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📈college algebra review

key term - Included Angle

Citation:

Definition

The included angle is the angle formed between two sides of a polygon or a triangle. It is the angle that is enclosed or 'included' between the two specified sides.

5 Must Know Facts For Your Next Test

  1. The included angle is one of the three angles in a triangle, and its value is essential for applying the Law of Cosines.
  2. The Law of Cosines is used to solve for unknown sides or angles in non-right triangles, where the included angle is a key piece of information.
  3. Knowing the included angle allows you to use the Law of Cosines to find the length of the third side of a triangle, given the lengths of the other two sides and the included angle.
  4. The included angle is measured in degrees and can be acute, obtuse, or right, depending on the triangle.
  5. The sum of the three angles in a triangle, including the included angle, is always 180 degrees.

Review Questions

  • Explain the role of the included angle in the Law of Cosines for solving non-right triangles.
    • The included angle is a crucial piece of information required to apply the Law of Cosines. The Law of Cosines states that the square of one side of a triangle is equal to the sum of the squares of the other two sides, minus twice the product of those two sides multiplied by the cosine of the included angle. This formula allows you to solve for an unknown side or angle in a non-right triangle, given the lengths of the other two sides and the measure of the included angle between them.
  • Describe how the included angle is related to the other angles in a triangle.
    • The included angle is one of the three angles that make up a triangle. The sum of the three angles in a triangle, including the included angle, is always 180 degrees. This relationship means that if you know the measure of two angles in a triangle, you can calculate the measure of the third angle, which would be the included angle. Understanding this connection between the included angle and the other angles in a triangle is essential for solving problems involving non-right triangles.
  • Analyze how the value of the included angle affects the solution when applying the Law of Cosines.
    • $$\c^2 = a^2 + b^2 - 2ab\cos C$$ The value of the included angle, represented by $C$ in the Law of Cosines formula, directly impacts the solution. As the included angle $C$ increases, the $\cos C$ term decreases, which in turn increases the value of the third side $c$. Conversely, as the included angle $C$ decreases, the $\cos C$ term increases, resulting in a smaller value for the third side $c$. This relationship between the included angle and the calculated side length is crucial to understand when applying the Law of Cosines to solve for unknown sides or angles in non-right triangles.

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