key term - Obtuse Triangle

Definition

An obtuse triangle is a triangle in which one of the angles is greater than 90 degrees. This type of triangle is particularly relevant in the context of non-right triangles, where the Law of Sines and the Law of Cosines are applied to solve for unknown sides or angles.

5 Must Know Facts For Your Next Test

1. In an obtuse triangle, the angle greater than 90 degrees is called the obtuse angle, while the other two angles are acute angles.
2. The Law of Sines and the Law of Cosines are used to solve for unknown sides or angles in obtuse triangles, where the given information may include a combination of sides and angles.
3. Obtuse triangles are often encountered in real-world applications, such as in surveying, navigation, and various engineering problems.
4. The sum of the angles in an obtuse triangle is always greater than 180 degrees, with the obtuse angle being the largest of the three.
5. Obtuse triangles have unique properties that differentiate them from acute and right triangles, which must be considered when applying the appropriate trigonometric identities and formulas.

Review Questions

• Explain how the Law of Sines is used to solve for unknown sides or angles in an obtuse triangle.
  • The Law of Sines states that the ratio of the sine of an angle to the length of the opposite side is constant for all triangles. In the context of an obtuse triangle, the Law of Sines can be used to solve for unknown sides or angles by setting up a proportion using the known information. This allows you to find the missing values by applying the appropriate trigonometric functions and solving the equation.
• Describe the differences between an obtuse triangle and a right triangle in terms of their properties and the application of the Law of Cosines.
  • The primary difference between an obtuse triangle and a right triangle is the presence of an angle greater than 90 degrees in the obtuse triangle. This affects the application of the Law of Cosines, which is used to solve for unknown sides or angles in non-right triangles. In a right triangle, the Law of Cosines simplifies to the Pythagorean Theorem, whereas in an obtuse triangle, the full Law of Cosines formula must be used to account for the obtuse angle and its relationship to the other sides and angles of the triangle.
• Analyze how the properties of an obtuse triangle, such as the sum of the angles and the relative sizes of the sides, influence the solutions obtained using the Law of Sines and the Law of Cosines.
  • The unique properties of an obtuse triangle, such as the sum of the angles being greater than 180 degrees and the obtuse angle being the largest of the three, directly impact the solutions obtained using the Law of Sines and the Law of Cosines. These properties affect the range of possible solutions, the number of solutions, and the interpretation of the results. For example, in an obtuse triangle, the Law of Cosines may yield two possible solutions for an unknown side, requiring additional information or analysis to determine the appropriate solution. Similarly, the Law of Sines may produce ambiguous or multiple solutions depending on the given information and the relationships between the sides and angles of the obtuse triangle.

Related terms

Acute Triangle:

A triangle in which all three angles are less than 90 degrees.

Right Triangle: A triangle in which one of the angles is exactly 90 degrees.

Angle of Elevation:

The angle between the horizontal line of sight and the line of sight to an object above the observer.