Non-right Triangle

Definition

A non-right triangle is a triangle in which none of the angles are 90 degrees, or right angles. This type of triangle is the focus of the Law of Cosines, which provides a method for solving for unknown sides or angles in a triangle when the given information does not include a right angle.

5 Must Know Facts For Your Next Test

1. The Law of Cosines is used to solve for unknown sides or angles in a non-right triangle when the given information does not include a right angle.
2. 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 the triangle, and $C$ is the angle between sides $a$ and $b$.
3. Non-right triangles can be classified as acute (all angles less than 90 degrees) or obtuse (one angle greater than 90 degrees).
4. The Ambiguous Case in the Law of Cosines occurs when solving for an unknown angle, and there are two possible solutions based on the given information.
5. Solving non-right triangles using the Law of Cosines requires the use of inverse trigonometric functions, such as $\cos^{-1}$, to find the unknown angle.

Review Questions

• Explain the purpose of the Law of Cosines and how it is used to solve for unknown sides or angles in a non-right triangle.
  • The Law of Cosines is a formula used to solve for an unknown side or angle in a non-right triangle when the lengths of two sides and the included angle, or the lengths of all three sides, are known. This is necessary because the simpler trigonometric ratios (sine, cosine, and tangent) that are used to solve right triangles cannot be applied to non-right triangles. The Law of Cosines provides an alternative method for determining the unknown side or angle by using the known information about the triangle's sides and angles.
• Describe the Ambiguous Case that can occur when using the Law of Cosines to solve for an unknown angle in a non-right triangle.
  • The Ambiguous Case in the Law of Cosines occurs when solving for an unknown angle in a non-right triangle, and there are two possible solutions based on the given information. This happens when the sum of the squares of the known sides is equal to the square of the known side opposite the unknown angle. In this situation, the inverse cosine function can yield two different angle values that satisfy the Law of Cosines equation. Understanding the Ambiguous Case is important when applying the Law of Cosines, as it requires the solver to consider both possible solutions and determine which one is valid based on the context of the problem.
• Analyze how the classification of a non-right triangle as acute or obtuse can affect the application of the Law of Cosines.
  • The classification of a non-right triangle as acute or obtuse can impact the way the Law of Cosines is applied to solve for unknown sides or angles. In an acute non-right triangle, all three angles are less than 90 degrees, and the Law of Cosines can be used straightforwardly to determine the unknown information. However, in an obtuse non-right triangle, where one angle is greater than 90 degrees, the Law of Cosines must be applied with additional consideration. The obtuse angle can affect the sign of the cosine value, which in turn impacts the calculation of the unknown side or angle. Recognizing the triangle's classification is crucial when using the Law of Cosines to ensure the correct application of the formula and the interpretation of the results.

Related terms

Law of Cosines: A formula used to solve for an unknown side or angle in a non-right triangle when the lengths of two sides and the included angle, or the lengths of all three sides, are known.

Oblique Triangle: Another term for a non-right triangle, where all three angles are acute (less than 90 degrees) or obtuse (greater than 90 degrees).

Ambiguous Case: A situation that can occur when using the Law of Cosines to solve for an unknown angle in a non-right triangle, where there are two possible solutions.