Law of Sines Calculator

**Law of Sines:** a/sin(A) = b/sin(B) = c/sin(C) = 2R Where: - a, b, c = sides of triangle - A, B, C = opposite angles - R = circumradius (radius of circle through all vertices) **Common usage forms:** To find a side: a = (sin(A) / sin(B)) × b To find an angle: sin(A) = (a / b) × sin(B), then A = arcsin(...) **Worked example: AAS** Given: A = 40°, B = 60°, a = 10. Find C, b, c. C = 180° − 40° − 60° = 80° (angles sum to 180°). b: a/sin(A) = b/sin(B) 10/sin(40°) = b/sin(60°) b = 10 × sin(60°)/sin(40°) b = 10 × 0.866/0.643 ≈ 13.47 c: 10/sin(40°) = c/sin(80°) c = 10 × sin(80°)/sin(40°) c = 10 × 0.985/0.643 ≈ 15.32 Complete triangle: A = 40°, B = 60°, C = 80°; a = 10, b = 13.47, c = 15.32. **Worked example: ASA** Given: A = 40°, c = 10, B = 60°. Find C, a, b. C = 80° (same as above). c is between A and B (ASA configuration). 10/sin(80°) = a/sin(40°) a = 10 × sin(40°)/sin(80°) = 10 × 0.643/0.985 ≈ 6.53 b = 10 × sin(60°)/sin(80°) ≈ 8.79 **Worked example: SSA (ambiguous case)** Given: a = 8, b = 10, A = 40°. Find B. sin(B)/b = sin(A)/a sin(B) = 10 × sin(40°)/8 = 10 × 0.643/8 ≈ 0.804 B = arcsin(0.804) ≈ 53.5° (one solution) OR B = 180° − 53.5° = 126.5° (also valid since sin is positive in both quadrants 1 and 2) Two possible triangles! Check each: For B = 53.5°: C = 180° − 40° − 53.5° = 86.5° (valid). For B = 126.5°: C = 180° − 40° − 126.5° = 13.5° (valid). Both triangles are possible. This is the SSA ambiguous case. **SSA cases:** | Condition | Solutions | |---|---| | a ≥ b | 1 solution (only B ≤ 90°) | | a < b sin(A) | 0 solutions (impossible) | | a = b sin(A) | 1 solution (right triangle at B) | | b sin(A) < a < b | 2 solutions | | a = b | 1 solution (isosceles) | **Cases where Law of Sines works:** | Case | Description | Use Law of Sines? | |---|---|---| | AAS | 2 angles + side opposite one | Yes (unique) | | ASA | 2 angles + included side | Yes (unique) | | SSA | 2 sides + angle opposite one | Yes (0, 1, or 2 solutions) | | SAS | 2 sides + included angle | No (use Law of Cosines) | | SSS | 3 sides | No (use Law of Cosines) | **Triangle area from Law of Sines:** Area = (1/2) × a × b × sin(C) = (1/2) × a × c × sin(B) = (1/2) × b × c × sin(A) Or in terms of circumradius: Area = (abc) / (4R) **Circumradius from Law of Sines:** R = a / (2 × sin(A)) = b / (2 × sin(B)) = c / (2 × sin(C)) Used for circle/triangle problems. **Sum of angles:** A + B + C = 180° (always for a triangle). Find third angle from two others. **Common applications:** - **Surveying**: triangulating positions from known points. - **Navigation**: sextant + known landmarks → ship position. - **Astronomy**: stellar parallax measurements. - **Engineering**: truss design with non-right angles. - **Construction**: angled rafters, hipped roofs. - **Sports**: angles in pool, archery, basketball. - **Computer graphics**: triangle mesh calculations. **SSA ambiguous case visualization:** Picture angle A at one vertex with side b extending out. Side a hangs from the other end of b. Depending on length of a: - a too short: doesn't reach baseline (0 triangles). - a = b sin(A): exactly reaches perpendicular (1 right triangle). - b sin(A) < a < b: reaches baseline in two places (2 triangles). - a = b: special case (1 triangle, isosceles). - a > b: only one valid configuration (1 triangle). **Numerical considerations:** For very obtuse angles: sin(180° − A) = sin(A). Use arcsin carefully — gives value in [-90°, 90°] only. In SSA, check if 180° − arcsin(value) also gives valid triangle. **Extended Law of Sines:** For triangle inscribed in circle of radius R: a/sin(A) = b/sin(B) = c/sin(C) = 2R Equivalent: any chord of length L subtends angle θ at circumference, with L = 2R sin(θ). **Connection to inscribed angles:** Inscribed angle theorem: angle subtended by chord at circumference is half the angle subtended at center. Law of Sines is consequence of this inscribed-angle property. **Worked example: surveying** Two observers (A, B) are 200 m apart. Both observe object at C. Angle BAC = 65°, angle ABC = 75°. Find ACB: ACB = 180° − 65° − 75° = 40°. Distance from A to C using Law of Sines: AC/sin(75°) = 200/sin(40°) AC = 200 × sin(75°)/sin(40°) = 200 × 0.966/0.643 ≈ 300.4 m Distance from B to C: BC = 200 × sin(65°)/sin(40°) = 200 × 0.906/0.643 ≈ 281.8 m **Common pitfalls:** - **Wrong correspondence**: ensure side opposite the angle. - **Degrees vs radians**: sin() typically uses radians in programming. - **SSA ambiguity**: check for two solutions. - **arcsin range**: only returns values 0-90° (need to check obtuse). - **Angle sum**: third angle = 180° − other two. **Software:** - **Excel**: SIN(RADIANS(angle)) for conversion. - **Python**: math.sin(math.radians(angle)) - **MATLAB**: built-in trig functions (use degrees with sind, asind). - **Calculator**: most scientific calculators have degree mode. - **Geometric calculators**: dedicated triangle solvers. **Triangle inequality:** For valid triangle: each side < sum of other two. Verify before solving. For Law of Sines to give valid solution: - All angles must be positive. - All sides must be positive. - Angle sum must equal 180°. **Connection to Law of Cosines:** - **Law of Sines**: relates sides and opposite angles. - **Law of Cosines**: relates one side to other two and included angle. Use whichever fits the given information. **Common applications:** - **Land surveying**: distances and bearings. - **Navigation**: maritime and aerial. - **Construction**: trusses, roofing, framing. - **Architecture**: structural angles. - **Astronomy**: positional astronomy. - **Sports**: trajectory analysis. - **Robotics**: forward kinematics. - **GPS**: trilateration with angles. **Pitfalls:** - **AAS vs SAA**: same setup, just notation. Treat the same. - **SSA ambiguity**: must check both possible solutions. - **Degree/radian conversion**: programming languages use radians. - **Verifying angle sum**: useful sanity check. - **Choosing right form**: solve for unknown side or angle. - **Significant figures**: input precision affects output.