Sine Law

In a triangle, the name given to the relationship of proportionality between the measures of the sides and the sine values of the angles opposite each of these sides.

Property

Consider the radius [latex]r[/latex] of any circumscribed circle of a triangle with sides [latex]a[/latex], [latex]b[/latex] and [latex]c[/latex] and angles A, B and C; this gives us the following relationship: [latex]\dfrac{a}{\sin(\textrm{A})} = \dfrac{b}{\sin(\textrm{B})} = \dfrac{c}{\sin(\textrm{C})} = 2r[/latex].

Example

Consider triangle ABC in which side AB measures 15 cm and angles ABC and BAC measure 60° and 50° respectively. We can deduce that angle BCA measures 70°, since 180° − (60° + 50°), then we can calculate the measure a of side BC by posing [latex]\dfrac{a}{\sin(50)} = \dfrac{15}{\sin(70)}[/latex]. Which gives a ≈ 12.23 cm.

Sine Law

Property

Example

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