Andromeda
Note

Law of Sines

Definition

The Law of Sines (Theorem 11.4.2) relates the side lengths of any triangle (right, acute, or obtuse) to the sines of its opposite angles. It states that the ratio of a side’s length to the sine of its opposite angle is constant: sin⁡αa=sin⁡βb=sin⁡γcorasin⁡α=bsin⁡β=csin⁡γ\frac{\sin \alpha}{a} = \frac{\sin \beta}{b} = \frac{\sin \gamma}{c} \quad \text{or} \quad \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}

  • How to read: “The sine of alpha over a equals the sine of beta over b, which equals the sine of gamma over c.”
  • Meaning: Every side-to-sine ratio in a triangle shares the same constant value—the link between geometry and trigonometry for any triangle shape.

Why It Matters

Proportion is the key to geometry. The law of sines allows us to solve for missing sides and angles by simply knowing the ratios between them, providing the essential toolkit for measuring what we cannot directly reach.

Core Concepts

  • The Law of Sines (two equivalent writings) sin⁡αa=sin⁡βb=sin⁡γc\frac{\sin \alpha}{a} = \frac{\sin \beta}{b} = \frac{\sin \gamma}{c} or asin⁡α=bsin⁡β=csin⁡γ\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}

    • How to read (first form): “The sine of alpha over a equals the sine of beta over b, which equals the sine of gamma over c.”
    • How to read (second form): “The side a over the sine of alpha equals b over the sine of beta, which equals c over the sine of gamma.”
    • Meaning (second form): Same constant ratio; this form is often easier when solving for a side length.
    • Meaning: The ratio of any side to the sine of its opposite angle is the same constant for the whole triangle. This is a direct consequence of area being expressible three different ways (see First Principles).

  • When Law of Sines is the right tool

    • ASA or AAS (two angles + any side): Find the third angle first (180° sum), then use the proportion to find the remaining sides. Usually the easiest.
    • SSA (two sides + angle opposite one of them): This is the famous ambiguous case — can produce 0, 1, or 2 triangles. Always draw the height h = b sin α from the vertex and compare the given side a to h and to b.

  • Ambiguous Case (SSA) decision tree (memorize the logic) Given angle α (acute), side a opposite it, and another side b:

    • If a < h (= b sin α) → no triangle (side too short to reach).
    • If a = h → exactly one right triangle.
    • If h < a < b → two possible triangles (the ambiguous case).
    • If a ≥ b → exactly one triangle.
    • If the given angle is obtuse, the rules change (a must be > b for any triangle to exist).

  • Law of Sines vs Law of Cosines decision rule

    • Know two angles + side → Law of Sines (fast).
    • Know SAS or SSS → Law of Cosines (safer, no ambiguous case).
    • Know SSA → Law of Sines but be extremely careful with the ambiguous case (or use Law of Cosines on the third side first to avoid ambiguity).

Connected Concepts

Connected notes