Andromeda
Note

Law of Cosines

Definition

The Law of Cosines (Theorem 11.4.3) is a general formula that relates the sides and one angle of any triangle (right or oblique). It is essentially a generalization of the Pythagorean theorem.

a2=b2+c2−2bccos⁡αa^2 = b^2 + c^2 - 2bc \cos \alpha b2=a2+c2−2accos⁡βb^2 = a^2 + c^2 - 2ac \cos \beta c2=a2+b2−2abcos⁡γc^2 = a^2 + b^2 - 2ab \cos \gamma

  • How to read: “The square of a equals b squared plus c squared minus two b c cosine alpha, and similarly for the other sides.”
  • Meaning: Generalizes the Pythagorean theorem to any triangle; the −2bccos⁡α-2bc\cos\alpha term corrects for non-right angles.

Why It Matters

Right triangles are the exception; the law of cosines is the rule. It provides the universal “correction factor” needed to calculate distance and angles in any triangle, making it the workhorse of navigation, surveying, and structural design.

Core Concepts

  • Side forms (SAS / SSS workhorse) a2=b2+c2−2bccos⁡αa^2 = b^2 + c^2 - 2bc \cos \alpha b2=a2+c2−2accos⁡βb^2 = a^2 + c^2 - 2ac \cos \beta c2=a2+b2−2abcos⁡ gammac^2 = a^2 + b^2 - 2ab \cos \ gamma

    • How to read (for side a): “The square of a equals b squared plus c squared minus two b c cosine alpha.”
    • Meaning / when to use: Angle α\alpha is opposite side aa (included between sides bb and cc). The term −2bccos⁡α-2bc \cos \alpha corrects for the angle not being 90°. Use for:
      • SAS: you know two sides and the included angle → solve directly for the third side.
      • SSS: you know all three sides → rearrange to solve for any angle (see angle form below). Always solve for the largest angle first when using SSS to avoid ambiguous case issues later with Law of Sines.

  • Angle form (solving for unknown angle) cos⁡α=b2+c2−a22bc\cos \alpha = \frac{b^2 + c^2 - a^2}{2bc}

    • How to read: “The cosine of alpha equals the fraction b squared plus c squared minus a squared all over two b c.”
    • Meaning / when to use: Direct rearrangement of the side form. The numerator is “how much the sides fail to satisfy Pythagorean.” Positive numerator → acute angle; negative → obtuse angle. This is the preferred way to find an angle when you know all three sides (SSS case).

  • Pythagorean Connection (special case)

    • When γ=90∘\gamma = 90^\circ, cos⁡90∘=0\cos 90^\circ = 0, so c2=a2+b2c^2 = a^2 + b^2.
    • How to read: “When gamma equals ninety degrees, the cosine of gamma is zero, so c squared equals a squared plus b squared.”
    • Meaning: Right-angle limit—cosine term vanishes and the law reduces to the Pythagorean theorem.

Connected Concepts

Connected notes