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Math • Law of Sines & Cosines (SAS Triangle Solver)

Law of Sines & Cosines (SAS Triangle Solver)

Solve an oblique (non-right) triangle given two sides and the included angle (SAS) using the Law of Cosines and Law of Sines.

Side-Angle-Side (SAS) Triangle Solver
Opposite Side c
6.245
Law of Cosines
Angle ∠A
43.90°
Law of Sines
Angle ∠B
76.10°
180° - ∠A - ∠C
Triangle Area
15.16
½ab sin(C)
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Calculation Guide & Reference

Law of Cosines & Law of Sines (SAS) Mathematics

Law of Sines and Cosines calculator solves an oblique (non-right) triangle given two sides and the included angle (SAS).

Standardized Mathematical Formula
c² = a² + b² − 2ab·cos(C)

The Law of Cosines generalizes the Pythagorean theorem to any triangle, not just right triangles — subtracting a correction term that accounts for the included angle. Once the third side is known, the Law of Sines finds the remaining angles.

Variables:
a, b:Two known side lengths
C:The included angle between sides a and b
c:The unknown third side, opposite angle C
How It Works (Step-by-Step)
  • 1Enter two side lengths (a, b) and the included angle C between them, in degrees.
  • 2View the third side c, the remaining two angles, and the triangle's area.
Real-World Numerical Example
Sides 5 and 7 with a 60° Included Angle

A triangle has sides a = 5 and b = 7, with an included angle C = 60°.

•c² = 5² + 7² − 2(5)(7)cos(60°) = 25 + 49 − 70(0.5) = 74 − 35 = 39.
•c = √39 ≈ 6.24.
•sin(A)/a = sin(C)/c → sin(A) = 5 × sin(60°) ÷ 6.24 ≈ 0.693 → A ≈ 43.9°.
•B = 180° − 60° − 43.9° ≈ 76.1°.
•Area = (1/2) × 5 × 7 × sin(60°) ≈ 15.16.
Result: The third side is ≈6.24, the other two angles are ≈43.9° and ≈76.1°, and the triangle's area is ≈15.16 square units.
Calculation Best Practices & Tips
Examine the discriminant (b² - 4ac) before solving quadratics to instantly know if roots are real, distinct, repeated, or complex.
In statistical analysis, always check both the mean and median to identify if data distribution is skewed by outliers.
When calculating percentage changes, ensure you divide by the original starting baseline value: ((New - Old) / Old) * 100.

Frequently Asked Questions (FAQ)

Use the Law of Cosines for any triangle that isn't a right triangle — it works for all triangles, while the Pythagorean theorem only works when one angle is exactly 90°.

SAS stands for "Side-Angle-Side" — you know two sides and the angle between them (the included angle), which is enough information to solve the entire triangle uniquely.

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