## Law of Cosines

In this lesson, students use right triangle trigonometry and the Pythagorean theorem to develop the law of cosines.

Prior to this lesson, it is likely that students have used the law of sines to solve problems involving certain non-right triangles. During this lesson, students will discover how the law of cosines can be used to solve problems involving non-right triangles for which the law of sines cannot be used.

Introduce the lesson with diagrams such as the following:

Pose these questions to students:

- Are there unique triangles with the side and angle measures given
above? Why?

[Yes. SSS and SAS are two methods that prove triangles congruent. Therefore, if three sides of a triangle are known, or if two sides and an included angle are known, the triangle is unique.] - Why can't the law of sines be used to determine the measures
of the missing angles and/or sides?

[To use the law of sines, you need to know the measure of at least one angle in the triangle. In addition, you need to know the measure of an angle and an opposite side to use the law of sines. Since that information is not given, the law of sines cannot be used.]

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 a copy of the Law of Cosines Activity Sheet to each student, and have students read the introduction. After students read the introduction, make the connection between the problems posed at the beginning of class and the purpose of the activity. Explain to students that during this activity, they will develop the law of cosines, which addresses the cases of triangles for which the law of sines cannot be used.

The Law of Cosines 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. Before students begin the questions on
the activity sheet, the teacher's role is *explainer*. To begin, draw triangle *ABC* with altitude *k* on the chalkboard or overhead projector, as shown below. Then, describe the diagram to students. Point out how the altitude *k* is perpendicular to side *c*.

Ask students why the parts of side *c* can be represented as *x* and *c* – *x*.

[Since the two parts add to equalc, if we use a variable to represent one of the parts, we know that the other part can be represented ascminus the first part.]

For Questions 1‑8 on the activity sheet, the teacher's role changes to *facilitator*.
Students should work with a partner as they complete these questions.
The teacher should circulate around the room to facilitate discussion
between students and to answer questions.

To facilitate partner discussion, the teacher should ask the following questions while circulating around the room:

- If
*k*^{2}=*b*^{2}‑ (*c*‑*x*)^{2}and*k*^{2}=*a*^{2}‑*x*^{2}, what conclusions can be drawn about*b*^{2}‑ (*c*‑*x*)^{2}and*a*^{2}‑*x*^{2}? [The transitive property allows us to conclude that they must be equal.] - Why use cos
*B*to eliminate*x*from the equation*b*^{2}=*a*^{2}+*c*^{2}‑ 2*cx*? [Students may be tempted to say that they used cos*B*because they are developing the law of cosines. However, push students to think more deeply. Cos*B*is the logical choice because it involves*x*, which we want to eliminate, and*a*, which is a side of triangle*ABC*.]

After students have completed Questions 1‑8, instruct them to post the equations they wrote. In this way, the teacher can check the work and make certain that all students have the correct equations.

After reviewing Questions 1‑8, the teacher's role returns to *explainer*.
Point out to students the different parts of the law of cosines.
Explain how the law of cosines written in Question 8 is only one form
of the law of cosines, as different angles and sides can be found by
substituting different variables into the equation. After explaining,
instruct students to work on Question 9 and to check their answers with
a partner. Once students have finished, ask them to state the equations
they wrote.

After students complete the Law of Cosines Activity Sheet, have them use the law of cosines to determine the
measures of the missing sides and angles in the triangles discussed at
the beginning of the lesson. While students are using the law of
cosines, the teacher's role again becomes *facilitator*. The
teacher should circulate around the room while students determine the
missing measures. (Students may have difficulty solving for the angles
at first. The teacher may wish to discuss the algebraic techniques
involved before students begin working.) 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 for the triangles are shown above, with the measures of the unknown sides and angles in green.

To conclude the lesson, ask students to describe the relationship between the given sides and/or angles of a triangle that would require the use of the law or sines or the law of cosines to determine the measures of the missing sides and/or angles in the triangle.

[Law of Sines: Two angles and a side (AAS or ASA); two sides and a non-included angle (SSA). Law of Cosines: Three sides (SSS); two sides and an included angle (SAS).]

**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 and for which triangles the law of cosines can be used to determine the missing angles/sides. Have students justify their answers.
- Using the set of triangles in option 1, have the students determine the missing angles and sides of the triangles.

**Extensions**

- Pose the following question: Why can't the law of sines or the law
of cosines be used to determine the missing sides of a triangle when
the measures of all three angles (AAA) are known?
[There are infinitely many triangles that have congruent angles—these are called

*similar triangles*.] - Provide students with a diagram of a right triangle with the
lengths of both legs given. Ask students to find the length of the
hypotenuse using the Pythagorean theorem. Then, have them find the
length of the hypotenuse using the Law of Cosines. Compare and contrast
the law of cosines with the Pythagorean theorem.
[The Pythagorean theorem could be considered a special case of the law of cosines. Since cos 90° = 0, the law of cosines reduces to

*c*^{2}=*a*^{2}+*b*^{2}for a right angle.]

**Questions for Students**

1. Given ΔABC with a = 4.15 cm, b = 7.65 cm, c = 6.85 cm and ΔABC with a = 7.27 cm, m∠C = 41.62°, b = 5.11 cm, are there unique triangles with the given side and angle measures? Why?

[Yes. SSS and SAS are two methods for which we can prove triangle congruent. Therefore, if three sides of a triangle are known, or if two sides and an included angle of a triangle are known, the triangle is unique.]

2. Why can't the law of sines be used to determine the measures of the missing angles or sides in triangles where three sides or two sides and an included angle are known?

[To use the law of sines, you need to know the measure of at least one angle in the triangle. Therefore, the law of sines cannot be used to determine the measures of the missing angles in the triangle with only three sides given. In addition, to use the law of sines, you need to know the measure of an angle and an opposite side. Therefore, the law of sines cannot be used to determine the measures of the missing angles and side in the triangle with the given sides and included angle, because the side opposite the given angle is unknown.]

3. What relationship between the given sides or angles of a triangle would require the use of the law or sines? Which would require the use of the law of cosines to determine the measures of the missing sides or angles?

[Law of Sines: Two angles and a side (AAS or ASA), two sides and a non-included angle (SSA).

Law of Cosines: Three sides (SSS), two sides and an included angle (SAS).]

**Teacher Reflection**

- How did the students respond as you engaged in different roles throughout the lesson?
- What were some of the ways that the 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?

### Law of Sines and Law of Cosines

Learn the Law of Sines and the Law of Cosines and determine when each can be used to find a side length or angle of a triangle.

### Law of Sines

### Learning Objectives

- Use right triangle trigonometry to develop the law of cosines.
- Use the law of cosines to solve problems.

### NCTM Standards and Expectations

- Analyze properties and determine attributes of two- and three-dimensional objects.

- Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them.

- Use trigonometric relationships to determine lengths and angle measures.