Formulas & tables/Mathematics/Plane geometry/Law of sines

Law of sines

asin⁡α=bsin⁡β=csin⁡γ=2R\dfrac{a}{\sin\alpha}=\dfrac{b}{\sin\beta}=\dfrac{c}{\sin\gamma}=2R

Result unit: length units

What it describes

The ratio of a side to the sine of the opposite angle is constant and equals the diameter of the circumscribed circle.

Symbols

a,b,ca, b, c
sides of the triangle
α,β,γ\alpha,\beta,\gamma
angles opposite the sides
RR
radius of the circumscribed circle

Where it comes from

In any triangle, the ratio of a side to the sine of the opposite angle is constant and equals the diameter of the circumscribed circle (2R2R). It links sides with angles.

When to use it

Triangles (not necessarily right), when you know one side and two angles, or two sides and an angle.

Example

In a triangle a=10a=10, α=30∘\alpha=30^\circ, β=45∘\beta=45^\circ. Find side bb.
Substitutionb=asin⁡βsin⁡α=10⋅sin⁡45∘sin⁡30∘b=\dfrac{a\sin\beta}{\sin\alpha}=\dfrac{10\cdot\sin45^\circ}{\sin30^\circ}.
Resultb=10⋅0.7070.5≈14.1b=10\cdot\dfrac{0.707}{0.5}\approx14.1.

Related formulas

Law of cosines Area of a triangle (two sides and the included angle)

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