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11.2.1: Resources and Key Concepts

  • Page ID
    197626
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    Key Concepts

    Theorems

    • Law of Cosines: If the angles of a triangle are \(A\), \(B\), and \(C\), and the opposite sides are respectively \(a\), \(b\), and \(c\), then:
      • \(a^2 = b^2 + c^2 - 2bc\cos(A)\)
      • \(b^2 = a^2 + c^2 - 2ac\cos(B)\)
      • \(c^2 = a^2 + b^2 - 2ab\cos(C)\)

    Common Mistakes

    • Order of Operations: When simplifying the Law of Cosines, follow the order of operations carefully. In an expression like \(b^2 + c^2 - 2bc\cos(A)\), the term \(2bc\cos(A)\) must be calculated before subtracting from the sum of squares. Do not subtract \(2bc\) from \(b^2+c^2\) first.
    • Incorrectly Applying Law of Sines After Law of Cosines: After using the Law of Cosines to find an angle, you can use the Law of Sines to find a second angle. To avoid the ambiguous case, you should always choose to find the angle opposite the smallest remaining side, as this angle is guaranteed to be acute.

    This page titled 11.2.1: Resources and Key Concepts was last modified on Tue, 08 Jul 2025 17:46:37 GMT and is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.

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