Law of Cosines Calculator

**Law of Cosines:** c² = a² + b² − 2ab × cos(C) Where: - a, b = two sides - c = third side (opposite angle C) - C = angle between sides a and b **All three forms (any side opposite any angle):** c² = a² + b² − 2ab × cos(C) a² = b² + c² − 2bc × cos(A) b² = a² + c² − 2ac × cos(B) **Worked example: SAS** Triangle with a = 5, b = 7, included angle C = 60°. Find c: c² = 25 + 49 − 2 × 5 × 7 × cos(60°) c² = 74 − 70 × 0.5 c² = 74 − 35 = 39 c = √39 ≈ 6.24 **Worked example: SSS** Triangle with a = 5, b = 7, c = 9. Find angles. cos(C) = (a² + b² − c²) / (2ab) = (25 + 49 − 81)/(70) = -7/70 = -0.1 C = arccos(-0.1) ≈ 95.74° By Law of Sines or repeat: cos(A) = (b² + c² − a²)/(2bc) = (49 + 81 − 25)/(126) = 105/126 ≈ 0.833 A = arccos(0.833) ≈ 33.56° B = 180° − A − C = 180° − 33.56° − 95.74° = 50.70°. Verify: arctan checks all match. **Relation to Pythagorean theorem:** If C = 90°: cos(C) = 0. c² = a² + b² − 0 = a² + b² Pythagorean is special case of Law of Cosines. **For angles 90°:** C < 90° (acute): cos(C) > 0, so c² < a² + b² (triangle less elongated). C = 90° (right): c² = a² + b². C > 90° (obtuse): cos(C) < 0, so c² > a² + b² (triangle more elongated). **Solving for angle (rearranged):** cos(C) = (a² + b² − c²) / (2ab) C = arccos((a² + b² − c²) / (2ab)) **Worked example: surveying** Two points A and B on opposite sides of a river. Surveyor stands at C and measures: distance from C to A = 200 m, from C to B = 150 m, angle at C = 75°. Width of river (distance A to B)? Side c (across river) = √(a² + b² − 2ab cos(C)): a = 200, b = 150, C = 75°. c² = 40,000 + 22,500 − 60,000 × cos(75°) c² = 62,500 − 60,000 × 0.2588 c² = 62,500 − 15,528 = 46,972 c ≈ 216.7 m. River is ~217 m wide at the measurement points. **Triangle area from SAS:** Area = (1/2) × a × b × sin(C) For our example: Area = 0.5 × 200 × 150 × sin(75°) = 15,000 × 0.966 ≈ 14,489 m². **Triangle area from SSS (Heron's formula):** s = (a + b + c) / 2 Area = √(s(s-a)(s-b)(s-c)) For a = 5, b = 7, c = 9: s = 21/2 = 10.5 Area = √(10.5 × 5.5 × 3.5 × 1.5) = √303.19 ≈ 17.41 **Common applications:** - **Surveying**: measuring distances across obstacles. - **Navigation**: triangulation with bearings. - **Astronomy**: stellar/planetary distances. - **Engineering**: truss member analysis. - **Robotics**: forward kinematics. - **CAD**: triangle mesh calculations. - **Game development**: NPC sight lines. - **GPS/GIS**: distance calculations. **Triangle inequality:** For any triangle: each side < sum of other two. If a + b ≤ c (or any permutation), the "triangle" is degenerate (line) or impossible. Check before applying Law of Cosines: ensure given sides satisfy triangle inequality. **Numerical accuracy:** For very acute or obtuse triangles, Law of Cosines can lose precision (subtraction of nearly equal values). Alternative formulations exist for high-precision work. **Inverse: finding angle from sides:** cos(C) = (a² + b² − c²) / (2ab) If cos(C) = 1: C = 0° (degenerate). If cos(C) = -1: C = 180° (degenerate, straight line). If 0 < cos(C) < 1: 0° < C < 90° (acute). If cos(C) = 0: C = 90° (right). If -1 < cos(C) < 0: 90° < C < 180° (obtuse). **Vector form:** For vectors u and v: u · v = |u| |v| cos(C). If u, v are sides of triangle from common vertex: third side w = u - v. |w|² = |u|² + |v|² - 2 u · v = |u|² + |v|² - 2|u||v|cos(C). Same as Law of Cosines, derived from vector dot product. **Common applications:** - **Surveying**: triangulating land boundaries. - **Navigation**: ship positioning from bearings. - **Aviation**: aircraft heading and ground course. - **Astronomy**: parallax calculations. - **Robotics**: arm joint angles from end-effector position. - **Civil engineering**: bridge and truss design. - **Architecture**: roof angles, rafters. - **Sports**: angles in pool/billiards, ball trajectory. **Software:** - **Excel**: combine with COS, ACOS functions. - **Python**: math.acos((a**2 + b**2 - c**2) / (2*a*b)) - **MATLAB**: built-in trig functions. - **CAD**: parametric triangle solving. - **Geometric calculators**: dedicated triangle solvers. **SSS vs SAS:** SSS (3 sides): Always solvable if triangle inequality holds. Unique triangle. SAS (2 sides + included angle): Always solvable. Unique triangle. Both give unique triangle, no ambiguity (unlike SSA). **SSA ambiguity:** For SSA (2 sides + non-included angle): 0, 1, or 2 triangles possible — "ambiguous case". Need Law of Sines and additional logic. **Worked example: angle wedge in machinery** Truss analysis: two members 4 m and 3 m meet at 110° angle. Third member length? c² = 16 + 9 − 2 × 4 × 3 × cos(110°) c² = 25 − 24 × (-0.342) c² = 25 + 8.21 = 33.21 c ≈ 5.76 m If C is obtuse (>90°), c > √(a²+b²) (more than Pythagorean would give). **Pitfalls:** - **Wrong angle/side correspondence**: angle C must be opposite side c. - **Confusing radians and degrees**: most calculators expect radians for trig. - **Numerical precision**: very small angles give precision issues. - **Triangle inequality**: must hold for valid triangle. - **Confusing SSS with SAS**: different input requirements. - **Negative cosine**: angle > 90° gives cos < 0 (obtuse triangle).