The Law of Cosines: Solving Oblique Triangles

In the world of trigonometry, we often encounter triangles that don't fit the neat right-angle mold. These are called oblique triangles, and they require special tools to solve for their unknown sides and angles. One such tool is the Law of Cosines, a powerful formula that allows us to tackle triangles where the Pythagorean theorem doesn't apply.

Understanding the Law of Cosines

The Law of Cosines is a generalization of the Pythagorean theorem. It states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the included angle.

Mathematically, this can be expressed as:

a² = b² + c² - 2bc cos A

where:

• a, b, and c are the lengths of the sides of the triangle
• A is the angle opposite side a

Solving Oblique Triangles with the Law of Cosines

The Law of Cosines is particularly useful when we have:

• SAS (Side-Angle-Side): Two sides and the included angle are known.
• SSS (Side-Side-Side): All three side lengths are known.

Example 1: SAS Case

Let's say we have a triangle with sides b = 5 units, c = 7 units, and the included angle A = 60 degrees. We want to find the length of side a.

Using the Law of Cosines:

a² = 5² + 7² - 2(5)(7) cos 60°

a² = 25 + 49 - 70(0.5)

a² = 34

a = √34 ≈ 5.83 units

Example 2: SSS Case

Suppose we have a triangle with sides a = 8 units, b = 6 units, and c = 10 units. We want to find the measure of angle A.

Using the Law of Cosines:

8² = 6² + 10² - 2(6)(10) cos A

64 = 36 + 100 - 120 cos A

120 cos A = 72

cos A = 0.6

A = cos^-1(0.6) ≈ 53.13 degrees

Heron's Area Formula

The Law of Cosines is closely related to Heron's Area Formula, which provides a way to calculate the area of a triangle given its side lengths. The formula is:

Area = √(s(s-a)(s-b)(s-c))

where:

• s is the semi-perimeter of the triangle (s = (a + b + c) / 2)
• a, b, and c are the lengths of the sides of the triangle

Conclusion

The Law of Cosines is a powerful tool in trigonometry that allows us to solve for unknown sides and angles in oblique triangles. It's a fundamental concept that finds applications in various fields, including engineering, physics, and surveying. By understanding and applying the Law of Cosines, we can unlock the secrets of these non-right-angled triangles and explore the fascinating world of geometry.