Solve ASA, AAS, and SSA (ambiguous) triangle cases
Enter any combination of sides and angles (minimum 3 values including at least 1 side).
Example: ASA case (A=40°, B=60°, c=10)
Enter any known values above to calculate the rest
The Law of Sines is one of the fundamental relationships in trigonometry, providing a powerful method for solving triangles when you know certain combinations of sides and angles. It establishes an elegant proportional relationship between the sides of a triangle and the sines of their opposite angles. For a comprehensive introduction, see the Law of Sines explained at Khan Academy.
For any triangle with sides a, b, c and opposite angles A, B, C:
a/sin(A) = b/sin(B) = c/sin(C)
This can also be written as: sin(A)/a = sin(B)/b = sin(C)/c. Both forms are equivalent and express the same relationship - the ratio of any side to the sine of its opposite angle is constant for all three sides of the triangle.
The Law of Sines is ideal for solving these triangle configurations:
When you know two angles and the side between them. Since the angles in a triangle sum to 180°, you can immediately find the third angle. Then use the Law of Sines to find the remaining sides. This case always produces a unique solution.
Example: A = 40°, B = 60°, c = 10
C = 180° - 40° - 60° = 80°
a = c × sin(A) / sin(C) = 10 × sin(40°) / sin(80°)
When you know two angles and a non-included side. Like ASA, you can find the third angle immediately. Then apply the Law of Sines to find the remaining sides. This case also always has a unique solution.
When you know two sides and an angle opposite one of them. This is called the "ambiguous case" because it can produce zero, one, or two valid triangles depending on the measurements:
Our calculator automatically detects the ambiguous case and alerts you when two solutions exist, displaying the first valid solution while noting that an alternative exists.
To solve a triangle using the Law of Sines:
When should you use the Law of Sines versus the Law of Cosines?
The Law of Sines appears in many practical applications:
To use the Law of Sines Calculator:
The calculator handles all edge cases, validates your input, and provides complete triangle solutions including area and perimeter. The visualization clearly labels all sides and angles for easy verification.
The Law of Sines is used to solve triangles when you know certain angle-side combinations, specifically ASA (two angles and the included side), AAS (two angles and a non-included side), or SSA (two sides and an angle opposite one of them). It establishes the proportion: a/sin(A) = b/sin(B) = c/sin(C), allowing you to find missing sides or angles.
The ambiguous case occurs in SSA (Side-Side-Angle) situations where you know two sides and an angle opposite one of them. Depending on the measurements, this can result in zero, one, or two valid triangles. When sin(opposite angle) gives a value between 0 and 1, there might be two possible angles (one acute and one obtuse) that satisfy the equation, each creating a different valid triangle.
Use the Law of Sines when you know ASA, AAS, or SSA (angle-side combinations where the angle is opposite one of the known sides). Use the Law of Cosines when you know SSS (all three sides) or SAS (two sides and the included angle between them). The Law of Cosines is more versatile but requires more complex calculations, while the Law of Sines is simpler when applicable.
Yes, the Law of Sines works perfectly for right triangles, but basic trigonometry (SOH-CAH-TOA) is usually simpler. For a right triangle, sin(90°) = 1, so the Law of Sines becomes: a/sin(A) = b/sin(B) = c/1 = c. This means the hypotenuse equals the ratio value, which matches what you'd get from sin(A) = a/c.
When solving SSA, calculate sin(opposite angle) = (opposite side × sin(known angle)) / adjacent side. If this value is between 0 and 1, two angles could work: θ₁ = arcsin(value) and θ₂ = 180° - θ₁. Check if both produce valid triangles (all angles positive and summing to 180°). If both work, you have the ambiguous case with two solutions.
The Law of Sines emerges from the relationship between a triangle's area and its sides and angles. The area can be expressed as ½×b×c×sin(A) or ½×a×c×sin(B) or ½×a×b×sin(C). Since all three expressions equal the same area, you can set them equal and derive the proportion a/sin(A) = b/sin(B) = c/sin(C). This relationship also equals the diameter of the triangle's circumscribed circle. For a deeper understanding, watch the proof of the Law of Sines video tutorial.
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