Stewart Precalc 6e Section 6.5: The Law of Sines

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Stewart Precalc 6e Section 6.5: The Law of Sines 0/50

1 Concept · Level 1

In triangle \(A B C\) with sides \(a\), \(b\), and \(c\) the Law of Sines states that ____

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2 Concept · Level 1

In which of the following cases can we use the Law of Sines to solve a triangle? ASA, SSS, SAS, SSA

3 Skill - Find side or angle · Level 2

Use the Law of Sines to find the indicated side \(x\) or angle \(\theta\).

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4 Skill - Find side or angle · Level 2

Use the Law of Sines to find the indicated side \(x\) or angle \(\theta\).

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5 Skill - Find side or angle · Level 2

Use the Law of Sines to find the indicated side \(x\) or angle \(\theta\).

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6 Skill - Find side or angle · Level 2

Use the Law of Sines to find the indicated side \(x\) or angle \(\theta\).

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7 Skill - Find side or angle · Level 2

Use the Law of Sines to find the indicated side \(x\) or angle \(\theta\).

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8 Skill - Find side or angle · Level 2

Use the Law of Sines to find the indicated side \(x\) or angle \(\theta\).

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9 Skill - Solve triangle · Level 2

Solve the triangle using the Law of Sines.

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10 Skill - Solve triangle · Level 2

Solve the triangle using the Law of Sines.

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11 Skill - Solve triangle · Level 2

Solve the triangle using the Law of Sines.

12 Skill - Solve triangle · Level 2

Solve the triangle using the Law of Sines.

13 Skill - Sketch and solve · Level 2

Sketch the triangle, and then solve the triangle using the Law of Sines. \(\angle A = 50^{\circ}\), \(\angle B = 68^{\circ}\), \(c = 230\).

14 Skill - Sketch and solve · Level 2

Sketch the triangle, and then solve the triangle using the Law of Sines. \(\angle A = 23^{\circ}\), \(\angle B = 110^{\circ}\), \(c = 50\).

15 Skill - Sketch and solve · Level 2

Sketch the triangle, and then solve the triangle using the Law of Sines. \(\angle A = 30^{\circ}\), \(\angle C = 65^{\circ}\), \(b = 10\).

16 Skill - Sketch and solve · Level 2

Sketch the triangle, and then solve the triangle using the Law of Sines. \(\angle A = 22^{\circ}\), \(\angle B = 95^{\circ}\), \(a = 420\).

17 Skill - Sketch and solve · Level 2

Sketch the triangle, and then solve the triangle using the Law of Sines. \(\angle B = 29^{\circ}\), \(\angle C = 51^{\circ}\), \(b = 44\).

18 Skill - Sketch and solve · Level 2

Sketch the triangle, and then solve the triangle using the Law of Sines. \(\angle B = 10^{\circ}\), \(\angle C = 100^{\circ}\), \(c = 115\).

19 Skill - All possible triangles · Level 3

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. \(a = 28\), \(b = 15\), \(\angle A = 110^{\circ}\).

20 Skill - All possible triangles · Level 3

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. \(a = 30\), \(c = 40\), \(\angle A = 37^{\circ}\).

21 Skill - All possible triangles · Level 3

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. \(a = 20\), \(c = 45\), \(\angle A = 125^{\circ}\).

22 Skill - All possible triangles · Level 3

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. \(b = 45\), \(c = 42\), \(\angle C = 38^{\circ}\).

23 Skill - All possible triangles · Level 3

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. \(b = 25\), \(c = 30\), \(\angle B = 25^{\circ}\).

24 Skill - All possible triangles · Level 3

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. \(a = 75\), \(b = 100\), \(\angle A = 30^{\circ}\).

25 Skill - All possible triangles · Level 3

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. \(a = 50\), \(b = 100\), \(\angle A = 50^{\circ}\).

26 Skill - All possible triangles · Level 3

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. \(a = 100\), \(b = 80\), \(\angle A = 135^{\circ}\).

27 Skill - All possible triangles · Level 3

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. \(a = 26\), \(c = 15\), \(\angle C = 29^{\circ}\).

28 Skill - All possible triangles · Level 3

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. \(b = 73\), \(c = 82\), \(\angle B = 58^{\circ}\).

29 Skill - Multi-part triangle · Level 3

For the triangle shown, find (a) \(\angle B C D\) and (b) \(\angle D C A\).

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30 Skill - Multi-part triangle · Level 3

For the triangle shown, find the length \(A D\).

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31 Skill - Proof · Level 4

In triangle \(A B C\), \(\angle A = 40^{\circ}\), \(a = 15\), and \(b = 20\). (a) Show that there are two triangles, \(A B C\) and \(A' B' C'\), that satisfy these conditions. (b) Show that the areas of the triangles in part (a) are proportional to the sines of the angles \(C\) and \(C'\), that is, \( \dfrac{\text{area of } \triangle A B C}{\text{area of } \triangle A' B' C'} = \dfrac{\sin C}{\sin C'} \)

32 Skill - Proof · Level 4

Show that, given the three angles \(A\), \(B\), \(C\) of a triangle and one side, say \(a\), the area of the triangle is \( \text{area} = \dfrac{a^2 \sin B \sin C}{2 \sin A} \)

33 Application - Tracking a Satellite · Level 3

Tracking a Satellite. The path of a satellite orbiting the earth causes it to pass directly over two tracking stations \(A\) and \(B\), which are 50 mi apart. When the satellite is on one side of the two stations, the angles of elevation at \(A\) and \(B\) are measured to be \(87.0^{\circ}\) and \(84.2^{\circ}\), respectively. (a) How far is the satellite from station \(A\)? (b) How high is the satellite above the ground?

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34 Application - Flight of a Plane · Level 3

Flight of a Plane. A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 5 mi apart, to be \(32^{\circ}\) and \(48^{\circ}\), as shown in the figure. (a) Find the distance of the plane from point \(A\). (b) Find the elevation of the plane.

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35 Application - Distance Across a River · Level 3

Distance Across a River. To find the distance across a river, a surveyor chooses points \(A\) and \(B\), which are 200 ft apart on one side of the river. She then chooses a reference point \(C\) on the opposite side of the river and finds that \(\angle B A C \approx 82^{\circ}\) and \(\angle A B C \approx 52^{\circ}\). Approximate the distance from \(A\) to \(C\).

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36 Application - Distance Across a Lake · Level 3

Distance Across a Lake. Points \(A\) and \(B\) are separated by a lake. To find the distance between them, a surveyor locates a point \(C\) on land such that \(\angle C A B = 48.6^{\circ}\). He also measures \(C A\) as 312 ft and \(C B\) as 527 ft. Find the distance between \(A\) and \(B\).

37 Application - Leaning Tower of Pisa · Level 3

The Leaning Tower of Pisa. The bell tower of the cathedral in Pisa, Italy, leans \(5.6^{\circ}\) from the vertical. A tourist stands 105 m from its base, with the tower leaning directly toward her. She measures the angle of elevation to the top of the tower to be \(29.2^{\circ}\). Find the length of the tower to the nearest meter.

38 Application - Radio Antenna · Level 3

Radio Antenna. A short-wave radio antenna is supported by two guy wires, 165 ft and 180 ft long. Each wire is attached to the top of the antenna and anchored to the ground, at two anchor points on opposite sides of the antenna. The shorter wire makes an angle of \(67^{\circ}\) with the ground. How far apart are the anchor points?

39 Application - Height of a Tree · Level 3

Height of a Tree. A tree on a hillside casts a shadow 215 ft down the hill. If the angle of inclination of the hillside is \(22^{\circ}\) to the horizontal and the angle of elevation of the sun is \(52^{\circ}\), find the height of the tree.

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40 Application - Length of a Guy Wire · Level 3

Length of a Guy Wire. A communications tower is located at the top of a steep hill. The angle of inclination of the hill is \(58^{\circ}\). A guy wire is to be attached to the top of the tower and to the ground, 100 m downhill from the base of the tower. The angle \(\alpha\) in the figure is determined to be \(12^{\circ}\). Find the length of cable required for the guy wire.

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41 Application - Calculating a Distance · Level 3

Calculating a Distance. Observers at \(P\) and \(Q\) are located on the side of a hill that is inclined \(32^{\circ}\) to the horizontal, as shown. The observer at \(P\) determines the angle of elevation to a hot-air balloon to be \(62^{\circ}\). At the same instant the observer at \(Q\) measures the angle of elevation to the balloon to be \(71^{\circ}\). If \(P\) is 60 m down the hill from \(Q\), find the distance from \(Q\) to the balloon.

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42 Application - Calculating an Angle · Level 3

Calculating an Angle. A water tower 30 m tall is located at the top of a hill. From a distance of 120 m down the hill, it is observed that the angle formed between the top and base of the tower is \(8^{\circ}\). Find the angle of inclination of the hill.

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43 Application - Distances to Venus · Level 4

Distances to Venus. The elongation \(\alpha\) of a planet is the angle formed by the planet, earth, and sun. It is known that the distance from the sun to Venus is 0.723 AU. At a certain time the elongation of Venus is found to be \(39.4^{\circ}\). Find the possible distances from the earth to Venus at that time in astronomical units (AU).

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44 Application - Soap Bubbles · Level 4

Soap Bubbles. When two bubbles cling together in midair, their common surface is part of a sphere whose center \(D\) lies on the line passing through the centers of the bubbles. Also, angles \(A C B\) and \(A C D\) each have measure \(60^{\circ}\). (a) Show that the radius \(r\) of the common face is given by \(r = \dfrac{a b}{a - b}\). [Hint: Use the Law of Sines together with the fact that an angle \(\theta\) and its supplement \(180^{\circ} - \theta\) have the same sine.] (b) Find the radius of the common face if the radii of the bubbles are 4 cm and 3 cm. (c) What shape does the common face take if the two bubbles have equal radii?

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45 Discovery - Ambiguous Case · Level 4

Number of Solutions in the Ambiguous Case. When the Law of Sines is used to solve a triangle in the SSA case, there may be two, one, or no solution(s). For given \(\angle A\) and sides \(a\) and \(b\), the number of solutions is determined by: if \(a \geq b\), 1 solution; if \(b > a > b \sin A\), 2 solutions; if \(a = b \sin A\), 1 solution; if \(a < b \sin A\), 0 solutions. If \(\angle A = 30^{\circ}\) and \(b = 100\), use these criteria to find the range of values of \(a\) for which the triangle \(A B C\) has two solutions, one solution, or no solution.

46 Example - Tracking a Satellite (ASA) · Level 2

Tracking a Satellite (ASA). A satellite orbiting the earth passes directly overhead at observation stations in Phoenix and Los Angeles, 340 mi apart. At an instant when the satellite is between these two stations, its angle of elevation is simultaneously observed to be \(60^{\circ}\) at Phoenix and \(75^{\circ}\) at Los Angeles. How far is the satellite from Los Angeles?

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47 Example - Solving a Triangle (SAA) · Level 2

Solving a Triangle (SAA). Solve the triangle in which \(\angle A = 20^{\circ}\), \(\angle C = 25^{\circ}\), and \(c = 80.4\).

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48 Example - SSA One-Solution Case · Level 3

SSA, the One-Solution Case. Solve triangle \(A B C\), where \(\angle A = 45^{\circ}\), \(a = 7 \sqrt{2}\), and \(b = 7\).

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49 Example - SSA Two-Solution Case · Level 4

SSA, the Two-Solution Case. Solve triangle \(A B C\) if \(\angle A = 43.1^{\circ}\), \(a = 186.2\), and \(b = 248.6\).

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50 Example - SSA, the No-Solution Case · Level 3

Solve triangle \(A B C\), where \(\angle A = 42^{\circ}\), \(a = 70\), and \(b = 122\).

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