Stewart Precalc 6e Chapter 6: Review

1 Degree to Radian Conversion · Level 1

Find the radian measure that corresponds to the given degree measure. (a) \(60°\) (b) \(330°\) (c) \(-135°\) (d) \(-90°\)

2 Degree to Radian Conversion · Level 1

Find the radian measure that corresponds to the given degree measure. (a) \(24°\) (b) \(-330°\) (c) \(750°\) (d) \(5°\)

3 Radian to Degree Conversion · Level 1

Find the degree measure that corresponds to the given radian measure. (a) \(\dfrac{5 \pi}{2}\) (b) \(-\dfrac{\pi}{6}\) (c) \(\dfrac{9 \pi}{4}\) (d) \(3.1\)

4 Radian to Degree Conversion · Level 1

Find the degree measure that corresponds to the given radian measure. (a) \(8\) (b) \(-\dfrac{5}{2}\)

5 Arc Length · Level 1

Find the length of an arc of a circle of radius 8 m if the arc subtends a central angle of 1 rad.

6 Central Angle · Level 1

Find the measure of a central angle \(\theta\) in a circle of radius 5 ft if the angle is subtended by an arc of length 7 ft.

7 Radius from Arc · Level 2

A circular arc of length 100 ft subtends a central angle of \(70°\). Find the radius of the circle.

8 Application - Revolutions · Level 2

How many revolutions will a car wheel of diameter 28 in. make over a period of half an hour if the car is traveling at 60 mi/h?

9 Application - Angle Subtended · Level 2

New York and Los Angeles are 2450 mi apart. Find the angle that the arc between these two cities subtends at the center of the earth. (The radius of the earth is 3960 mi.)

10 Sector Area · Level 1

Find the area of a sector with central angle 2 rad in a circle of radius 5 m.

11 Sector Area · Level 1

Find the area of a sector with central angle \(52°\) in a circle of radius 200 ft.

12 Central Angle from Area · Level 2

A sector in a circle of radius 25 ft has an area of 125 ft². Find the central angle of the sector.

13 Application - Angular and Linear Speed · Level 2

A potter's wheel with radius 8 in. spins at 150 rpm. Find the angular and linear speeds of a point on the rim of the wheel.

14 Application - Gear Ratio · Level 3

In an automobile transmission a gear ratio \(g\) is the ratio \( g = \dfrac{\text{angular speed of engine}}{\text{angular speed of wheels}} \) The angular speed of the engine is shown on the tachometer (in rpm). A certain sports car has wheels with radius 11 in. Its gear ratios are: 1st = 4.1, 2nd = 3.0, 3rd = 1.6, 4th = 0.9, 5th = 0.7. Suppose the car is in fourth gear and the tachometer reads 3500 rpm. (a) Find the angular speed of the engine. (b) Find the angular speed of the wheels. (c) How fast (in mi/h) is the car traveling?

15 Trigonometric Ratios · Level 1

Find the values of the six trigonometric ratios of \(\theta\) (see figure).

16 Trigonometric Ratios · Level 1

Find the values of the six trigonometric ratios of \(\theta\) (see figure).

17 Right Triangle - Find Sides · Level 2

Find the sides labeled \(x\) and \(y\), rounded to two decimal places (see figure).

18 Right Triangle - Find Sides · Level 2

Find the sides labeled \(x\) and \(y\), rounded to two decimal places (see figure).

19 Right Triangle - Find Sides · Level 2

Find the sides labeled \(x\) and \(y\), rounded to two decimal places (see figure).

20 Right Triangle - Find Sides · Level 2

Find the sides labeled \(x\) and \(y\), rounded to two decimal places (see figure).

21 Solve Triangle · Level 2

Solve the triangle (see figure).

22 Solve Triangle · Level 2

Solve the triangle (see figure).

23 Solve Triangle · Level 2

Solve the triangle (see figure).

24 Solve Triangle · Level 2

Solve the triangle (see figure).

25 Express in Trig Ratios · Level 2

Express the lengths \(a\) and \(b\) in the figure in terms of the trigonometric ratios of \(\theta\).

26 Application - Angle of Elevation · Level 2

The highest free-standing tower in North America is the CN Tower in Toronto, Canada. From a distance of 1 km from its base, the angle of elevation to the top of the tower is \(28.81°\). Find the height of the tower.

27 Application - Polygon · Level 2

Find the perimeter of a regular hexagon that is inscribed in a circle of radius 8 m.

28 Application - Pistons · Level 3

The pistons in a car engine move up and down repeatedly to turn the crankshaft, as shown. Find the height of the point \(P\) above the center \(O\) of the crankshaft in terms of the angle \(\theta\).

29 Application - Moon Radius · Level 3

As viewed from the earth, the angle subtended by the full moon is \(0.518°\). Use this information and the fact that the distance \(A B\) from the earth to the moon is 236,900 mi to find the radius of the moon.

30 Application - Angles of Depression · Level 3

A pilot measures the angles of depression to two ships to be \(40°\) and \(52°\) (see the figure). If the pilot is flying at an elevation of 35,000 ft, find the distance between the two ships.

31 Exact Value · Level 1

Find the exact value of \(\sin 315°\).

32 Exact Value · Level 1

Find the exact value of \(\csc \dfrac{9 \pi}{4}\).

33 Exact Value · Level 1

Find the exact value of \(\tan(-135°)\).

34 Exact Value · Level 1

Find the exact value of \(\cos \dfrac{5 \pi}{6}\).

35 Exact Value · Level 2

Find the exact value of \(\cot\left(-\dfrac{22 \pi}{3}\right)\).

36 Exact Value · Level 1

Find the exact value of \(\sin 405°\).

37 Exact Value · Level 1

Find the exact value of \(\cos 585°\).

38 Exact Value · Level 2

Find the exact value of \(\sec \dfrac{22 \pi}{3}\).

39 Exact Value · Level 1

Find the exact value of \(\csc \dfrac{8 \pi}{3}\).

40 Exact Value · Level 1

Find the exact value of \(\sec \dfrac{13 \pi}{6}\).

41 Exact Value · Level 1

Find the exact value of \(\cot(-390°)\).

42 Exact Value · Level 2

Find the exact value of \(\tan \dfrac{23 \pi}{4}\).

43 Trig Ratios from Point · Level 2

Find the values of the six trigonometric ratios of the angle \(\theta\) in standard position if the point \((-5, 12)\) is on the terminal side of \(\theta\).

44 Trig Function from Point on Unit Circle · Level 2

Find \(\sin \theta\) if \(\theta\) is in standard position and its terminal side intersects the circle of radius 1 centered at the origin at the point \(\left(-\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right)\).

45 Angle from Line · Level 2

Find the acute angle that is formed by the line \(y - \sqrt{3} x + 1 = 0\) and the \(x\)-axis.

46 Trig Ratios from Line · Level 3

Find the six trigonometric ratios of the angle \(\theta\) in standard position if its terminal side is in Quadrant III and is parallel to the line \(4y - 2x - 1 = 0\).

47 Trig Identity Rewrite · Level 2

Write the first expression in terms of the second, for \(\theta\) in the given quadrant. \(\tan \theta\), \(\cos \theta\); \(\theta\) in Quadrant II.

48 Trig Identity Rewrite · Level 2

Write the first expression in terms of the second, for \(\theta\) in the given quadrant. \(\sec \theta\), \(\sin \theta\); \(\theta\) in Quadrant III.

49 Trig Identity Rewrite · Level 2

Write the first expression in terms of the second, for \(\theta\) in the given quadrant. \(\tan^2 \theta\), \(\sin \theta\); \(\theta\) in any quadrant.

50 Trig Identity Rewrite · Level 2

Write the first expression in terms of the second, for \(\theta\) in the given quadrant. \(\csc^2 \theta \cos^2 \theta\), \(\sin \theta\); \(\theta\) in any quadrant.

51 Find Six Trig Functions · Level 2

Find the values of the six trigonometric functions of \(\theta\) from the information given. \(\tan \theta = \dfrac{\sqrt{7}}{3}\), \(\sec \theta = \dfrac{4}{3}\).

52 Find Six Trig Functions · Level 2

Find the values of the six trigonometric functions of \(\theta\) from the information given. \(\sec \theta = \dfrac{41}{40}\), \(\csc \theta = -\dfrac{41}{9}\).

53 Find Six Trig Functions · Level 2

Find the values of the six trigonometric functions of \(\theta\) from the information given. \(\sin \theta = \dfrac{3}{5}\), \(\cos \theta < 0\).

54 Find Six Trig Functions · Level 2

Find the values of the six trigonometric functions of \(\theta\) from the information given. \(\sec \theta = -\dfrac{13}{5}\), \(\tan \theta > 0\).

55 Compute from Identity · Level 2

If \(\tan \theta = -\dfrac{1}{2}\) for \(\theta\) in Quadrant II, find \(\sin \theta + \cos \theta\).

56 Compute from Identity · Level 2

If \(\sin \theta = \dfrac{1}{2}\) for \(\theta\) in Quadrant I, find \(\tan \theta + \sec \theta\).

57 Pythagorean Identity · Level 1

If \(\tan \theta = -1\), find \(\sin^2 \theta + \cos^2 \theta\).

58 Double Angle · Level 2

If \(\cos \theta = -\dfrac{\sqrt{3}}{2}\) and \(\dfrac{\pi}{2} < \theta < \pi\), find \(\sin 2 \theta\).

59 Inverse Trig Exact Value · Level 1

Find the exact value of the expression \(\sin^{-1}\left(\dfrac{\sqrt{3}}{2}\right)\).

60 Inverse Trig Exact Value · Level 1

Find the exact value of the expression \(\tan^{-1}\left(\dfrac{\sqrt{3}}{3}\right)\).

61 Inverse Trig Composition · Level 2

Find the exact value of the expression \(\tan\left(\sin^{-1} \dfrac{2}{5}\right)\).

62 Inverse Trig Composition · Level 2

Find the exact value of the expression \(\sin\left(\cos^{-1} \dfrac{3}{8}\right)\).

63 Inverse Trig Algebraic · Level 2

Rewrite the expression \(\sin(\tan^{-1} x)\) as an algebraic expression in \(x\).

64 Inverse Trig Algebraic · Level 2

Rewrite the expression \(\sec(\sin^{-1} x)\) as an algebraic expression in \(x\).

65 Express Theta in Terms of x · Level 2

Express \(\theta\) in terms of \(x\) (see figure).