Trigonometric Triangle Solver

Trigonometry—derived from the Greek words for "triangle" and "measure"—is the scientific study of the relationships between the side lengths and interior angles of triangles. While right-angled triangles can be solved using simple Pythagorean calculations ($a^2 + b^2 = c^2$) and direct trigonometric functions (Sine, Cosine, Tangent), general oblique triangles require a much deeper mathematical framework. Oblique triangles possess no right angles ($90^\circ$). To solve these geometries, mathematicians use two core theorems: the Law of Sines and the Law of Cosines.

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📐 The Core Solving Theorems: Sines & Cosines

Oblique triangles are classified as either acute (all interior angles are strictly less than $90^\circ$) or obtuse (one interior angle is strictly greater than $90^\circ$). To solve for their missing dimensions, we apply theorems based on the configuration of our initial variables:

• The Law of Cosines: Established as a generalization of the Pythagorean Theorem, it handles configurations where two sides and the included angle (SAS) or all three sides (SSS) are known:
  c² = a² + b² - 2ab · cos(C)
  This equation allows us to bridge the gap between adjacent side lines and solve for the third side or isolated angles without needing opposing parameters.
• The Law of Sines: States that the ratio of the length of a side of a triangle to the sine of its opposite interior angle is constant across all three pairs:
  a / sin(A) = b / sin(B) = c / sin(C)
  This law is highly efficient for configurations where we have an angle and its opposing side, such as in ASA or AAS layouts.

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🔗 Boundary Constraints & Edge Validations

Not every set of random inputs can create a valid triangle. Before running trigonometric loops, rigorous boundary validations are required to confirm physical plausibility:

For a **SSS (Side-Side-Side)** layout, the inputs must strictly satisfy the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side:
a + b > c && a + c > b && b + c > a
If you attempt to solve a triangle with sides 3, 4, and 8, no physical triangle can exist. The two shorter sides (3 and 4) can never span far enough to meet above the long base line (8), producing a mathematical impossibility (an arccosine domain error during calculation).

For angle-based layouts (ASA, AAS), the sum of the two inputted interior angles must be strictly **less than 180°**. If two angles add up to 180° or more, the two opposing lines will remain parallel or diverge, never intersecting to form a closed third vertex.

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💻 Vertex Mapping & SVG Cartesian Plots

Plotting a solved general triangle on a digital monitor is a beautiful intersection of geometry and coordinate systems. Because SVG viewports use absolute pixel coordinates $(X, Y)$ relative to the top-left corner, we must translate our abstract lengths ($a, b, c$) and angles ($A, B, C$) into exact Cartesian points.

To achieve this, we align the triangle's base along the horizontal plane. By placing **Vertex A** at the local origin $(0, 0)$ and **Vertex B** along the positive x-axis at coordinate $(c, 0)$, we establish a firm base line. Using polar-to-Cartesian trigonometric formulas, we solve for **Vertex C**:
X_c = b · cos(A)
Y_c = b · sin(A)
To display this cleanly, we find the absolute bounding box enclosing these points, scale them proportionally to fit our display viewport, and invert the y-coordinates (since SVG's y-axis increases downwards). This produces a perfectly scaled, dynamically centered rendering of the exact triangle.