Exercises: 3.2 The Law of Sines

Homework 3.2

Exercise Group

For Problems 1–6, use the law of sines to find the indicated side. Round to two decimal places.
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Exercise Group

For Problems 7–12, use the law of sines to find the indicated angle. Round to two decimal places.
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Exercise Group

For Problems 13–18, sketch the triangle and solve. Round answers to two decimal places.
13. $b=7,A=23°,B=42°$
14. $c=34,A=53°,C=26°$
15. $a=1.8,c=2.1,C=44°$
16. $b=8.5,c=6.8,B=23°$
17. $c=75,A=35°,B=46°$
18. $a=94,B=29°,C=84°$

Exercise Group

For Problems 19–26, sketch and label a triangle to illustrate the problem. Solve the problem.
19. Maryam wants to know the height of a cliff on the other side of a ravine. The angle of elevation from her edge of the ravine to the cliff top is $84.6°.$ When she moves $30$ feet back from the ravine, the angle of elevation is $82.5°.$ How tall is the cliff?
20. Amir wants to know the height of a tree in the median strip of a highway. The angle of elevation from the highway shoulder to the treetop is $43.5°.$ When he moves 10 feet farther away from the tree, the angle of elevation is $37.2°.$ How tall is the tree?
21. Delbert and Francine are $10$ kilometers apart, both observing a satellite that passes directly over their heads. At a moment when the satellite is between them, Francine measures its angle of elevation as $84.6°,$ and Delbert measures an angle of $87°.$ How far is the satellite from Delbert?
22. Megan rows her kayak due east. When she began, she spotted a lighthouse $2000$ meters in the distance at an angle of $14°$ south of east. After traveling for an hour, the lighthouse was at an angle of $83°$ south of east. How far did Megan travel, and what was her average speed?
23. Chad is hiking along a straight path but needs to detour around a large pond. He turns $23°$ from his path until clear of the pond, then walks back to his original path, intercepting it at an angle of $29°$ and at a distance of $2$ miles from where he had left the path. How far did Chad walk in each of the two segments of his detour, and how much farther did his detour require compared with a straight line through the pond?

24. Bob is flying to Monterey but must change course to avoid a storm. He flies $19°$ off from his original direction until he clears the storm, then turns again to return back to his original flight path, intercepting it at an angle of $54.9°$ and at a distance of $50$ miles from where he had left it. How much farther did his detour require compared with his original course?

25. Geologists find an outcropping from an underground rock formation that normally indicates the presence of oil. The outcropping is on a hillside, and the formation itself dips another $17°$ from the surface. If an oil well is placed $1000$ meters downhill from the outcropping, how far will the well have to drill before it reaches the formation?

26. A proposed ski lift will rise from point near the base of the slope with an angle of $27°.$ At a distance of $400$ meters farther from the slope, the angle of elevation to the top of the ski lift is $19°.$ How long is the ski lift?

27. Thelma wants to measure the height of a hill. She first plants a $50$-foot-tall antenna at the hill’s peak. Then she descends the hill and finds a point where she can see the top and the bottom of the antenna. The angle of elevation to the bottom of the antenna is $23°,$ and the angle of elevation to the top of the antenna is $24°.$

1. Find $\mathrm{\angle }ACB.$
2. Find$\mathrm{\angle }CAB,$ at the top of the antenna.
3. How long is $BC,$ the distance from the bottom of the antenna to $C?$
4. How tall is the hill?

28. A billboard of California’s gubernatorial candidate Angelyne is located on the roof of a building. At a distance of $180$ feet from the building, the angles of elevation to the bottom and top of the billboard are, respectively, $39.8°$ and $47.3°.$ How tall is the billboard?

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For Problems 29–32, compute the following distances in astronomical units (AUs). Then convert to kilometers, using the fact that 1 AU $\approx 1.5$ times ${10}^8$ km.
29. When observed from opposite sides of Earth’s orbit, the star Alpha Centauri has a parallax of ${0.76}^{\prime \prime }.$ How far from the Sun is Alpha Centauri?
30. How far from the Sun is Barnard’s Star, which has a parallax of ${1.1}^{\prime \prime }$ when observed at opposite ends of Earth’s orbit?
31. How far from the Sun is Tau Ceti, which has a parallax of ${0.55}^{\prime \prime }$ when observed from opposite ends of Earth’s orbit?
32. How far from the Sun is Sirius, which has a parallax of ${0.75}^{\prime \prime }$ when observed from opposite ends of Earth’s orbit?

Exercise Group

Problems 33–38 consider the ambiguous case of the law of sines, when two sides and an angle opposite one of them are known.
33. In the right triangle $\mathrm{△}ABC$ shown, $\mathrm{\angle }A=30°,\mathrm{\angle }C=90°,$ and $c=3$ inches.

1. Use the definition of $\sin A$ to solve for $a$ (the length of side $\overline{BC}$).
2. Can you draw a triangle $\mathrm{△}ABC$ with $\mathrm{\angle }A=30°$ and $c=3$ if $a<{\displaystyle \frac{3}{2}}?$ Why or why not?
3. How many triangles are possible if ${\displaystyle \frac{3}{2}}<a<3?$
4. How many triangles are possible if $a>3?$

34.
In this problem we show that there are two different triangles $\mathrm{△}ABC$ with $\mathrm{\angle }A=30°,a=2$ and $c=3.$

1. Use a protractor to draw an angle $\mathrm{\angle }A=30°.$ Mark point $B$ on one side of the angle so that $\overline{AB}$ is 3 inches long.
2. Locate two distinct points on the other side of the angle that are each 2 inches from point $B.$ These points are both possible locations for point $C.$
3. Use the law of sines to find two distinct possible measures for $\mathrm{\angle }C.$

35. In $\mathrm{△}ABC,\mathrm{\angle }A=30°$ and $c=12.$ How many triangles are possible for each of the following lengths for side $a?$ Sketch the solutions in each case.

1. $a=6$
2. $a=4$
3. $a=9$
4. $a=15$

36. Consider the triangle $\mathrm{△}ABC$ shown below.

1. Express the length of the altitude in terms of $\mathrm{\angle }A$ and $c.$
2. Now suppose we keep $\mathrm{\angle }A$ and side $c$ fixed but allow $a$ to vary in length. What is the smallest value $a$ can have and still be long enough to make a triangle?
3. What are the largest and smallest values that $a$ can have in order to produce two distinct triangles $\mathrm{△}ABC$ (without changing $\mathrm{\angle }A$ and side $c$)?

37. For the triangle in Problem 36, suppose $\mathrm{\angle }A=40°$ and $c=8.$

1. Sketch and solve the triangle if $a=12.$
2. Sketch and solve the triangle if $a=6.$
3. Sketch and solve the triangle if $a=4.$
4. For what value of $a$ is $c$ the hypotenuse of a right triangle?

38. For the figure in Problem 36, suppose $\mathrm{\angle }A=70°$ and $c=20.$

1. For what value of $a$ is the triangle a right triangle?
2. For what values of $a$ are there two solutions for the triangle?
3. For what values of $a$ is there one obtuse solution for the triangle?
4. For what value of $a$ is there no solution?

Exercise Group

For Problems 39–42, find the remaining angles of the triangle. Round answers to two decimal places. (These problems involve the ambiguous case.)
39. $a=66,c=43,\mathrm{\angle }C=25°$
40. $b=10,c=14,\mathrm{\angle }B=20°$
41. $b=100,c=80,\mathrm{\angle }B=49°$
42. $b=4.7,c=6.3,\mathrm{\angle }C=54°$
43. Delbert and Francine are $1000$ yards apart. The angle Delbert sees between Francine and a certain tree is $38°.$ If the tree is $800$ yards from Francine, how far is it from Delbert? (There are two possible answers.)
44. From the lookout point on Fabrick Rock, Ann can see not only the famous “Crooked Spire” in Chesterfield, which is $8$ miles away, but also the red phone box in the village of Alton. Chesterfield and Alton are $7$ miles apart. Fabrick Rock has a plaque that shows directions to famous sites, and from the plaque Ann determines that the angle between the lines to the spire and the phone box measures $19°.$ How far is Fabrick Rock from the phone box? (There are two possible answers.)
45.

1. Sketch a triangle with $\mathrm{\angle }A=25°,\mathrm{\angle }B=35°,$ and $b=16.$
2. Use the law of sines to find $a.$
3. Use the law of sines to find $c.$
4. Find $c$ without using the law of sines. (Hint: Sketch the altitude, $h,$ from $\mathrm{\angle }C$ to make two right triangles. Find $h,$ then use $h$ to find $c$.)

46.

1. Sketch a triangle with $\mathrm{\angle }A=75°$, $a=15,$ and $b=6.$
2. Use the law of sines to find $c.$
3. Find $c$ without using the law of sines.

Exercise Group

Problems 47–48 prove the law of sines using the formula for the area of a triangle. (See Section 3.1 for the appropriate formula.)
47.

1. 1. Sketch a triangle with angles $\mathrm{\angle }A,\mathrm{\angle }B$ and $\mathrm{\angle }C$ and opposite sides of lengths respectively $a,b$ and $c.$
   2. Write the area of the triangle in terms of $a,b,$ and angle $\mathrm{\angle }C.$
   3. Write the area of the triangle in terms of $a,c,$ and angle $\mathrm{\angle }B.$
   4. Write the area of the triangle in terms of [latex]b

   48. Equate the three different expressions from Problem 47 for the area of the triangle. Multiply through by ${\displaystyle \frac{2}{abc}}$ and simplify to deduce the law of sines.
   49. Here is a method for solving certain oblique triangles by dividing them into two right triangles. In the triangle shown, we know two angles, $\mathrm{\angle }A$ and $\mathrm{\angle }B,$ and the side opposite one of them, say $a.$ We would like to find side $b.$

   

   1. Draw the altitude $h$ from angle $\mathrm{\angle }C.$
   2. Write an expression for $b$ in terms of $h$ and angle $\mathrm{\angle }A.$
   3. Write an expression for $h$ in terms of angle $\mathrm{\angle }B.$
   4. Substitute your expression for $h$ into your expression for $b.$
   5. Which of the following is equivalent to the formula you wrote in part $d$?
      1. $a\sin A=b\sin B$
      2. ${\displaystyle \frac{a}{\sin A}}={\displaystyle \frac{b}{\sin B}}$
      3. ${\displaystyle \frac{a}{\sin B}}={\displaystyle \frac{b}{\sin A}}$