Lesson 33: Applying the Laws of Sines and Cosines
Transcript of Lesson 33: Applying the Laws of Sines and Cosines
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying the Laws of Sines and Cosines
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Lesson 33: Applying the Laws of Sines and Cosines
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)
Once students have made their decisions, have them turn to a partner and compare their
choices. Pairs that disagree should discuss why and, if necessary, bring their arguments to
the whole class so they may be critiqued. Ask students to provide justification for their
choices.
Opening Exercise
For each triangle shown below, identify the method (Pythagorean theorem, law of sines, law of
cosines) you would use to find each length ๐.
law of sines
Scaffolding:
Consider having students make a graphic organizer to clearly distinguish between the law of sines and the law of cosines.
Ask advanced students to write a letter to a younger student that explains the law of sines and the law of cosines and how to apply them to solve problems.
MP.3
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying the Laws of Sines and Cosines
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law of cosines
Pythagorean
theorem
law of sines
or
law of cosines
law of sines
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying the Laws of Sines and Cosines
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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 โณ ๐จ๐ฉ๐ช.
Which method should we use to find length ๐ด๐ต 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 โ ๐ถ, and then we can use that
information about the value of the pair of ratios sin ๐ด
๐=
sin ๐ถ
๐. Since the values are equivalent, we can
solve to find the missing length.
Why canโt 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 canโt 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 ๐ด๐ตฬ ฬ ฬ ฬ .
Let ๐ฅ represent the length of ๐ด๐ตฬ ฬ ฬ ฬ .
sin 75
2.93=
sin 23
๐ฅ
๐๐ฅ =2.93 sin 23
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 ๐ด๐ตฬ ฬ ฬ ฬ is approximately 1.2.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying the Laws of Sines and Cosines
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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 โณ ๐จ๐ฉ๐ช.
Which method should we use to find side ๐ด๐ถฬ ฬ ฬ ฬ 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
๐2 = ๐2 + ๐2 โ 2๐๐ cos ๐ต because we are given the lengths of sides ๐ and ๐ and we know the angle
measure for โ ๐ต.
Write the equation that can be used to find the length of ๐ด๐ถฬ ฬ ฬ ฬ , and determine the length using one calculation.
Round your answer to the tenths place.
Exercises 1โ6 (16 minutes)
All students should be able to complete Exercises 1 and 2 independently. These exercises can be used to informally
assess studentsโ 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 3โ4 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.
๐ฅ2 = 5.812 + 5.952 โ 2(5.81)(5.95) cos 16
๐ฅ = โ5.812 + 5.952 โ 2(5.81)(5.95) cos 16
๐ฅ โ 1.6
Scaffolding:
It may be necessary to demonstrate to students how to use a calculator to determine the answer in one step.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying the Laws of Sines and Cosines
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Exercises 1โ6
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 ๐จ๐ชฬ ฬ ฬ ฬ and ๐จ๐ฉฬ ฬ ฬ ฬ .
By the triangle sum theorem, ๐โ ๐จ = ๐๐ยฐ.
Let ๐ represent the length of side ๐จ๐ชฬ ฬ ฬ ฬ .
๐ฌ๐ข๐ง ๐๐
๐. ๐๐=
๐ฌ๐ข๐ง ๐๐
๐
๐ =๐. ๐๐ ๐ฌ๐ข๐ง๐๐
๐ฌ๐ข๐ง๐๐
๐ โ ๐. ๐
Let ๐ represent the length of side ๐จ๐ฉฬ ฬ ฬ ฬ .
๐ฌ๐ข๐ง ๐๐
๐. ๐๐=
๐ฌ๐ข๐ง ๐๐
๐
๐ =๐. ๐๐ ๐ฌ๐ข๐ง ๐๐
๐ฌ๐ข๐ง ๐๐
๐ โ ๐. ๐
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 ๐ miles. School officials want the race to end at the place it began, so after the ๐-mile
bike ride, racers must turn ๐๐ยฐ and run ๐. ๐ 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 ๐ represent the length of the swim portion of the triathalon.
๐๐ = ๐๐ + ๐. ๐๐ โ ๐(๐)(๐. ๐) ๐๐จ๐ฌ ๐๐
๐ = โ๐๐ + ๐. ๐๐ โ ๐(๐)(๐. ๐) ๐๐จ๐ฌ ๐๐
๐ = ๐. ๐๐๐๐๐๐๐๐๐ โฆ
๐ โ ๐
Total length of the race: ๐ + ๐. ๐ + (๐) โ ๐. ๐
The total length of the race is approximately ๐. ๐ miles.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying the Laws of Sines and Cosines
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3. Two lighthouses are ๐๐ 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 ๐๐ยฐ east of south, and
the other lighthouse keeper marks the boat as ๐๐ยฐ 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 72ยฐ, 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 ๐ be the distance from the southern lighthouse to the boat.
๐ฌ๐ข๐ง ๐๐
๐๐=
๐ฌ๐ข๐ง ๐๐
๐
๐ =๐๐ ๐ฌ๐ข๐ง ๐๐
๐ฌ๐ข๐ง ๐๐
๐ = ๐๐. ๐๐๐๐๐๐๐๐โฆ
The southern lighthouse is approximately ๐๐ ๐ฆ๐ข. 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.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying the Laws of Sines and Cosines
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Let ๐ be the distance from the northern lighthouse to the boat.
๐ฌ๐ข๐ง ๐๐
๐๐=
๐ฌ๐ข๐ง ๐๐
๐
๐ =๐๐ ๐ฌ๐ข๐ง ๐๐
๐ฌ๐ข๐ง ๐๐
๐ = ๐๐. ๐๐๐โ๐๐๐โ๐๐โฆ
The northern lighthouse is approximately ๐๐ ๐ฆ๐ข. from the boat.
Let ๐ be the distance from the northern lighthouse to the boat.
๐๐ = ๐๐๐ + ๐๐๐ โ ๐(๐๐)(๐๐) โ ๐๐จ๐ฌ ๐๐
๐ = โ๐๐๐ + ๐๐๐ โ ๐(๐๐)(๐๐) โ ๐๐จ๐ฌ ๐๐
๐ = ๐๐. ๐๐๐โ๐๐๐โ๐๐โฆ
The northern lighthouse is approximately ๐๐ ๐ฆ๐ข. 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.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying the Laws of Sines and Cosines
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Scaffolding:
Allow students time to struggle with the problem and, if necessary, guide them to drawing the horizontal and vertical dashed lines shown below.
4. A pendulum ๐๐ ๐ข๐ง. in length swings ๐๐ยฐ 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
explain how you found it.
At the bottom of a swing, the pendulum is perpendicular to the
ground, and it bisects the ๐๐ยฐ angle; therefore, the pendulum
currently forms an angle of ๐๐ยฐ with the vertical. By the
triangle sum theorem, the angle formed by the pendulum and
the horizontal is ๐๐ยฐ. The ๐ฌ๐ข๐ง ๐๐ will give us the length from the
top of the pendulum to where the horizontal and vertical lines
intersect, ๐, which happens to be the highest point of the
pendulum.
๐ฌ๐ข๐ง ๐๐ =๐
๐๐
๐๐ ๐ฌ๐ข๐ง ๐๐ = ๐
๐๐. ๐๐๐โ๐๐๐โ๐โฆ = ๐
The lowest point would be when the pendulum is perpendicular
to the ground, which would be exactly ๐๐ ๐ข๐ง. Then, the
difference between those two points is approximately ๐. ๐๐ ๐ข๐ง.
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)
๐๐ยฐ
๐
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying the Laws of Sines and Cosines
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Name Date
Lesson 33: Applying the Laws of Sines and Cosines
Exit Ticket
1. Given triangle ๐๐ฟ๐พ, ๐พ๐ฟ = 8, ๐พ๐ = 7, and ๐โ ๐พ = 75ยฐ, find the length of the unknown side to the nearest tenth.
Justify your method.
2. Given triangle ๐ด๐ต๐ถ, ๐โ ๐ด = 36ยฐ, ๐โ ๐ต = 79ยฐ, and ๐ด๐ถ = 9, find the lengths of the unknown sides to the nearest
tenth.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying the Laws of Sines and Cosines
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Exit Ticket Sample Solutions
1. Given triangle ๐ด๐ณ๐ฒ, ๐ฒ๐ณ = ๐, ๐ฒ๐ด = ๐, and ๐โ ๐ฒ = ๐๐ยฐ, 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:
๐๐ = ๐๐ + ๐๐ โ ๐(๐)(๐)(๐๐จ๐ฌ ๐๐)
๐๐ = ๐๐ + ๐๐ โ ๐๐๐(๐๐จ๐ฌ ๐๐)
๐๐ = ๐๐๐ โ ๐๐๐(๐๐จ๐ฌ ๐๐)
๐ = โ๐๐๐ โ ๐๐๐(๐๐จ๐ฌ ๐๐)
๐ โ ๐. ๐
2. Given triangle ๐จ๐ฉ๐ช, ๐โ ๐จ = ๐๐ยฐ, โ ๐ฉ = ๐๐ยฐ, and ๐จ๐ช = ๐, find the lengths of the unknown sides to the nearest
tenth.
By the angle sum of a triangle, ๐โ ๐ช = ๐๐ยฐ.
Using the law of sines:
๐ฌ๐ข๐ง ๐จ
๐=
๐ฌ๐ข๐ง ๐ฉ
๐=
๐ฌ๐ข๐ง ๐ช
๐
๐ฌ๐ข๐ง ๐๐
๐=
๐ฌ๐ข๐ง ๐๐
๐=
๐ฌ๐ข๐ง ๐๐
๐
๐ =๐ ๐ฌ๐ข๐ง ๐๐
๐ฌ๐ข๐ง ๐๐
๐ โ ๐. ๐
๐ =๐ ๐ฌ๐ข๐ง ๐๐
๐ฌ๐ข๐ง ๐๐
๐ โ ๐. ๐
๐จ๐ฉ โ ๐. ๐ and ๐ฉ๐ช โ ๐. ๐.
Problem Set Sample Solutions
1. Given triangle ๐ฌ๐ญ๐ฎ, ๐ญ๐ฎ = ๐๐, angle ๐ฌ has a measure of ๐๐ยฐ, and angle ๐ญ has a measure of ๐๐ยฐ, 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 ๐ฎ has a measure of ๐๐ยฐ.
The given triangle provides two angles and one side opposite a given angle, so it is
appropriate to apply the law of sines.
๐ฌ๐ข๐ง ๐๐
๐๐=
๐ฌ๐ข๐ง ๐๐
๐ฌ๐ฎ=
๐ฌ๐ข๐ง ๐๐
๐ฌ๐ญ
๐ฌ๐ข๐ง ๐๐
๐๐=
๐ฌ๐ข๐ง ๐๐
๐ฌ๐ฎ
๐ฌ๐ฎ =๐๐ ๐ฌ๐ข๐ง ๐๐
๐ฌ๐ข๐ง ๐๐
๐ฌ๐ฎ โ ๐๐. ๐
๐ฌ๐ข๐ง ๐๐
๐๐=
๐ฌ๐ข๐ง ๐๐
๐ฌ๐ญ
๐ฌ๐ญ =๐๐ ๐ฌ๐ข๐ง ๐๐
๐ฌ๐ข๐ง ๐๐
๐ฌ๐ญ โ ๐๐. ๐
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying the Laws of Sines and Cosines
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2. Given triangle ๐จ๐ฉ๐ช, angle ๐จ has a measure of ๐๐ยฐ, ๐จ๐ช = ๐๐. ๐, and ๐จ๐ฉ = ๐๐, find ๐ฉ๐ช 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.
๐ฉ๐ช๐ = ๐๐๐ + ๐๐. ๐๐ โ ๐(๐๐)(๐๐. ๐)(๐๐จ๐ฌ ๐๐)
๐ฉ๐ช๐ = ๐๐๐ + ๐๐๐. ๐๐ โ ๐๐๐. ๐ ๐๐จ๐ฌ ๐๐
๐ฉ๐ช๐ = ๐๐๐. ๐๐ โ ๐๐๐. ๐ ๐๐จ๐ฌ ๐๐
๐ฉ๐ช = โ๐๐๐. ๐๐ โ ๐๐๐. ๐ ๐๐จ๐ฌ ๐๐
๐ฉ๐ช โ ๐๐. ๐
3. James flies his plane from point ๐จ at a bearing of ๐๐ยฐ east of north, averaging a speed of ๐๐๐ miles per hour for
๐ hours, to get to an airfield at point ๐ฉ. He next flies ๐๐ยฐ west of north at an average speed of ๐๐๐ miles per hour
for ๐. ๐ hours to a different airfield at point ๐ช.
a. Find the distance from ๐จ to ๐ฉ.
distance = rate โ time ๐ = ๐๐๐ โ ๐ ๐ = ๐๐๐
The distance from ๐จ to ๐ฉ is ๐๐๐ ๐ฆ๐ข.
b. Find the distance from ๐ฉ to ๐ช.
distance = rate โ time ๐ = ๐๐๐ โ ๐. ๐ ๐ = ๐๐๐. ๐
The distance from ๐ฉ to ๐ช is ๐๐๐. ๐ ๐ฆ๐ข.
c. Find the measure of angle ๐จ๐ฉ๐ช.
All lines pointing to the north/south are parallel;
therefore, by alternate interior โ โs, the return path
from ๐ฉ to ๐จ is ๐๐ยฐ west of south. Using angles on a
line, this mentioned angle, the measure of the angle
formed by the path from ๐ฉ to ๐ช with north (๐๐ยฐ)
and ๐โ ๐จ๐ฉ๐ช sum to ๐๐๐; thus, ๐โ ๐จ๐ฉ๐ช = ๐๐ยฐ.
d. Find the distance from ๐ช to ๐จ.
The triangle shown provides two sides and the included
angle, so using the law of cosines,
๐๐ = ๐๐ + ๐๐ โ ๐๐๐(๐๐จ๐ฌ ๐)
๐๐ = (๐๐๐. ๐)๐ + (๐๐๐)๐ โ ๐(๐๐๐. ๐)(๐๐๐)(๐๐จ๐ฌ ๐๐)
๐๐ = ๐๐๐โ๐๐๐. ๐๐ + ๐๐๐โ๐๐๐ โ ๐๐๐โ๐๐๐(๐๐จ๐ฌ ๐๐)
๐๐ = ๐๐๐โ๐๐๐. ๐๐ โ ๐๐๐โ๐๐๐(๐๐จ๐ฌ ๐๐)
๐ = โ๐๐๐โ๐๐๐. ๐๐ โ ๐๐๐โ๐๐๐(๐๐จ๐ฌ ๐๐)
๐ โ ๐๐๐. ๐
The distance from ๐ช to ๐จ is approximately ๐๐๐. ๐ ๐ฆ๐ข.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying the Laws of Sines and Cosines
500
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
e. What length of time can James expect the return trip from ๐ช to ๐จ to take?
distance = rate โ time
๐๐๐. ๐ = ๐๐๐ โ ๐๐
๐. ๐ โ ๐๐
distance = rate โ time
๐๐๐. ๐ = ๐๐๐๐๐
๐. ๐ โ ๐๐
James can expect the trip from ๐ช to ๐จ to take between ๐. ๐ and ๐. ๐ hours.
4. Mark is deciding on the best way to get from point ๐จ to point ๐ฉ 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 ๐จ๐ชฬ ฬ ฬ ฬ and ๐ฉ๐ชฬ ฬ ฬ ฬ ). His other option is to cross the creek at ๐จ and take a straight path to ๐ฉ, which
he knows to be a distance of ๐. ๐ ๐ฆ๐ข. 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.
๐จ๐ฉฬ ฬ ฬ ฬ is ๐. ๐๐ยฐ north of east, and ๐จ๐ชฬ ฬ ฬ ฬ is ๐๐. ๐๐ยฐ east of
north. The directions north and east are
perpendicular, so the angles at point ๐จ form a right
angle. Therefore, ๐โ ๐ช๐จ๐ฉ = ๐๐. ๐๐ยฐ.
By the angle sum of a triangle, ๐โ ๐ท๐ช๐จ = ๐๐๐. ๐๐ยฐ.
โ ๐ท๐ช๐จ and โ ๐ฉ๐ช๐จ are angles on a line with a sum of
๐๐๐ยฐ, so ๐โ ๐ฉ๐ช๐จ = ๐๐. ๐๐ยฐ.
Also, by the angle sum of a triangle, ๐โ ๐จ๐ฉ๐ช = ๐๐. ๐๐ยฐ.
Using the law of sines:
๐ฌ๐ข๐ง ๐จ
๐=
๐ฌ๐ข๐ง ๐ฉ
๐=
๐ฌ๐ข๐ง ๐ช
๐
๐ฌ๐ข๐ง ๐๐. ๐๐
๐=
๐ฌ๐ข๐ง ๐๐. ๐๐
๐=
๐ฌ๐ข๐ง ๐๐. ๐๐
๐. ๐
๐
๐ โ ๐. ๐๐๐๐
๐
๐ โ ๐. ๐๐๐๐
Let ๐ represent the distance from point ๐จ to ๐ฉ through point ๐ช in miles.
๐ = ๐จ๐ช + ๐ฉ๐ช
๐ โ ๐. ๐๐๐๐ + ๐. ๐๐๐๐
๐ โ ๐. ๐
The distance from point ๐จ to ๐ฉ through point ๐ช is approximately ๐. ๐ ๐ฆ๐ข. This distance is approximately ๐. ๐ ๐ฆ๐ข.
longer than walking a straight line from ๐จ to ๐ฉ.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying the Laws of Sines and Cosines
501
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
5. If you are given triangle ๐จ๐ฉ๐ช 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:
Given two angles and two sides, one of the angles being
an included angle, it is appropriate to use either the law
of sines or the law of cosines.
Case 2:
Given two angles and the sides opposite those angles,
the law of sines can be applied as the remaining angle
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.