Lesson 33: Applying the Laws of Sines and Cosines

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 33 M2 GEOMETRY

Student Outcomes . Students understand that the law of sines can be used to find missing side lengths in a triangle when the measures of the angles and one side length are known. . Students understand that the law of cosines can be used to find a missing side length in a triangle when the angle opposite the side and the other two side lengths are known. . Students solve triangle problems using the laws of sines and cosines.

Lesson Notes In this lesson, students apply the laws of sines and cosines learned in the previous lesson to find missing side lengths of triangles. The goal of this lesson is to clarify when students can apply the law of sines and when they can apply the law of cosines. Students are not prompted to use one law or the other; they must determine that on their own.

Classwork

Opening Exercise (10 minutes) Scaffolding: Once students have made their decisions, have them turn to a partner and compare their . Consider having students choices. Pairs that disagree should discuss why and, if necessary, bring their arguments to make a graphic organizer MP.3 the whole class so they may be critiqued. Ask students to provide justification for their to clearly distinguish choices. between the law of sines and the law of cosines.

Opening Exercise . Ask advanced students to write a letter to a younger For each triangle shown below, identify the method (Pythagorean theorem, law of sines, law of cosines) you would use to find each length 풙. student that explains the law of sines and the law of cosines and how to apply them to solve problems.

law of sines

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 33 M2 GEOMETRY

law of cosines

Pythagorean theorem

law of sines or law of cosines

law of sines

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Example 1 (5 minutes)

Students use the law of sines to find missing side lengths in a triangle.

Example 1

Find the missing side length in △ 푨푩푪.

. Which method should we use to find length 퐴퐵 in the triangle shown below? Explain. Provide a minute for students to discuss in pairs.  Yes. The law of sines can be used because we are given information about two of the angles and one side. We can use the triangle sum theorem to find the measure of ∠퐶, and then we can use that sin 퐴 sin 퐶 information about the value of the pair of ratios = . Since the values are equivalent, we can 푎 푐 solve to find the missing length. . Why can’t we use the Pythagorean theorem with this problem?  We can only use the Pythagorean theorem with right triangles. The triangle in this problem is not a right triangle. . Why can’t we use the law of cosines with this problem?  The law of cosines requires that we know the lengths of two sides of the triangle. We are only given information about the length of one side. . Write the equation that allows us to find the length of ̅퐴퐵̅̅̅.  Let 푥 represent the length of ̅퐴퐵̅̅̅. sin 75 sin 23 = 2.93 푥 2.93 sin 23 푐푥 = sin 75 . We want to perform one calculation to determine the answer so that it is most accurate and rounding errors are avoided. In other words, we do not want to make approximations at each step. Perform the calculation, and round the length to the tenths place.  The length of ̅퐴퐵̅̅̅ is approximately 1.2.

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Example 2 (5 minutes)

Students use the law of cosines to find missing side lengths in a triangle.

Example 2

Find the missing side length in △ 푨푩푪.

. Which method should we use to find side 퐴퐶̅̅̅̅ in the triangle shown below? Explain. Provide a minute for students to discuss in pairs.  We do not have enough information to use the law of sines because we do not have enough information to write any of the ratios related to the law of sines. However, we can use 푏2 = 푎2 + 푐2 − 2푎푐 cos 퐵 because we are given the lengths of sides 푎 and 푐 and we know the angle measure for ∠퐵. . Write the equation that can be used to find the length of 퐴퐶̅̅̅̅, and determine the length using one calculation. Round your answer to the tenths place.  Scaffolding: 푥2 = 5.812 + 5.952 − 2(5.81)(5.95) cos 16 It may be necessary to

푥 = √5.812 + 5.952 − 2(5.81)(5.95) cos 16 demonstrate to students how 푥 ≈ 1.6 to use a calculator to determine the answer in one

step.

Exercises 1–6 (16 minutes) All students should be able to complete Exercises 1 and 2 independently. These exercises can be used to informally assess students’ understanding in how to apply the laws of sines and cosines. Information gathered from these problems can inform how to present the next two exercises. Exercises 3–4 are challenging, and students should be allowed to work in pairs or small groups to discuss strategies to solve them. These problems can be simplified by having students remove the triangle from the context of the problem. For some groups of students, they may need to see a model of how to simplify the problem before being able to do it on their own. It may also be necessary to guide students to finding the necessary pieces of information, for example, angle measures, from the context or the diagram. Students who need a challenge should have the opportunity to make sense of the problems and persevere in solving them. The last two exercises are debriefed as part of the Closing.

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Exercises 1–6

Use the laws of sines and cosines to find all missing side lengths for each of the triangles in the exercises below. Round your answers to the tenths place. 1. Use the triangle to the right to complete this exercise. a. Identify the method (Pythagorean theorem, law of sines, law of cosines) you would use to find each of the missing lengths of the triangle. Explain why the other methods cannot be used.

Law of sines

The Pythagorean theorem requires a right angle, which is not applicable to this problem. The law of cosines requires information about two side lengths, which is not given. Applying the law of sines requires knowing the measure of two angles and the length of one side.

b. Find the lengths of 푨푪̅̅̅̅ and 푨푩̅̅̅̅.

By the triangle sum theorem, 풎∠푨 = ퟓퟒ°. Let 풄 represent the length of side 푨푩̅̅̅̅. Let 풃 represent the length of side 푨푪̅̅̅̅. 퐬퐢퐧 ퟓퟒ 퐬퐢퐧 ퟓퟐ = 퐬퐢퐧 ퟓퟒ 퐬퐢퐧 ퟕퟒ ퟑ. ퟑퟏ 풄 = ퟑ. ퟑퟏ 풃 ퟑ. ퟑퟏ 퐬퐢퐧 ퟓퟐ 풄 = ퟑ. ퟑퟏ 퐬퐢퐧ퟕퟒ 퐬퐢퐧 ퟓퟒ 풃 = 퐬퐢퐧ퟓퟒ 풄 ≈ ퟑ. ퟐ 풃 ≈ ퟑ. ퟗ

2. Your school is challenging classes to compete in a triathlon. The race begins with a swim along the shore and then continues with a bike ride for ퟒ miles. School officials want the race to end at the place it began, so after the ퟒ-mile bike ride, racers must turn ퟑퟎ° and run ퟑ. ퟓ miles directly back to the starting point. What is the total length of the race? Round your answer to the tenths place. a. Identify the method (Pythagorean theorem, law of sines, law of cosines) you would use to find the total length of the race. Explain why the other methods cannot be used.

Law of cosines

The Pythagorean theorem requires a right angle, which is not applicable to this problem because we do not know if we have a right triangle. The law of sines requires information about two angle measures, which is not given. Applying the law of cosines requires knowing the measure of two sides and the included angle measure.

b. Determine the total length of the race. Round your answer to the tenths place.

Let 풂 represent the length of the swim portion of the triathalon.

풂ퟐ = ퟒퟐ + ퟑ. ퟓퟐ − ퟐ(ퟒ)(ퟑ. ퟓ) 퐜퐨퐬 ퟑퟎ 풂 = √ퟒퟐ + ퟑ. ퟓퟐ − ퟐ(ퟒ)(ퟑ. ퟓ) 퐜퐨퐬 ퟑퟎ 풂 = ퟐ. ퟎퟎퟎퟑퟐퟐퟏퟒퟖ … 풂 ≈ ퟐ

Total length of the race: ퟒ + ퟑ. ퟓ + (ퟐ) ≈ ퟗ. ퟓ

The total length of the race is approximately ퟗ. ퟓ miles.

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3. Two lighthouses are ퟑퟎ miles apart on each side of shorelines running north and south, as shown. Each lighthouse keeper spots a boat in the distance. One lighthouse keeper notes the location of the boat as ퟒퟎ° east of south, and the other lighthouse keeper marks the boat as ퟑퟐ° west of south. What is the distance from the boat to each of the lighthouses at the time it was spotted? Round your answers to the nearest mile.

Students must begin by identifying the angle formed by one lighthouse, the boat, and the other lighthouse. This may be accomplished by drawing auxiliary lines and using facts about parallel lines cut by a transversal and the triangle sum theorem (or knowledge of exterior angles of a triangle). Once students know the angle is 72°, then the other angles in the triangle formed by the lighthouses and the boat can be found. The following calculations lead to the solution.

Let 풙 be the distance from the southern lighthouse to the boat. 퐬퐢퐧 ퟕퟐ 퐬퐢퐧 ퟓퟎ = ퟑퟎ 풙 ퟑퟎ 퐬퐢퐧 ퟓퟎ 풙 = 퐬퐢퐧 ퟕퟐ 풙 = ퟐퟒ. ퟏퟔퟒퟎퟎퟑퟖퟐ…

The southern lighthouse is approximately ퟐퟒ 퐦퐢. from the boat.

With this information, students may choose to use the law of sines or the law of cosines to find the other distance. Shown below are both options.

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Let 풚 be the distance from the northern lighthouse to the boat. 퐬퐢퐧 ퟕퟐ 퐬퐢퐧 ퟓퟖ = ퟑퟎ 풚 ퟑퟎ 퐬퐢퐧 ퟓퟖ 풚 = 퐬퐢퐧 ퟕퟐ 풚 = ퟐퟔ. ퟕퟓퟎ ퟕퟏퟔ ퟏퟐ…

The northern lighthouse is approximately ퟐퟕ 퐦퐢. from the boat.

Let 풂 be the distance from the northern lighthouse to the boat.

풂ퟐ = ퟑퟎퟐ + ퟐퟒퟐ − ퟐ(ퟑퟎ)(ퟐퟒ) ⋅ 퐜퐨퐬 ퟓퟖ 풂 = √ퟑퟎퟐ + ퟐퟒퟐ − ퟐ(ퟑퟎ)(ퟐퟒ) ⋅ 퐜퐨퐬 ퟓퟖ 풂 = ퟐퟔ. ퟕퟎퟎ ퟒퟗퟏ ퟕퟓ…

The northern lighthouse is approximately ퟐퟕ 퐦퐢. from the boat.

If groups of students work the problem both ways using the law of sines and the law of cosines, it may be a good place to have a discussion about why the answers were close but not exactly the same. When rounded to the nearest mile, the answers are the same, but if asked to round to the nearest tenths place, the results would be slightly different. The reason for the difference is that in the solution using the law of cosines, one of the values had already been approximated (24), leading to an even more approximated and less precise answer.

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4. A pendulum ퟏퟖ 퐢퐧. in length swings ퟕퟐ° from right to left. What is the difference between the highest and lowest point of the pendulum? Round your answer to the hundredths place, and Scaffolding: explain how you found it. Allow students time to struggle with the problem and, if necessary, guide them to drawing the horizontal and vertical dashed lines shown below.

At the bottom of a swing, the pendulum is perpendicular to the ground, and it bisects the ퟕퟐ° angle; therefore, the pendulum currently forms an angle of ퟑퟔ° with the vertical. By the triangle sum theorem, the angle formed by the pendulum and the horizontal is ퟓퟒ°. The 퐬퐢퐧 ퟓퟒ will give us the length from the top of the pendulum to where the horizontal and vertical lines intersect, 풙, which happens to be the highest point of the pendulum. 풙 퐬퐢퐧 ퟓퟒ = ퟏퟖ ퟏퟖ 퐬퐢퐧 ퟓퟒ = 풙 ퟏퟒ. ퟓퟔퟐ ퟑퟎퟓ ퟗ… = 풙 풙 The lowest point would be when the pendulum is perpendicular ퟓퟒ° to the ground, which would be exactly ퟏퟖ 퐢퐧. Then, the difference between those two points is approximately ퟑ. ퟒퟒ 퐢퐧.

5. What appears to be the minimum amount of information about a triangle that must be given in order to use the law of sines to find an unknown length?

To use the law of sines, you must know the measures of two angles and at least the length of one side.

6. What appears to be the minimum amount of information about a triangle that must be given in order to use the law of cosines to find an unknown length?

To use the law of cosines, you must know at least the lengths of two sides and the angle measure of the included angle.

Closing (4 minutes) Ask students to summarize the key points of the lesson. Additionally, consider asking students the following questions. Have students respond in writing, to a partner, or to the whole class. . What is the minimum amount of information that must be given about a triangle in order to use the law of sines to find missing lengths? Explain. . What is the minimum amount of information that must be given about a triangle in order to use the law of cosines to find missing lengths? Explain.

Exit Ticket (5 minutes)

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Name Date

Lesson 33: Applying the Laws of Sines and Cosines

Exit Ticket

1. Given triangle 푀퐿퐾, 퐾퐿 = 8, 퐾푀 = 7, and 푚∠퐾 = 75°, find the length of the unknown side to the nearest tenth. Justify your method.

2. Given triangle 퐴퐵퐶, 푚∠퐴 = 36°, 푚∠퐵 = 79°, and 퐴퐶 = 9, find the lengths of the unknown sides to the nearest tenth.

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Exit Ticket Sample Solutions

1. Given triangle 푴푳푲, 푲푳 = ퟖ, 푲푴 = ퟕ, and 풎∠푲 = ퟕퟓ°, find the length of the unknown side to the nearest tenth. Justify your method.

The triangle provides the lengths of two sides and their included angles, so using the law of cosines:

풌ퟐ = ퟖퟐ + ퟕퟐ − ퟐ(ퟖ)(ퟕ)(퐜퐨퐬 ퟕퟓ) 풌ퟐ = ퟔퟒ + ퟒퟗ − ퟏퟏퟐ(퐜퐨퐬 ퟕퟓ) 풌ퟐ = ퟏퟏퟑ − ퟏퟏퟐ(퐜퐨퐬 ퟕퟓ) 풌 = √ퟏퟏퟑ − ퟏퟏퟐ(퐜퐨퐬 ퟕퟓ) 풌 ≈ ퟗ. ퟐ

2. Given triangle 푨푩푪, 풎∠푨 = ퟑퟔ°, ∠푩 = ퟕퟗ°, and 푨푪 = ퟗ, find the lengths of the unknown sides to the nearest tenth.

By the angle sum of a triangle, 풎∠푪 = ퟔퟓ°.

Using the law of sines: 퐬퐢퐧 푨 퐬퐢퐧 푩 퐬퐢퐧 푪 = = 풂 풃 풄 퐬퐢퐧 ퟑퟔ 퐬퐢퐧 ퟕퟗ 퐬퐢퐧 ퟔퟓ = = 풂 ퟗ 풄

ퟗ 퐬퐢퐧 ퟑퟔ ퟗ 퐬퐢퐧 ퟔퟓ 풂 = 풄 = 퐬퐢퐧 ퟕퟗ 퐬퐢퐧 ퟕퟗ 풂 ≈ ퟓ. ퟒ 풄 ≈ ퟖ. ퟑ

푨푩 ≈ ퟖ. ퟑ and 푩푪 ≈ ퟓ. ퟒ.

Problem Set Sample Solutions

1. Given triangle 푬푭푮, 푭푮 = ퟏퟓ, angle 푬 has a measure of ퟑퟖ°, and angle 푭 has a measure of ퟕퟐ°, find the measures of the remaining sides and angle to the nearest tenth. Justify your method.

Using the angle sum of a triangle, the remaining angle 푮 has a measure of ퟕퟎ°.

The given triangle provides two angles and one side opposite a given angle, so it is appropriate to apply the law of sines. 퐬퐢퐧 ퟑퟖ 퐬퐢퐧 ퟕퟐ 퐬퐢퐧 ퟕퟎ = = ퟏퟓ 푬푮 푬푭

퐬퐢퐧 ퟑퟖ 퐬퐢퐧 ퟕퟐ 퐬퐢퐧 ퟑퟖ 퐬퐢퐧 ퟕퟎ = = ퟏퟓ 푬푮 ퟏퟓ 푬푭 ퟏퟓ 퐬퐢퐧 ퟕퟐ ퟏퟓ 퐬퐢퐧 ퟕퟎ 푬푮 = 푬푭 = 퐬퐢퐧 ퟑퟖ 퐬퐢퐧 ퟑퟖ 푬푮 ≈ ퟐퟑ. ퟐ 푬푭 ≈ ퟐퟐ. ퟗ

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2. Given triangle 푨푩푪, angle 푨 has a measure of ퟕퟓ°, 푨푪 = ퟏퟓ. ퟐ, and 푨푩 = ퟐퟒ, find 푩푪 to the nearest tenth. Justify your method.

The given information provides the lengths of the sides of the triangle and an included angle, so it is appropriate to use the law of cosines.

푩푪ퟐ = ퟐퟒퟐ + ퟏퟓ. ퟐퟐ − ퟐ(ퟐퟒ)(ퟏퟓ. ퟐ)(퐜퐨퐬 ퟕퟓ) 푩푪ퟐ = ퟓퟕퟔ + ퟐퟑퟏ. ퟎퟒ − ퟕퟐퟗ. ퟔ 퐜퐨퐬 ퟕퟓ 푩푪ퟐ = ퟖퟎퟕ. ퟎퟒ − ퟕퟐퟗ. ퟔ 퐜퐨퐬 ퟕퟓ 푩푪 = √ퟖퟎퟕ. ퟎퟒ − ퟕퟐퟗ. ퟔ 퐜퐨퐬 ퟕퟓ 푩푪 ≈ ퟐퟒ. ퟗ

3. James flies his plane from point 푨 at a bearing of ퟑퟐ° east of north, averaging a speed of ퟏퟒퟑ miles per hour for ퟑ hours, to get to an airfield at point 푩. He next flies ퟔퟗ° west of north at an average speed of ퟏퟐퟗ miles per hour for ퟒ. ퟓ hours to a different airfield at point 푪. a. Find the distance from 푨 to 푩.

distance = rate ⋅ time 풅 = ퟏퟒퟑ ⋅ ퟑ 풅 = ퟒퟐퟗ The distance from 푨 to 푩 is ퟒퟐퟗ 퐦퐢.

b. Find the distance from 푩 to 푪.

distance = rate ⋅ time 풅 = ퟏퟐퟗ ⋅ ퟒ. ퟓ 풅 = ퟓퟖퟎ. ퟓ The distance from 푩 to 푪 is ퟓퟖퟎ. ퟓ 퐦퐢.

c. Find the measure of angle 푨푩푪.

All lines pointing to the north/south are parallel; therefore, by alternate interior ∠’s, the return path from 푩 to 푨 is ퟑퟐ° west of south. Using angles on a line, this mentioned angle, the measure of the angle formed by the path from 푩 to 푪 with north (ퟔퟗ°) and 풎∠푨푩푪 sum to ퟏퟖퟎ; thus, 풎∠푨푩푪 = ퟕퟗ°.

d. Find the distance from 푪 to 푨.

The triangle shown provides two sides and the included angle, so using the law of cosines, 풃ퟐ = 풂ퟐ + 풄ퟐ − ퟐ풂풄(퐜퐨퐬 퐁) 풃ퟐ = (ퟓퟖퟎ. ퟓ)ퟐ + (ퟒퟐퟗ)ퟐ − ퟐ(ퟓퟖퟎ. ퟓ)(ퟒퟐퟗ)(퐜퐨퐬 ퟕퟗ) 풃ퟐ = ퟑퟑퟔ ퟗퟖퟎ. ퟐퟓ + ퟏퟖퟒ ퟎퟒퟏ − ퟒퟗퟖ ퟎퟔퟗ(퐜퐨퐬 ퟕퟗ) 풃ퟐ = ퟓퟐퟏ ퟎퟐퟏ. ퟐퟓ − ퟒퟗퟖ ퟎퟔퟗ(퐜퐨퐬 ퟕퟗ) 풃 = √ퟓퟐퟏ ퟎퟐퟏ. ퟐퟓ − ퟒퟗퟖ ퟎퟔퟗ(퐜퐨퐬 ퟕퟗ) 풃 ≈ ퟔퟓퟐ. ퟕ The distance from 푪 to 푨 is approximately ퟔퟓퟐ. ퟕ 퐦퐢.

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e. What length of time can James expect the return trip from 푪 to 푨 to take? distance = rate ⋅ time distance = rate ⋅ time

ퟔퟓퟐ. ퟕ = ퟏퟐퟗ ⋅ 풕ퟏ ퟔퟓퟐ. ퟕ = ퟏퟒퟑ풕ퟐ

ퟓ. ퟏ ≈ 풕ퟏ ퟒ. ퟔ ≈ 풕ퟐ James can expect the trip from 푪 to 푨 to take between ퟒ. ퟔ and ퟓ. ퟏ hours.

4. Mark is deciding on the best way to get from point 푨 to point 푩 as shown on the map of Crooked Creek to go fishing. He sees that if he stays on the north side of the creek, he would have to walk around a triangular piece of private property (bounded by 푨푪̅̅̅̅ and 푩푪̅̅̅̅). His other option is to cross the creek at 푨 and take a straight path to 푩, which he knows to be a distance of ퟏ. ퟐ 퐦퐢. The second option requires crossing the water, which is too deep for his boots and very cold. Find the difference in distances to help Mark decide which path is his better choice.

푨푩̅̅̅̅ is ퟒ. ퟖퟐ° north of east, and 푨푪̅̅̅̅ is ퟐퟒ. ퟑퟗ° east of north. The directions north and east are perpendicular, so the angles at point 푨 form a right angle. Therefore, 풎∠푪푨푩 = ퟔퟎ. ퟕퟗ°.

By the angle sum of a triangle, 풎∠푷푪푨 = ퟏퟎퟑ. ퟔퟖ°.

∠푷푪푨 and ∠푩푪푨 are angles on a line with a sum of ퟏퟖퟎ°, so 풎∠푩푪푨 = ퟕퟔ. ퟑퟐ°.

Also, by the angle sum of a triangle, 풎∠푨푩푪 = ퟒퟐ. ퟖퟗ°.

Using the law of sines: 퐬퐢퐧 푨 퐬퐢퐧 푩 퐬퐢퐧 푪 = = 풂 풃 풄 퐬퐢퐧 ퟔퟎ. ퟕퟗ 퐬퐢퐧 ퟒퟐ. ퟖퟗ 퐬퐢퐧 ퟕퟔ. ퟑퟐ = = 풂 풃 ퟏ. ퟐ 풃 풂 풃 ≈ ퟎ. ퟖퟒퟎퟔ 풂 ≈ ퟏ. ퟎퟕퟖퟎ

Let 풅 represent the distance from point 푨 to 푩 through point 푪 in miles. 풅 = 푨푪 + 푩푪 풅 ≈ ퟎ. ퟖퟒퟎퟔ + ퟏ. ퟎퟕퟖퟎ 풅 ≈ ퟏ. ퟗ The distance from point 푨 to 푩 through point 푪 is approximately ퟏ. ퟗ 퐦퐢. This distance is approximately ퟎ. ퟕ 퐦퐢. longer than walking a straight line from 푨 to 푩.

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5. If you are given triangle 푨푩푪 and the measures of two of its angles and two of its sides, would it be appropriate to apply the law of sines or the law of cosines to find the remaining side? Explain.

Case 1: Case 2: Given two angles and two sides, one of the angles being Given two angles and the sides opposite those angles, an included angle, it is appropriate to use either the law the law of sines can be applied as the remaining angle of sines or the law of cosines. can be calculated using the angle sum of a triangle. The law of cosines then can also be applied as the previous calculation provides an included angle.

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