Interactive triangle demonstrating the law of sines and cosines. Drag vertices and watch ratios and formulas update.
Move a vertex to reshape the triangle and observe the law of sines and law of cosines update for the new geometry. The visualizer connects each side length with its opposite angle, making it useful for seeing why both laws describe the same triangle from different sets of measurements. The most common mistake is pairing a side with an adjacent angle in the sine ratio. Each side must be paired with the angle directly opposite it.
Dragging any vertex changes the three side lengths and usually all three interior angles. The displayed relationships are recalculated from that single drawn triangle. Moving a vertex toward the opposite side produces a flatter triangle, while moving it away changes the balance between the opposite side and angle. The diagram therefore acts as the input rather than merely illustrating a fixed calculation.
The sine relationship compares opposite pairs as a/sin(A) = b/sin(B) = c/sin(C). The cosine relationship combines all three sides with one included angle, such as c^2 = a^2 + b^2 - 2ab cos(C). As the shape changes, the numerical terms change while these equalities continue to describe the current nondegenerate triangle.
Not from one displayed configuration alone. It can help illustrate the ambiguity by showing differently shaped triangles, but the current drawing represents only one triangle at a time. To decide whether zero, one, or two triangles satisfy particular SSA measurements, analyze the given constraints and check every candidate angle and side against them.
Use the cosine relationship when the known data are three sides or two sides with their included angle. Use the sine relationship when a complete opposite side-angle pair is known along with another side or angle. For SSA data, the sine calculation may produce a supplementary angle that also needs checking against the remaining measurements.
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