Law of sines

The law of sines states that each side of a triangle divided by the sine of its opposite angle equals the same ratio, equal to the triangle’s circumdiameter.

Circumcircle: The circle passing through all three vertices of a triangle. Each side is a chord whose length equals the circumcircle’s diameter times the sine of its opposite angle.
Triangle: A polygon with three sides, three vertices, and three interior angles. The law relates the three sides to their three opposite angles.
Triangulation: A method of locating a point by measuring angles from known positions. The law converts measured triangle angles and a baseline into distances to a target.
Law of tangents: A triangle identity relating sums and differences of two sides to tangents of half-sums and half-differences of opposite angles. It offers a less commonly used alternative relation among triangle sides and angles.
Inscribed angle theorem: A theorem stating that an inscribed angle measures half the central angle subtending the same arc. It connects a triangle’s side chord to the angle opposite that side.
Sine rule notation: The notation a/sin A = b/sin B = c/sin C for a triangle’s sides and opposite angles. It expresses the law by pairing each side with its opposite angle.
Surveying: The measurement and mapping of land, boundaries, and physical features. Surveyors use triangle measurements to determine inaccessible distances.
Pythagorean theorem: A theorem stating that the square of a right triangle’s hypotenuse equals the sum of the squares of its legs. It solves right triangles through side lengths, while the law of sines applies to triangles of any shape.
Sine: A trigonometric function that gives the ratio of a right triangle’s opposite side to its hypotenuse. The law compares each side with the sine of its opposite angle.
Sine function: A periodic function that assigns a value to an angle and is central to trigonometry. Its values supply the ratios used to compare the triangle’s sides.

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