Solving for a Side

Definition

Solving for a side refers to the process of determining the length of an unknown side in a non-right triangle using the Law of Cosines. This technique allows you to find the length of a side given the measurements of the other two sides and the angle between them.

5 Must Know Facts For Your Next Test

1. The Law of Cosines states that $c^2 = a^2 + b^2 - 2ab\cos(C)$, where $a$, $b$, and $c$ are the lengths of the sides of a triangle, and $C$ is the angle between sides $a$ and $b$.
2. To solve for an unknown side length in a non-right triangle, you need to know the lengths of the other two sides and the angle between them.
3. Rearranging the Law of Cosines formula allows you to isolate the unknown side length and solve for it.
4. Solving for a side using the Law of Cosines is particularly useful when the triangle does not have a right angle, as the Pythagorean Theorem cannot be applied.
5. The Law of Cosines can be used to solve for any of the three sides of a non-right triangle, not just the unknown side.

Review Questions

• Explain the purpose and application of the Law of Cosines in solving for a side of a non-right triangle.
  • The Law of Cosines is a formula that allows you to solve for an unknown side length in a non-right triangle. This is useful when the triangle does not have a right angle, as the Pythagorean Theorem cannot be applied. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles, enabling you to calculate the length of a side given the measurements of the other two sides and the angle between them.
• Describe the step-by-step process of using the Law of Cosines to solve for a side in a non-right triangle.
  • To solve for a side using the Law of Cosines, you need to know the lengths of the other two sides and the angle between them. First, you would identify the unknown side and the given information. Then, you would rearrange the Law of Cosines formula to isolate the unknown side length. This may involve solving a quadratic equation or using the inverse cosine function. Finally, you would plug the known values into the formula and solve for the unknown side length.
• Analyze the relationship between the Law of Cosines and the Pythagorean Theorem, and explain how this relationship is used in solving for a side in a non-right triangle.
  • The Law of Cosines is a generalization of the Pythagorean Theorem, which can only be applied to right triangles. In a right triangle, the cosine of the angle between the two known sides is 0, simplifying the Law of Cosines formula to the Pythagorean Theorem. However, in non-right triangles, the cosine of the angle between the sides is not necessarily 0, and the Law of Cosines must be used to solve for the unknown side length. This relationship between the two formulas highlights the versatility of the Law of Cosines in solving a wider range of triangle problems.

Related terms

Law of Cosines:

A formula that relates the lengths of the sides of a triangle to the cosine of one of its angles, allowing for the calculation of an unknown side length.

Non-right Triangle:

A triangle in which none of the angles are 90 degrees, requiring the use of the Law of Cosines to solve for unknown side lengths.

Trigonometric Ratios:

The ratios of the sides of a right triangle, such as sine, cosine, and tangent, which are used in the Law of Cosines to solve for unknown side lengths.