Acute Triangle

Definition

An acute triangle is a triangle in which all three interior angles are less than 90 degrees. This type of triangle is an essential concept in the study of non-right triangles and the application of the Law of Sines and Law of Cosines.

5 Must Know Facts For Your Next Test

1. In an acute triangle, the sum of the three interior angles is always less than 180 degrees.
2. The Law of Sines and Law of Cosines are particularly useful for solving problems involving acute triangles, as they provide equations to find unknown sides or angles.
3. Acute triangles have a unique property where the longest side is always opposite the largest angle, and the shortest side is always opposite the smallest angle.
4. The trigonometric functions, such as sine, cosine, and tangent, are defined for acute angles and can be used to solve for unknown values in acute triangles.
5. Acute triangles are commonly used in various applications, including surveying, engineering, and navigation, where precise angle and side measurements are required.

Review Questions

• Explain how the properties of an acute triangle relate to the application of the Law of Sines.
  • The Law of Sines is a fundamental equation used to solve for unknown sides or angles in non-right triangles, including acute triangles. In an acute triangle, the relationship between the sides and angles is such that the ratio of any side to the sine of its opposite angle is constant. This property allows the Law of Sines to be applied to acute triangles to determine missing information, making it a valuable tool for solving a variety of problems involving these types of triangles.
• Describe how the characteristics of an acute triangle influence the use of the Law of Cosines.
  • The Law of Cosines is another important equation used to solve for unknown sides or angles in non-right triangles, including acute triangles. Unlike the Law of Sines, the Law of Cosines involves the cosine of an angle and the lengths of the sides of the triangle. The fact that all three angles in an acute triangle are less than 90 degrees means that the cosine function can be readily applied to these triangles, allowing the Law of Cosines to be used effectively to determine missing information in acute triangle problems.
• Analyze how the unique properties of an acute triangle, such as the relationship between sides and angles, contribute to the versatility of the Law of Sines and Law of Cosines in solving a variety of non-right triangle problems.
  • The defining characteristic of an acute triangle, where all three angles are less than 90 degrees, gives rise to several unique properties that make acute triangles particularly well-suited for the application of the Law of Sines and Law of Cosines. Specifically, the fact that the longest side is always opposite the largest angle, and the shortest side is always opposite the smallest angle, allows these trigonometric laws to be applied in a straightforward manner to solve for unknown sides or angles. Furthermore, the sum of the three angles being less than 180 degrees provides additional constraints that can be leveraged when using these laws to solve complex non-right triangle problems. The versatility of the Law of Sines and Law of Cosines is greatly enhanced by the distinctive characteristics of acute triangles.

Related terms

Obtuse Triangle: A triangle in which one interior angle is greater than 90 degrees.

Right Triangle: A triangle in which one interior angle is exactly 90 degrees.

Included Angle: The angle between two sides of a triangle.