key term - Oblique Triangle

Definition

An oblique triangle is a triangle in which none of the angles are right angles, meaning all three angles are acute or obtuse. These triangles are the focus of the Law of Sines and Law of Cosines, which provide methods for solving for unknown sides and angles in non-right triangles.

5 Must Know Facts For Your Next Test

1. Oblique triangles can be solved using the Law of Sines or the Law of Cosines, which provide formulas for finding unknown sides and angles.
2. The Law of Sines states that the ratio of the sine of an angle to the length of its opposite side is constant for all angles and sides of an oblique triangle.
3. The Law of Cosines provides a formula for finding the length of an unknown side in an oblique triangle given the lengths of the other two sides and the included angle.
4. Oblique triangles have a wide range of applications, including surveying, navigation, and engineering, where the ability to solve for unknown sides and angles is crucial.
5. Solving oblique triangles often requires the use of trigonometric functions, such as sine, cosine, and tangent, to find the missing information.

Review Questions

• Explain the key differences between an oblique triangle and a right triangle, and how these differences impact the methods used to solve for unknown sides and angles.
  • The primary difference between an oblique triangle and a right triangle is that an oblique triangle has no right angles, while a right triangle has one angle that is 90 degrees. This distinction is crucial because it determines the methods used to solve for unknown sides and angles. For right triangles, the Pythagorean Theorem and trigonometric ratios (sine, cosine, and tangent) can be used to solve for missing information. However, for oblique triangles, the Law of Sines and the Law of Cosines must be employed, as these triangles lack the 90-degree angle that simplifies the calculations for right triangles.
• Describe the Law of Sines and how it is used to solve for unknown sides and angles in an oblique triangle.
  • The Law of Sines states that the ratio of the sine of an angle to the length of its opposite side is constant for all angles and sides of an oblique triangle. Mathematically, this can be expressed as: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$, where A, B, and C are the angles of the triangle, and a, b, and c are the lengths of the opposite sides. By knowing any three of these values, the fourth can be calculated using the Law of Sines. This formula allows for the solution of oblique triangles when at least one side and two angles, or two sides and one angle, are known.
• Explain the Law of Cosines and how it can be used to solve for unknown sides in an oblique triangle when the lengths of two sides and the included angle are known.
  • The Law of Cosines provides a formula for finding the length of an unknown side in an oblique triangle given the lengths of the other two sides and the included angle. The formula is: $c^2 = a^2 + b^2 - 2ab\cos C$, where a, b, and c are the lengths of the sides, and C is the angle between the sides a and b. This formula is useful when you know the lengths of two sides and the angle between them, and need to find the length of the third side. The Law of Cosines is an alternative method to the Law of Sines for solving oblique triangles, and it allows for the calculation of missing sides when the necessary information is available.

Related terms

Acute Triangle:

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

Obtuse Triangle:

A triangle in which one angle is greater than 90 degrees.

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