Law of Cosines

Definition

The Law of Cosines is a formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is used primarily to find unknown side lengths or angles in any triangle, not just right triangles. This law is particularly useful when dealing with oblique triangles, where the angles are not 90 degrees, and complements the Law of Sines in solving triangle problems.

5 Must Know Facts For Your Next Test

1. The Law of Cosines can be expressed with the formula: $$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$, where 'c' is the side opposite angle C and 'a' and 'b' are the other two sides.
2. This law allows you to calculate a missing side when two sides and the included angle are known, making it essential for triangle calculations.
3. The Law of Cosines is especially handy in navigation and engineering fields where precise measurements are needed for non-right triangles.
4. It reduces to the Pythagorean theorem when dealing with right triangles because, in that case, $$\cos(90^\circ) = 0$$.
5. You can use the Law of Cosines to find angles as well by rearranging the formula to solve for the cosine value and then applying the inverse cosine function.

Review Questions

• How would you apply the Law of Cosines to find an unknown side in a triangle when given two sides and their included angle?
  • To find an unknown side using the Law of Cosines, use the formula $$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$. Here, 'a' and 'b' are the lengths of the two known sides, and 'C' is the included angle between them. You can plug in these values to calculate 'c', which is the length of the unknown side. This approach is particularly useful in oblique triangles where traditional right triangle methods don't apply.
• What is the relationship between the Law of Cosines and other trigonometric principles, like the Law of Sines?
  • The Law of Cosines complements the Law of Sines in solving triangles. While the Law of Sines helps find unknown angles or sides when given an angle-side pair or two angles, the Law of Cosines allows for solving cases where two sides and an included angle are known. Both laws enable us to work with any type of triangle, but they serve different purposes depending on what information is available.
• Evaluate how mastering the Law of Cosines can enhance problem-solving skills in real-world applications such as navigation or architecture.
  • Mastering the Law of Cosines significantly enhances problem-solving skills in fields like navigation or architecture by enabling precise calculations involving non-right triangles. In navigation, for instance, knowing how to calculate distances between points using only angles and limited information is crucial for plotting accurate routes. Similarly, in architecture, designing complex structures often requires working with triangular frameworks where applying trigonometric principles ensures stability and accuracy. Understanding this law empowers individuals to tackle practical challenges effectively.

Related terms

Oblique Triangle: A triangle that does not contain a right angle, which can be either acute (all angles are less than 90 degrees) or obtuse (one angle is greater than 90 degrees).

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.

Law of Sines: A formula that states the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides and angles in any triangle.