Law of Cosines

Definition

The Law of Cosines is a mathematical formula used to find a side or angle in any triangle when certain other sides or angles are known. This law extends the Pythagorean theorem, allowing calculations for non-right triangles by relating the lengths of the sides of the triangle to the cosine of one of its angles. It is particularly useful in situations where two sides and the included angle, or all three sides, are known.

5 Must Know Facts For Your Next Test

1. The Law of Cosines can be expressed with the formula: $$c^2 = a^2 + b^2 - 2ab\cos(C)$$, where 'c' is the side opposite angle 'C', and 'a' and 'b' are the other two sides.
2. When used in conjunction with known angles and sides, this law allows for solving triangles that are not right-angled.
3. In cases where all three sides are known, the Law of Cosines can help find each angle using rearranged formulas.
4. The Law of Cosines helps to derive the Law of Sines, making it easier to solve triangles by switching between these laws based on what information is available.
5. It is especially handy in real-world applications such as navigation, physics problems involving non-right angled components, and architecture.

Review Questions

• How does the Law of Cosines apply to finding unknown sides in oblique triangles?
  • The Law of Cosines is essential for determining unknown sides in oblique triangles by providing a relationship between the sides and angles. For example, if two sides and the included angle are known, you can use the formula $$c^2 = a^2 + b^2 - 2ab\cos(C)$$ to find the third side. This approach allows for solving triangles that do not contain right angles and makes it possible to use cosine relationships in various practical scenarios.
• Discuss how you would use the Law of Cosines to transition into using the Law of Sines for solving triangles.
  • You can start by applying the Law of Cosines to find one unknown angle or side in a triangle. Once you have at least one angle calculated, you can then switch to using the Law of Sines. For instance, after finding angle A using the cosine formula, you can utilize $$\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$$ to find any remaining angles or sides. This combined approach provides flexibility in tackling different types of triangle problems efficiently.
• Evaluate how well understanding the Law of Cosines equips you for practical applications in fields such as navigation or engineering.
  • Mastering the Law of Cosines significantly enhances problem-solving capabilities in practical applications like navigation and engineering. In navigation, determining distances across terrain without direct measurement often relies on solving oblique triangles using this law. Similarly, engineers may need to calculate forces acting on structures where angles are not 90 degrees. A solid grasp of this law ensures accurate calculations and informed decision-making in real-world scenarios where precision is critical.

Related terms

Triangle: A polygon with three edges and three vertices, commonly categorized as equilateral, isosceles, or scalene based on the lengths of its sides.

Cosine: A trigonometric function that relates the angle of a triangle to the ratio of the length of the adjacent side to the length of the hypotenuse.

Oblique Triangle: A triangle that does not contain a right angle; it can be either acute or obtuse.