Prior to this lesson, it is likely that students have only used
trigonometry to solve problems involving right triangles. During this
lesson, students will discover how trigonometry can be used to solve
problems involving non-right triangles as well. Introduce the lesson
with a diagram of triangle ABC drawn on the chalkboard or overhead
projector:
To make the discussion concrete, add the following measurements to the diagram: a = 7 cm, ∠B = 45°, and ∠C = 75°.
Pose the following questions to students:
- Is there a unique triangle with the given angle and side measures?
Why?
[Yes. ASA is one method for proving that triangles are congruent.
Therefore, if two angles and a side are known, the triangle is unique.]
- How might you determine the measures of the missing angle and
sides? (You don't actually have to find the missing measures; rather,
describe how you think you would determine them.)
[Since the sum of the three angles in every triangle is 180°, subtract
the two angle measures from 180° to determine the measure of the third
angle. Finding the side lengths is more difficult; finding a method for
determining the side lengths is the purpose of this lesson.]
Instruct students to think about the questions individually. Then,
allow them to discuss their ideas with a partner. Then discuss the
questions with the whole group.
After discussing the questions, distribute one Law of Sines Activity Sheet to each student, and have students read the
introduction. After students have read the introduction, make the
connection between the problem posed at the beginning of class and the
purpose of the activity. Explain to students that during this activity,
they will develop a method for solving problems that involve non‑right
triangles as well as right triangles.
Law Of Sines Activity Sheet
The activity sheet provides students with specific instructions
that guide them through the activity. The teacher will take on
different roles as students engage in the activity. For the first
six questions of the activity sheet, the teacher's role is explainer. Before students begin working on the questions from the activity sheet, explain the following:
- An altitude of a triangle extends from a vertex to the opposite side and forms a right angle with the opposite side.
- Drawing an altitude of triangle ABC creates two right triangles.
- Since
two right triangles are created, right triangle trigonometry can be
used to describe the relationships between the angles and sides of each
triangle.
- Because triangle ABC shares angles and sides with
the two right triangles, the relationships between the angles and sides
of the right triangles can be used to describe the relationships
between the angles and sides of triangle ABC.
Ask the students to work on Questions 1 and 2 on the activity
sheet. If you choose to project the figure onto the board and draw on it, be sure that the students understand that your role is the explainer. Thus, they should be telling you how to sketch the altitude. This will help to reinforce the relationships between the right triangles and
triangle ABC.
Instruct students to work on Question 3 and to check their
answers with a partner. After students have had time to work, ask
students to provide the equations that they wrote. Write the equations
on the board.
Explain to students that each equation contains the altitude k. Ask students why each equation contains the altitude k [k is the side opposite both angles, A and C,
in each right triangle]. Instruct students to work on Question 4 and to
check their answers with a partner. After students have had time to
work, ask students to provide the equations that they wrote. Write the
equations on the overhead.
Explain to students that since both equations in Question 4 are equal to k,
they can be set equal to each other. Ask students why is this true.
[This is true because of the transitive property—since two expressions
are equal to the same expression, they are equal to each other.]
Instruct students to work on Question 5 and to check their answers with
a partner. After students have had time to work, ask students to
provide the equation that they wrote. Write the equation on the
overhead.
Explain to students that the equation in Question 5 no longer involves k.
Ask students why this is true. [This is the result of the transitive
property.] Instruct students to work on Question 6 and to check their
answers with a partner. After students have had time to work, ask them
to provide the equation that they wrote. Write the equation on the
overhead.
Explain to students that the equations in Questions 5 and 6 are
equivalent. Ask students why they think the equation in Question 6
might be preferable to the equation in Question 5. [The angles and
opposite sides are grouped together in the equation.]
For Questions 7‑13 on the activity sheet, the teacher's role changes to facilitator.
Students should work with a partner as they complete Questions 7‑13.
The teacher should circulate around the room to facilitate discussion
between students and to answer questions. After students finish
working, the teacher's role changes to connector. It is
important that students connect the different parts of the law of sines
into one cohesive whole. To assist students in connecting the various
parts of the law of sines, ask the following questions:
- Why can we conclude that sin B/b = sin C/c?
[Again, the transitive property allows us to conclude that they are equal to one another because they are both equal to sin A/a.] - The law of sines can also be written as sin A/a = sin B/b = sin C/c.
How does this extended equation illustrate the three equations that
were developed during the lesson?
[Each of the three pairwise
equalities represented by this extended equation illustrates one of the
equations developed on the activity sheet.]
After students complete the Law of Sines Activity Sheet, have them use the law of sines to determine the measures of ∠C, b, and c
in the triangle discussed at the beginning of the lesson. While
students are using the law of sines, the teacher's role again becomes facilitator.
The teacher should circulate around the room while students determine
the missing measures. After students have had time to determine the
measures and check answers with their partner, the teacher can choose
students to post solutions on the board.
The solutions to the triangle are shown above. The measure of ∠A is found by subtracting the two known angle measures from 180°; the measures of sides b and c are found using the Law of Sines.
Once students have developed the law of sines, it is important
that they realize the potential of this law as well as its limitations.
At this point in the lesson, the teacher's role becomes questioner.
Remind the students that for the triangle at the beginning of class,
two angles and a non‑included side were given. Therefore, if two angles
and a non‑included side of a triangle are given, the law of sines can
be used to determine the missing angle and sides, and since ASA is a
valid method for proving triangles congruent, the triangle will be
unique.
Pose the following questions to students:
- Besides ASA, for what other combinations of angles and sides could
the law of sines be used to determine the missing angles and sides of a
triangle?
[AAS (two angles and a non-included side) and SSA (two sides
and a non‑included angle).]
- Of the combinations of angles and sides, AAS and SSA, which
guarantee a unique triangle? Why?
[AAS guarantees a unique triangle because ASA can be used to prove
triangles congruent. However, SSA does not guarantee a unique triangle
because SSA provides insufficient information to prove triangles
congruent.]
To close the lesson, describe the importance of the law of sines, in
that it can be used to solve problems involving non‑right triangles.
However, emphasize that the law of sines cannot solve all problems
involving non‑right triangles. If two angles and a side or two sides
and a non‑included angle of a triangle are known, the law of sines can
be used to determine the missing angles and sides of the triangle.
However, when two sides and a non‑included angle of a triangle are
known, the missing angles and sides may not be unique.
Assessment Options
- Provide students with a set of triangles with given angles and
sides. Ask students to determine for which triangles the law of sines
can be used to determine the missing angles and sides. Have students
justify their answers.
- Using the same set of triangles as above, have the students
determine the missing angles and sides of the triangles for which the
law of sines is applicable.
Extensions
1. The following extension relates to the SSA case, for which the law of
sines can be used, but for which there is no guarantee of a unique
triangle. Refer to ΔABC which is not drawn to scale.
- Use the law of sines to determine the missing angles and sides if m∠A = 41°, a = 24, and b = 10.
[There is only one possible triangle. Although two values, 15.86° and 164.14°, result for the measure of ∠B, the second answer is impossible. Therefore, m∠b = 15.86, m∠C = 123.14°, and c = 30.63]
- Use the law of sines to determine the missing angles and sides if m∠A = 32°, a = 6.5;, and b = 9.2.
[There are two possible triangles, because the measure of ∠B could be either 48.59° or 131.41°. If m∠B = 48.59°, then m∠C = 99.41° and c = 12.1. If m∠B = 131.41°, then m∠C = 16.59° and c = 3.5.
- Use the law of sines to determine the missing angles and sides if m∠B = 58°, a = 5, and b = 3.4.
[There are no solutions, because the law of sines would yield that sin A = (5 × sin 58°) / 3.4 = 1.2471, which is impossible.]
2. Move on to the next lesson,
Law of Cosines.
Questions for Students
1. Given: ΔABC with m∠A = 79.39°, m∠B = 44.77°, and a = 7.07 cm. Is there a unique triangle with the following angle and side measures? Why?
[Yes. AAS is one method for which we can prove two triangles are congruent. Therefore, if two angles and a side are known, the triangle is unique.]
2. The two equations below are equivalent.
Why might one form be preferred over another?
[The second equation groups sides and angles together, so it might be preferable. However, either will work when attempting to find the sides or angles of a triangle, if three values within the equation are known.]
3. If two angles and a non‑included side of a triangle are known, the law of sines can be used to determine the missing angle and sides. For what other combinations of angles and sides could the law of sines be used to determine the missing angles and sides of a triangle?
[ASA (two angles and an included side) and SSA (two sides and a non‑included angle).]
4. Of ASA and SSA, which guarantees a unique triangle? Why?
[ASA guarantees a unique triangle because ASA can be used to prove triangles congruent. However, SSA does not guarantee a unique triangle because SSA is insufficient information to prove triangles congruent.]
Teacher Reflection
- How did the students respond as you engaged in different roles throughout the lesson?
- What were some ways that students illustrated that they were actively engaged in the learning process?
- Did
you find it necessary to make adjustments while teaching the lesson? If
so, what adjustments, and were these adjustments effective?
- What worked with classroom behavior management? What didn't work? How would you change what didn't work?