Stewart Precalc 6e Section 6.2: Trigonometry of Right Triangles

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Stewart Precalc 6e Section 6.2: Trigonometry of Right Triangles 0/71

1 Concepts - Right Triangle Trigonometry · Level 1

A right triangle with an angle \(\theta\) is shown in the figure.

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(a) Label the "opposite" and "adjacent" sides of \(\theta\) and the hypotenuse of the triangle.

Answer for 1(a)

(b) The trigonometric functions of the angle \(\theta\) are defined as follows: \(\sin \theta = ?\), \(\cos \theta = ?\), \(\tan \theta = ?\)

Answer for 1(b)

(c) The trigonometric ratios do not depend on the size of the triangle. This is because all right triangles with an acute angle \(\theta\) are _____.

Answer for 1(c)

Enter your answer directly below each part above.

2 Concepts - Reciprocal Identities · Level 1

The reciprocal identities state that \(\sec \theta = \dfrac{1}{\cos \theta}\), \(\csc \theta = ?\), and \(\cot \theta = ?\).

3 Skills - Six Trigonometric Ratios · Level 2

Find the exact values of the six trigonometric ratios of the angle \(\theta\) in the triangle.

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4 Skills - Six Trigonometric Ratios · Level 2

Find the exact values of the six trigonometric ratios of the angle \(\theta\) in the triangle.

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5 Skills - Six Trigonometric Ratios · Level 2

Find the exact values of the six trigonometric ratios of the angle \(\theta\) in the triangle.

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6 Skills - Six Trigonometric Ratios · Level 2

Find the exact values of the six trigonometric ratios of the angle \(\theta\) in the triangle.

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7 Skills - Six Trigonometric Ratios · Level 2

Find the exact values of the six trigonometric ratios of the angle \(\theta\) in the triangle.

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8 Skills - Six Trigonometric Ratios · Level 2

Find the exact values of the six trigonometric ratios of the angle \(\theta\) in the triangle.

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9 Skills - Complementary Angles · Level 2

Find (a) \(\sin \alpha\) and \(\cos \beta\), (b) \(\tan \alpha\) and \(\cot \beta\), and (c) \(\sec \alpha\) and \(\csc \beta\).

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10 Skills - Complementary Angles · Level 2

Find (a) \(\sin \alpha\) and \(\cos \beta\), (b) \(\tan \alpha\) and \(\cot \beta\), and (c) \(\sec \alpha\) and \(\csc \beta\).

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11 Skills - Finding a Side · Level 2

Find the side labeled \(x\).

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12 Skills - Finding a Side · Level 2

Find the side labeled \(x\).

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13 Skills - Finding a Side · Level 3

Find the side labeled \(x\). State your answer rounded to five decimal places.

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14 Skills - Finding a Side · Level 3

Find the side labeled \(x\). State your answer rounded to five decimal places.

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15 Skills - Finding a Side · Level 2

Find the side labeled \(x\).

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16 Skills - Finding a Side · Level 2

Find the side labeled \(x\).

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17 Skills - Express Sides via Trig Ratios · Level 3

Express \(x\) and \(y\) in terms of trigonometric ratios of \(\theta\).

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18 Skills - Express Sides via Trig Ratios · Level 3

Express \(x\) and \(y\) in terms of trigonometric ratios of \(\theta\).

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19 Skills - Find Other Ratios · Level 2

Sketch a triangle that has acute angle \(\theta\), and find the other five trigonometric ratios of \(\theta\) given \(\sin \theta = \dfrac{3}{5}\).

20 Skills - Find Other Ratios · Level 2

Sketch a triangle that has acute angle \(\theta\), and find the other five trigonometric ratios of \(\theta\) given \(\cos \theta = \dfrac{9}{40}\).

21 Skills - Find Other Ratios · Level 2

Sketch a triangle that has acute angle \(\theta\), and find the other five trigonometric ratios of \(\theta\) given \(\cot \theta = 1\).

22 Skills - Find Other Ratios · Level 2

Sketch a triangle that has acute angle \(\theta\), and find the other five trigonometric ratios of \(\theta\) given \(\tan \theta = \sqrt{3}\).

23 Skills - Find Other Ratios · Level 2

Sketch a triangle that has acute angle \(\theta\), and find the other five trigonometric ratios of \(\theta\) given \(\sec \theta = \dfrac{7}{2}\).

24 Skills - Find Other Ratios · Level 2

Sketch a triangle that has acute angle \(\theta\), and find the other five trigonometric ratios of \(\theta\) given \(\csc \theta = \dfrac{13}{12}\).

25 Skills - Evaluate Expressions · Level 2

Evaluate the expression without using a calculator: \(\sin \dfrac{\pi}{6} + \cos \dfrac{\pi}{6}\).

26 Skills - Evaluate Expressions · Level 2

Evaluate the expression without using a calculator: \(\sin 30^{\circ} \cdot \csc 30^{\circ}\).

27 Skills - Evaluate Expressions · Level 2

Evaluate the expression without using a calculator: \(\sin 30^{\circ} \cos 60^{\circ} + \sin 60^{\circ} \cos 30^{\circ}\).

28 Skills - Evaluate Expressions · Level 2

Evaluate the expression without using a calculator: \((\sin 60^{\circ})^2 + (\cos 60^{\circ})^2\).

29 Skills - Evaluate Expressions · Level 2

Evaluate the expression without using a calculator: \(\left(\sin \dfrac{\pi}{3} \cos \dfrac{\pi}{4} + \sin \dfrac{\pi}{4} \cos \dfrac{\pi}{3}\right)^2\).

30 Skills - Evaluate Expressions · Level 2

Evaluate the expression without using a calculator: \(\left(\sin \dfrac{\pi}{3} \cos \dfrac{\pi}{4} - \sin \dfrac{\pi}{4} \cos \dfrac{\pi}{3}\right)^2\).

31 Skills - Solve Right Triangle · Level 3

Solve the right triangle.

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32 Skills - Solve Right Triangle · Level 3

Solve the right triangle.

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33 Skills - Solve Right Triangle · Level 3

Solve the right triangle.

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34 Skills - Solve Right Triangle · Level 3

Solve the right triangle.

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35 Skills - Solve Right Triangle · Level 3

Solve the right triangle.

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36 Skills - Solve Right Triangle · Level 3

Solve the right triangle.

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37 Skills - Solve Right Triangle · Level 3

Solve the right triangle.

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38 Skills - Solve Right Triangle · Level 3

Solve the right triangle.

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39 Skills - Measurement Estimate · Level 2

Use a ruler to carefully measure the sides of the triangle, and then use your measurements to estimate the six trigonometric ratios of \(\theta\).

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40 Skills - Measurement Estimate · Level 2

Using a protractor, sketch a right triangle that has the acute angle \(40^{\circ}\). Measure the sides carefully, and use your results to estimate the six trigonometric ratios of \(40^{\circ}\).

41 Skills - Find x Rounded · Level 3

Find \(x\) rounded to one decimal place.

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42 Skills - Find x Rounded · Level 3

Find \(x\) rounded to one decimal place.

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43 Skills - Find x Rounded · Level 3

Find \(x\) rounded to one decimal place.

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44 Skills - Find x Rounded · Level 3

Find \(x\) rounded to one decimal place.

45 Skills - Express Length · Level 3

Express the length \(x\) in terms of the trigonometric ratios of \(\theta\).

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46 Skills - Express Lengths · Level 3

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

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47 Applications - Angle of Elevation · Level 2

Height of a Building. The angle of elevation to the top of the Empire State Building in New York is found to be \(11^{\circ}\) from the ground at a distance of \(1\) mi from the base of the building. Using this information, find the height of the Empire State Building.

48 Applications - Angle of Depression · Level 3

Gateway Arch. A plane is flying within sight of the Gateway Arch in St. Louis, Missouri, at an elevation of \(35{,}000\) ft. The pilot would like to estimate her distance from the Gateway Arch. She finds that the angle of depression to a point on the ground below the arch is \(22^{\circ}\).

(a) What is the distance between the plane and the arch?

Answer for 48(a)

(b) What is the distance between a point on the ground directly below the plane and the arch?

Answer for 48(b)

Enter your answer directly below each part above.

49 Applications - Laser Beam · Level 3

Deviation of a Laser Beam. A laser beam is to be directed toward the center of the moon, but the beam strays \(0.5^{\circ}\) from its intended path.

(a) How far has the beam diverged from its assigned target when it reaches the moon? (The distance from the earth to the moon is \(240{,}000\) mi.)

Answer for 49(a)

(b) The radius of the moon is about \(1000\) mi. Will the beam strike the moon?

Answer for 49(b)

Enter your answer directly below each part above.

50 Applications - Angle of Depression · Level 2

Distance at Sea. From the top of a \(200\)-ft lighthouse, the angle of depression to a ship in the ocean is \(23^{\circ}\). How far is the ship from the base of the lighthouse?

51 Applications - Leaning Ladder · Level 2

Leaning Ladder. A \(20\)-ft ladder leans against a building so that the angle between the ground and the ladder is \(72^{\circ}\). How high does the ladder reach on the building?

52 Applications - Guy Wire · Level 2

Height of a Tower. A \(600\)-ft guy wire is attached to the top of a communications tower. If the wire makes an angle of \(65^{\circ}\) with the ground, how tall is the communications tower?

53 Applications - Kite · Level 2

Elevation of a Kite. A man is lying on the beach, flying a kite. He holds the end of the kite string at ground level, and estimates the angle of elevation of the kite to be \(50^{\circ}\). If the string is \(450\) ft long, how high is the kite above the ground?

54 Applications - Distance to Pole · Level 4

Determining a Distance. A woman standing on a hill sees a flagpole that she knows is \(60\) ft tall. The angle of depression to the bottom of the pole is \(14^{\circ}\), and the angle of elevation to the top of the pole is \(18^{\circ}\). Find her distance \(x\) from the pole.

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55 Applications - Tower from Window · Level 4

Height of a Tower. A water tower is located \(325\) ft from a building (see the figure). From a window in the building, an observer notes that the angle of elevation to the top of the tower is \(39^{\circ}\) and that the angle of depression to the bottom of the tower is \(25^{\circ}\). How tall is the tower? How high is the window?

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56 Applications - Two Cars · Level 3

Determining a Distance. An airplane is flying at an elevation of \(5150\) ft, directly above a straight highway. Two motorists are driving cars on the highway on opposite sides of the plane, and the angle of depression to one car is \(35^{\circ}\) and to the other is \(52^{\circ}\). How far apart are the cars?

57 Applications - Two Cars Same Side · Level 3

Determining a Distance. If both cars in Exercise 56 are on one side of the plane and if the angle of depression to one car is \(38^{\circ}\) and to the other car is \(52^{\circ}\), how far apart are the cars?

58 Applications - Hot-Air Balloon · Level 4

Height of a Balloon. A hot-air balloon is floating above a straight road. To estimate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be \(20^{\circ}\) and \(22^{\circ}\). How high is the balloon?

59 Applications - Mountain Height · Level 4

Height of a Mountain. To estimate the height of a mountain above a level plain, the angle of elevation to the top of the mountain is measured to be \(32^{\circ}\). One thousand feet closer to the mountain along the plain, it is found that the angle of elevation is \(35^{\circ}\). Estimate the height of the mountain.

60 Applications - Cloud Cover · Level 4

Height of Cloud Cover. To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle \(75^{\circ}\) from the horizontal. An observer \(600\) m away measures the angle of elevation to the spot of light to be \(45^{\circ}\). Find the height \(h\) of the cloud cover.

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61 Applications - Earth-Sun Distance · Level 4

Distance to the Sun. When the moon is exactly half full, the earth, moon, and sun form a right angle (see the figure). At that time the angle formed by the sun, earth, and moon is measured to be \(89.85^{\circ}\). If the distance from the earth to the moon is \(240{,}000\) mi, estimate the distance from the earth to the sun.

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62 Applications - Moon Distance · Level 4

Distance to the Moon. To find the distance to the sun as in Exercise 61, we needed to know the distance to the moon. Here is a way to estimate that distance: When the moon is seen at its zenith at a point \(A\) on the earth, it is observed to be at the horizon from point \(B\) (see the figure). Points \(A\) and \(B\) are \(6155\) mi apart, and the radius of the earth is \(3960\) mi.

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(a) Find the angle \(\theta\) in degrees.

Answer for 62(a)

(b) Estimate the distance from point \(A\) to the moon.

Answer for 62(b)

Enter your answer directly below each part above.

63 Applications - Earth Radius · Level 4

Radius of the Earth. From a satellite \(600\) mi above the earth, it is observed that the angle formed by the vertical and the line of sight to the horizon is \(60.276^{\circ}\). Use this information to find the radius of the earth.

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64 Applications - Parallax · Level 5

Parallax. To find the distance to nearby stars, the method of parallax is used. The idea is to find a triangle with the star at one vertex and with a base as large as possible. To do this, the star is observed at two different times exactly \(6\) months apart, and its apparent change in position is recorded. From these two observations, \(\angle E_1 S E_2\) can be calculated. The angle \(E_1 S O\) is called the parallax of the star. Alpha Centauri, the star nearest the earth, has a parallax of \(0.000211^{\circ}\). Estimate the distance to this star. (Take the distance from the earth to the sun to be \(9.3 \times 10^7\) mi.)

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65 Applications - Venus Distance · Level 4

Distance from Venus to the Sun. The elongation \(\alpha\) of a planet is the angle formed by the planet, earth, and sun. When Venus achieves its maximum elongation of \(46.3^{\circ}\), the earth, Venus, and the sun form a triangle with a right angle at Venus. Find the distance between Venus and the sun in astronomical units (AU). (By definition the distance between the earth and the sun is \(1\) AU.)

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66 Discovery/Discussion/Writing · Level 2

Similar Triangles. If two triangles are similar, what properties do they share? Explain how these properties make it possible to define the trigonometric ratios without regard to the size of the triangle.

67 Example - Finding Trigonometric Ratios · Level 1

Find the six trigonometric ratios of the acute angle \(\theta\) in a right triangle in which the side opposite \(\theta\) has length \(2\), the side adjacent to \(\theta\) has length \(\sqrt{5}\), and the hypotenuse has length \(3\).

68 Example - Finding Trigonometric Ratios · Level 2

If \(\cos \alpha = \dfrac{3}{4}\), sketch a right triangle with acute angle \(\alpha\), and find the other five trigonometric ratios of \(\alpha\).

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69 Example - Solving a Right Triangle · Level 2

Solve the right triangle ABC in which \(\angle A = 30^{\circ}\), \(\angle C = 90^{\circ}\), and the hypotenuse has length \(12\). Find \(\angle B\) and the lengths of sides \(a\) and \(b\).

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70 Example - Application: Height of a Tree · Level 2

A giant redwood tree casts a shadow \(532\) ft long. Find the height of the tree if the angle of elevation of the sun is \(25.7^{\circ}\).

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71 Example - Application: Building and Flagpole · Level 3

From a point on the ground \(500\) ft from the base of a building, an observer finds that the angle of elevation to the top of the building is \(24^{\circ}\) and that the angle of elevation to the top of a flagpole atop the building is \(27^{\circ}\). Find the height of the building and the length of the flagpole.

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