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Math & Conversions Law of Sines Calculator Law of Sines Calculator ($\frac{\sin(A)}{a} = \frac{\sin(B)}{b}$) This calculator solves any non-right triangle when provided with two angles and one side (AAS/ASA) or two sides and one non-included angle (SSA). Enter exactly three known values below. The tool will identify the case and provide all solutions. Enter Exactly 3 Known Values Side a Side b Side c Angle $\alpha$ (A) Angle $\beta$ (B) Angle $\gamma$ (C) Solve Triangle Solution 1: Solution 2 (Ambiguous Case): Step-by-Step Solution The Law of Sines Formula The Law of Sines is used to solve oblique triangles (non-right triangles). It establishes a direct proportionality between the side lengths and the sines of their opposite angles: $$\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}$$ To find a missing value, you must always use a complete ratio (a known side and its opposite angle) as your reference. Triangle Solution Cases for the Law of Sines The Law of Sines is effective in solving three specific input cases, based on the known parts of the triangle: **AAS (Angle-Angle-Side):** Two angles and a non-included side. (Always has one unique solution) **ASA (Angle-Side-Angle):** Two angles and the included side. (Always has one unique solution) **SSA (Side-Side-Angle):** Two sides and a non-included angle. (The **Ambiguous Case**) The Ambiguous Case (SSA) When solving the SSA case, the Law of Sines may yield **zero, one, or two** possible triangles. This ambiguity occurs because the sine function is positive in both the first and second quadrants ($\sin(\theta) = \sin(180^\circ - \theta)$). To determine the number of solutions, the height ($h$) of the triangle is compared to the known side opposite the known angle ($a$) and the adjacent side ($b$). $$h = b \sin(\alpha)$$ **Zero Solutions:** If $a < h$ or $a \le b$ and $\alpha \ge 90^\circ$. **One Solution:** If $a = h$ or $a \ge b$. **Two Solutions:** If $h < a < b$. Frequently Asked Questions (FAQ) What is the Law of Sines formula? The Law of Sines is a ratio relating the length of a side of a triangle to the sine of its opposite angle: $\\frac{a}{\\sin(\\alpha)} = \\frac{b}{\\sin(\\beta)} = \\frac{c}{\\sin(\\gamma)}$. What is the Ambiguous Case (SSA)? The Ambiguous Case (Side-Side-Angle) occurs when the known parts can create zero, one, or two different triangles. This ambiguity arises because the sine of an angle is equal to the sine of its supplement ($\sin(\\theta) = \\sin(180^\\circ - \\theta)$). How do you find the area using the Law of Sines? If you know two sides and the included angle (SAS), the area is calculated by: $\\mathcal{A} = \\frac{1}{2} ab \\sin(\\gamma)$. The Law of Sines is integral to finding the necessary angle for this formula if the full SAS set is not known initially. When should I use the Law of Cosines instead? You should use the Law of Cosines for the SSS (Side-Side-Side) and SAS (Side-Angle-Side) cases, as it is less prone to ambiguity than the Law of Sines in these scenarios. Law of Sines Formulas The Core Ratio $$\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)}$$ Find Angle $\beta$ $$\beta = \sin^{-1}\left(\frac{b \sin(\alpha)}{a}\right)$$ Ambiguous Case (h) $$h = b \sin(\alpha)$$ Related Triangle Tools Law of Cosines Calculator (SSS, SAS) Triangle Solver (General) Heron's Formula Calculator (Area) Right Triangle Calculator Degrees to Radians Converter
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