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Module 6: Trigonometry of General Triangles Application of Laws of Sine and Cosine

Applications Involving Law of Sine and Cosine

Lesson

Some people pref er to use the same f orm ev ery time, and some people pref er to use whichev er f orm results in the unknown in the numerator. With the unknown in the numerator, it is easy to solv e just by multipy ing both sides of the equation by the quantity div ided into the unknown.

Sometimes, either law can be applied to determine a missing side length or angle measure. In general, the law of sines is easier to use so alway s check to see if y ou can use it f irst. Use the law of sines f or cases AAS, ASA, or SSA. SSA is the ambiguous case and may create one triangle, two triangles, or no triangle. For the two-triangle case, use whichev er triangle best represents the situation. Use the law of cosines f or cases SSS and SAS.

The law of sines and the law of cosines hav e many practical applications, such as in surv ey ing. A surv ey or takes precise measurements to establish of f icial land airspace and water boundaries. A surv ey or of ten uses an instrument called a transit to measure angles. When giv en a situation, f irst determine which law should be used to solv e the problem. Sketch the situation if a diagram has not been prov ided. If a sketch is too complicated, draw the triangle that results f rom the situation. Determine which law(s) to use in order to solv e the problem. Af ter getting an answer, re-read the question to make sure that the original question has been answered, and include appropriate units with y our answer.

Examples

Example 1: Ref er to the f igure below.

Part a: State whether the law of sines or the law of cosines is the best choice to solv e f or x. Part b: Explain y our choice. Part c: Substitute the v alues into the appropriate f ormula. Do NOT solv e.

Solution: Part a: law of sines Part b: The case is SSA, which indicates the law of sines.

Part c:

Example 2: A surv ey or wants to f ind the distance across a lake as shown. He sets up his transit and measures an angle of 34º. He also measures the distance to each side of the lake, getting 9 miles and 8.3 miles respectiv ely. What is the distance across the lake?

Solution: Draw the triangle represented by the situation. The case is SAS, use the law of cosines.

The distance is approximately 5.1 miles. The answer is reasonable because 34º is likely the smallest angle; theref ore, the opposite side should be the smallest side.

Example 3: A tree growing on a hillside casts a 112-f oot shadow straight down the hill (see f igure). To the nearest f oot, f ind the v ertical height of the tree if , relativ e to the horizontal, the hill slopes 15° and the angle of elev ation of the sun is 65°.

Solution: When parallel lines are cut by a transv ersal, alternate interior angles are congruent. The diagram below shows the angle measures and the triangle resulting f rom the situation. Use the law of sines.

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Example 4: The tallest trees in the world grow in Redwood National Park in Calif ornia; they are taller than a f ootball f ield is long. Find the height of the tree giv en in the f igure.

Solution: There are two triangles in the picture, an obtuse triangle and a right triangle. Both must be used in order to f ind the height of the tree. Shown below is a diagram that represents the situation, and a diagram of the obtuse and right triangles drawn separately.

Find b on the obtuse triangle, then use right triangle trigonometry to f ind the height of the tree. First, f ind m < 1. The 42º angle and < 1 f orm a linear pair m < 1 = 180º - 42º = 138º Use the triangle sum theorem to f ind m < 2. m < 2 = 180º - 138º - 38º = 4º The case is AAS, use the law of sines.

Use right triangle trigonometry with the last triangle.

The tree is about 236 f eet tall.

Example 5: A salesperson is trav eling down Interstate 43 and has intermittent cell phone serv ice. There is a transmission tower near Interstate 43. The range of serv ice f rom the tower f orms a 53º angle and the range of serv ice is 28 miles to one section of I-43 and 32 miles to another point on I-43.

Part a: How many miles along I-43 can the salesperson get serv ice? Part b: If the salesperson is trav eling at a speed of 45 miles per hour, how long will she hav e serv ice? Part c: If she slows down to 35 mph, how much longer will she be able to hav e serv ice?

Solution: a. Draw the triangle and label the inf ormation. The case is SAS, use the law of cosines.

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She will get serv ice f or approximately 27 miles.

b: Use d = rt 27 = 45t 0.6 = t 0.6 hours (or, 0.60 hrs × 60 min/hr = 36 minutes)

c: d = rt 27 = 35t 27/35 = t 0.77 hours 0.77 - 0.6 = 0.17(or, 0.17 hrs × 60 min/hr = 10.2 minutes) She will hav e serv ice 0.17 hours longer.

Self-Check

Self-Check Problem 1

Ref er to ∆XYZ, where x = 16, y = 7 and z = 13.

Part a: State whether the law of sines or the law of cosines is the best choice to solv e f or Y.

Part b: Explain y our choice.

Part c: Substitute the v alues into the appropriate f ormula. Do NOT solv e.

Check Answer

Sorry, incorrect answer.

Part a: law of cosines

Part b: Because you were not given a side opposite an angle, you cannot use the law of sines. The case is SSS, you must use the law of cosines.

Part c:

The correct response(s): Law of Cosines because the case is SSS; 7² = 16² + 13² -2(16)(13)cosY

Self-Check Problem 2

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The course f or a boat race starts at point A and proceeds in the direction S 52º W to point B, then in the direction S 40º E to point C, and

f inally back to A. Point C lies 10 kilometers directly south of point A. Approximate the total distance of the race course (hint: ).

Check Answer

Sorry, incorrect answer.

Because , alternate interior angles are congruent and m<C = 40 degrees. By the triangle sum theorem, m<ABC = 180-52-40 = 88 degrees. Draw the triangle that

represents the situation.

Total distance = 6.43 km + 7.88 km + 10 km = 24.31 km

The correct response(s): 24.31 km

Self-Check Problem 3

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An architect is designing a kitchen f or a client. When designing a kitchen, the architect must pay special attention to the placement of the

stov e, sink, and ref rigerator. In order f or a kitchen to be utilized ef f ectiv ely, these three amenities must f orm a triangle with each other. This

is known as the "work triangle." By design, the three parts of the work triangle must be no less than 3 f eet apart and no more than 7 f eet

apart. Based on the dimensions of the current kitchen, the architect has determined that the sink will be 3.6 f eet away f rom the stov e and 5.7

f eet away f rom the ref rigerator. If the sink f orms a 103º with the stov e and the ref rigerator, will the distance between the stov e and the

ref rigerator remain within the conf ines of the work triangle? Explain y our answer.

Check Answer

Sorry, incorrect answer.

The correct response(s): The triangle does not conform to the definition of a work triangle. The sink and the refrigerator are too far apart by 0.4 feet.

Self-Check Problem 4

In the f igure below, m<C = 22º, DC = 12, BC = 14.3, m<BDA = 65º, and m<ABD = 11º. Find AB.

Check Answer

Sorry, incorrect answer.

The correct response(s): AB ≈ 5.14

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Self-Check Problem 5

You and a f riend liv e 8.4 miles apart. A hot air balloon is f loating between y our houses as shown in the f igure. Giv en the angles of elev ation,

approximate the height of the balloon. (Hint: The height of the balloon is the altitude of the triangle.)

Check Answer

Sorry, incorrect answer.

C = 180-24-48 = 108 degrees

The balloon is 3.9 miles above the ground.

The correct response(s): 3.9 miles

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