Level 3

Find the values of the trigonometric functions of \(\theta\) from the information given: \(\tan \theta = -4\), \(\sin \theta > 0\).

53 Function Notation Pitfalls · Level 3

If \(\theta = \dfrac{\pi}{3}\), find the value of each expression. (a) \(\sin 2 \theta\), \(2 \sin \theta\) (b) \(\sin \dfrac{1}{2} \theta\), \(\dfrac{1}{2} \sin \theta\) (c) \(\sin^2 \theta\), \(\sin(\theta^2)\)

54 Area of a Triangle · Level 2

Find the area of a triangle with sides of length 7 and 9 and included angle \(72^{\circ}\).

55 Area of a Triangle · Level 2

Find the area of a triangle with sides of length 10 and 22 and included angle \(10^{\circ}\).

56 Area of a Triangle · Level 2

Find the area of an equilateral triangle with side of length 10.

57 Inverse Application of Area Formula · Level 3

A triangle has an area of \(16\) \(\in^2\), and two of the sides of the triangle have lengths \(5\) in. and \(7\) in. Find the angle included by these two sides.

58 Inverse Application of Area Formula · Level 3

An isosceles triangle has an area of \(24\) \(cm^2\), and the angle between the two equal sides is \(\dfrac{5 \pi}{6}\). What is the length of the two equal sides?

59 Area of Shaded Region · Level 3

Find the area of the shaded region in the figure.

60 Area of Shaded Region · Level 3

Find the area of the shaded region in the figure.

61 Pythagorean Identity Proof · Level 3

Use the first Pythagorean identity to prove the second. [Hint: Divide by \(\cos^2 \theta\).]

62 Pythagorean Identity Proof · Level 3

Use the first Pythagorean identity to prove the third.

63 Application - Height of a Rocket · Level 3

Height of a Rocket. A rocket fired straight up is tracked by an observer on the ground a mile away. (a) Show that when the angle of elevation is \(\theta\), the height of the rocket in feet is \(h = 5280 \tan \theta\). (b) Complete the table to find the height of the rocket at the given angles of elevation: \(20^{\circ}\), \(60^{\circ}\), \(80^{\circ}\), \(85^{\circ}\).

64 Application - Rain Gutter · Level 4

Rain Gutter. A rain gutter is to be constructed from a metal sheet of width \(30\) cm by bending up one-third of the sheet on each side through an angle \(\theta\). (a) Show that the cross-sectional area of the gutter is modeled by the function \(A(\theta) = 100 \sin \theta + 100 \sin \theta \cos \theta\). (b) Graph the function \(A\) for \(0 \leq \theta \leq \dfrac{\pi}{2}\). (c) For what angle \(\theta\) is the largest cross-sectional area achieved?

65 Application - Wooden Beam · Level 4

Wooden Beam. A rectangular beam is to be cut from a cylindrical log of diameter \(20\) cm. (a) Express the cross-sectional area of the beam as a function of the angle \(\theta\) in the figures. (b) Graph the function you found in part (a). (c) Find the dimensions of the beam with largest cross-sectional area.

66 Application - Strength of a Beam · Level 4

Strength of a Beam. The strength of a beam is proportional to the width and the square of the depth. A beam is cut from a log as in Exercise 65. Express the strength of the beam as a function of the angle \(\theta\) in the figures.

67 Application - Shot Put Trajectory · Level 4

Throwing a Shot Put. The range \(R\) and height \(H\) of a shot put thrown with an initial velocity of \(v_0\) ft/s at an angle \(\theta\) are given by \(R = \dfrac{v_0^2 \sin(2 \theta)}{g}\) and \(H = \dfrac{v_0^2 \sin^2 \theta}{2 g}\). On the earth \(g \approx 32\) ft/s\(^2\) and on the moon \(g \approx 5.2\) ft/s\(^2\). Find the range and height of a shot put thrown under the given conditions. (a) On the earth with \(v_0 = 12\) ft/s and \(\theta = \dfrac{\pi}{6}\). (b) On the moon with \(v_0 = 12\) ft/s and \(\theta = \dfrac{\pi}{6}\).

68 Application - Sledding Down an Incline · Level 3

Sledding. The time in seconds that it takes for a sled to slide down a hillside inclined at an angle \(\theta\) is \(t = \sqrt{\dfrac{d}{16 \sin \theta}}\), where \(d\) is the length of the slope in feet. Find the time it takes to slide down a 2000-ft slope inclined at \(30^{\circ}\).

69 Application - Beehive Wax Optimization · Level 4

Beehives. In a beehive each cell is a regular hexagonal prism. The amount of wax \(W\) in the cell depends on the apex angle \(\theta\) and is given by \(W = 3.02 - 0.38 \cot \theta + 0.65 \csc \theta\). Bees instinctively choose \(\theta\) so as to use the least amount of wax possible. (a) Use a graphing device to graph \(W\) as a function of \(\theta\) for \(0 < \theta < \pi\). (b) For what value of \(\theta\) does \(W\) attain its minimum value?

70 Application - Turning a Corner with a Pipe · Level 4

Turning a Corner. A steel pipe is being carried down a hallway that is 9 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. (a) Show that the length of the pipe in the figure is modeled by the function \(L(\theta) = 9 \csc \theta + 6 \sec \theta\). (b) Graph the function \(L\) for \(0 < \theta < \dfrac{\pi}{2}\). (c) Find the minimum value of the function \(L\). (d) Explain why the value of \(L\) you found in part (c) is the length of the longest pipe that can be carried around the corner.

71 Application - Angle of Elevation of a Rainbow · Level 4

Rainbows. Rainbows are created when sunlight of different wavelengths (colors) is refracted and reflected in raindrops. The angle of elevation \(\theta\) of a rainbow is always the same. It can be shown that \(\theta = 4 \beta - 2 \alpha\), where \(\sin \alpha = k \sin \beta\) and \(\alpha = 59.4^{\circ}\) and \(k = 1.33\) is the index of refraction of water. Use the given information to find the angle of elevation \(\theta\) of a rainbow.

72 Discussion - Calculator Mode Error · Level 2

Using a Calculator. To solve a certain problem, you need to find the sine of 4 rad. Your study partner uses his calculator and tells you that \(\sin 4 = 0.0697564737\). On your calculator you get \(\sin 4 = -0.7568024953\). What is wrong? What mistake did your partner make?

73 Discovery - Viète's Trigonometric Diagram · Level 4

Viète's Trigonometric Diagram. Each of the six trigonometric functions of \(\theta\) is equal to the length of a line segment in the figure. For instance, \(\sin \theta = |\text{PR}|\) since from triangle \(\text{OPR}\) we have \(\sin \theta = \dfrac{\text{opp}}{\text{hyp}} = \dfrac{|\text{PR}|}{|\text{OR}|} = \dfrac{|\text{PR}|}{1} = |\text{PR}|\). For each of the five other trigonometric functions, find a line segment in the figure whose length equals the value of the function at \(\theta\). (Note: The radius of the circle is 1, the center is \(O\), segment \(\text{QS}\) is tangent to the circle at \(R\), and \(\angle \text{SOQ}\) is a right angle.)

74 Example - Finding Trigonometric Functions of Angles · Level 2

Find (a) \(\cos 135^{\circ}\) and (b) \(\tan 390^{\circ}\).

75 Example - Finding Reference Angles · Level 2

Find the reference angle \(\overline{\theta}\) for (a) \(\theta = \dfrac{5 \pi}{3}\) and (b) \(\theta = 870^{\circ}\).

76 Example - Using the Reference Angle to Evaluate Trigonometric Functions · Level 3

Find (a) \(\sin 240^{\circ}\) and (b) \(\cot 495^{\circ}\).

77 Example - Using the Reference Angle to Evaluate Trigonometric Functions · Level 3

Find (a) \(\sin \dfrac{16 \pi}{3}\) and (b) \(\sec\left(-\dfrac{\pi}{4}\right)\).

78 Example - Expressing One Trigonometric Function in Terms of Another · Level 3

(a) Express \(\sin \theta\) in terms of \(\cos \theta\). (b) Express \(\tan \theta\) in terms of \(\sin \theta\), where \(\theta\) is in Quadrant II.

79 Example - Evaluating a Trigonometric Function · Level 2

If \(\tan \theta = \dfrac{2}{3}\) and \(\theta\) is in Quadrant III, find \(\cos \theta\).

80 Example - Evaluating Trigonometric Functions · Level 2

If \(\sec \theta = 2\) and \(\theta\) is in Quadrant IV, find the other five trigonometric functions of \(\theta\).

81 Example - Finding the Area of a Triangle · Level 2

Find the area of triangle \(ABC\) with sides of length \(10\) cm and \(3\) cm and included angle \(120^{\circ}\).

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