Law Of Cosine Example Problems

Mastering the Law of Cosines: Example Problems and Deep Dive

The Law of Cosines is a fundamental theorem in trigonometry that extends the Pythagorean theorem to non-right-angled triangles. It provides a powerful tool for solving problems involving the sides and angles of any triangle, regardless of whether it's acute, obtuse, or right-angled. This article will delve into the intricacies of the Law of Cosines, exploring its applications through a series of example problems, covering various scenarios and complexities. We will also unpack the underlying mathematical principles to provide a comprehensive understanding. Understanding the Law of Cosines is crucial for fields ranging from surveying and navigation to engineering and computer graphics.

Understanding the Law of Cosines

The Law of Cosines states the relationship between the lengths of the sides of a triangle and the cosine of one of its angles. For a triangle with sides a, b, and c, and angles A, B, and C opposite to those sides respectively, the Law of Cosines can be expressed in three forms:

• a² = b² + c² - 2bc cos A
• b² = a² + c² - 2ac cos B
• c² = a² + b² - 2ab cos C

Notice that if angle A (or B or C) is 90 degrees (a right angle), cos A = 0, and the formula simplifies to the Pythagorean theorem: a² = b² + c². This highlights the Law of Cosines as a generalization of the Pythagorean theorem.

Example Problems: A Step-by-Step Approach

Let's work through a series of example problems to illustrate the application of the Law of Cosines in different contexts.

Example 1: Finding the Length of a Side

A triangle has sides b = 5 cm and c = 7 cm, and the angle between them, A, is 60 degrees. Find the length of side a.

Solution:

We use the formula: a² = b² + c² - 2bc cos A

Substituting the given values:

a² = 5² + 7² - 2(5)(7) cos 60°

a² = 25 + 49 - 70 (0.5)

a² = 74 - 35

a² = 39

a = √39 ≈ 6.24 cm

Therefore, the length of side a is approximately 6.24 cm.

Example 2: Finding an Angle

A triangle has sides a = 8 cm, b = 10 cm, and c = 12 cm. Find the measure of angle A.

Solution:

We use the formula: a² = b² + c² - 2bc cos A

Rearranging the formula to solve for cos A:

cos A = (b² + c² - a²) / 2bc

Substituting the given values:

cos A = (10² + 12² - 8²) / (2 * 10 * 12)

cos A = (100 + 144 - 64) / 240

cos A = 180 / 240

cos A = 0.75

A = cos⁻¹(0.75) ≈ 41.41°

Therefore, the measure of angle A is approximately 41.41 degrees.

Example 3: Solving an Obtuse Triangle

A triangle has sides a = 15 m, b = 20 m, and angle C = 120°. Find the length of side c.

Solution:

We use the formula: c² = a² + b² - 2ab cos C

Substituting the given values:

c² = 15² + 20² - 2(15)(20) cos 120°

c² = 225 + 400 - 600(-0.5)

c² = 625 + 300

c² = 925

c = √925 ≈ 30.41 m

Therefore, the length of side c is approximately 30.41 meters. This example demonstrates the Law of Cosines' effectiveness even with obtuse angles.

Example 4: Ambiguous Case (SSA)

This is a crucial scenario where the Law of Cosines plays a vital role. Consider a triangle with sides a = 10, b = 12, and angle A = 30°. Find the possible values of side c. Note that this is the ambiguous case (SSA) because we have two sides and an angle opposite one of them. The sine rule might lead to multiple solutions, but the Law of Cosines provides a more direct approach.

Solution:

First, we use the sine rule to find a possible value for angle B:

sin B / b = sin A / a

sin B / 12 = sin 30° / 10

sin B = (12 * 0.5) / 10 = 0.6

B = arcsin(0.6) ≈ 36.87° or B ≈ 143.13°

If B ≈ 36.87°, then C ≈ 180° - 30° - 36.87° ≈ 113.13°.

If B ≈ 143.13°, then C ≈ 180° - 30° - 143.13° ≈ 6.87°.

Now, use the Law of Cosines for each case to find c:

• Case 1 (B ≈ 36.87°, C ≈ 113.13°):

c² = a² + b² - 2ab cos C = 10² + 12² - 2(10)(12) cos 113.13° ≈ 337.96

c ≈ 18.38

• Case 2 (B ≈ 143.13°, C ≈ 6.87°):

c² = a² + b² - 2ab cos C = 10² + 12² - 2(10)(12) cos 6.87° ≈ 4.04

c ≈ 2.01

Therefore, there are two possible values for side c: approximately 18.38 and 2.01. This illustrates the importance of considering the ambiguous case and using the Law of Cosines for accurate solution.

Mathematical Derivation of the Law of Cosines

The Law of Cosines can be derived using the distance formula and basic trigonometry. Consider a triangle with vertices A, B, and C, and let's position vertex C at the origin (0,0) of a Cartesian coordinate system. Let the coordinates of vertex A be (b cos C, b sin C) and the coordinates of vertex B be (a, 0).

Using the distance formula, we can express the length of side c as:

c² = (a - b cos C)² + (0 - b sin C)²

Expanding and simplifying this equation, we get:

c² = a² - 2ab cos C + b² cos²C + b² sin²C

Since cos²C + sin²C = 1 (a fundamental trigonometric identity), the equation simplifies to:

c² = a² + b² - 2ab cos C

Similarly, we can derive the other two forms of the Law of Cosines by repositioning the vertices and applying the distance formula.

Frequently Asked Questions (FAQs)

Q1: When should I use the Law of Cosines instead of the Law of Sines?

A: Use the Law of Cosines when you know:

• Three sides of a triangle (SSS) and need to find an angle.
• Two sides and the included angle (SAS) and need to find the third side.

Use the Law of Sines when you know:

• Two angles and one side (AAS or ASA).
• Two sides and an angle opposite one of them (SSA), but be aware of the ambiguous case.

Q2: Can the Law of Cosines be used for right-angled triangles?

A: Yes, but it simplifies to the Pythagorean theorem in this case.

Q3: What if I have an obtuse angle in my triangle?

A: The Law of Cosines works perfectly well with obtuse angles. Just remember that the cosine of an obtuse angle is negative.

Q4: How can I check my answer?

A: Always verify your results using different methods or approximations. For instance, in solving a triangle, you can check if the angles add up to 180° and if the ratios of sides and angles are consistent with the Law of Sines.

Conclusion

The Law of Cosines is a versatile and powerful tool for solving a wide variety of problems involving triangles. Its ability to handle any type of triangle, including obtuse triangles, and its connection to the Pythagorean theorem make it an essential part of any trigonometry toolkit. By understanding its derivation and practicing its application through various examples, you'll gain a strong foundation in this fundamental mathematical concept. Remember to always carefully consider the given information and choose the most appropriate method – Law of Cosines or Law of Sines – to efficiently solve triangle problems. Mastering these techniques will greatly enhance your problem-solving skills in mathematics and related fields.