Law of tangents

In trigonometry, the law of tangents or tangent rule is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, a, b, and c are the lengths of the three sides of the triangle, and α, β, and γ are the angles opposite those three respective sides. The law of tangents states that The law of tangents, although not as commonly known as the law of sines or the law of cosines, is equivalent to the law of sines, and can be used in any case where two sides and the included angle, or two angles and a side, are known. To prove the law of tangents one can start with the law of sines: Let so that It follows that Using the trigonometric identity, the factor formula for sines specifically we get As an alternative to using the identity for the sum or difference of two sines, one may cite the trigonometric identity (see tangent half-angle formula). The law of tangents can be used to compute the missing side and angles of a triangle in which two sides a and b and the enclosed angle γ are given. From one can compute α − β; together with α + β = 180° − γ this yields α and β; the remaining side c can then be computed using the law of sines. In the time before electronic calculators were available, this method was preferable to an application of the law of cosines c = , as this latter law necessitated an additional lookup in a logarithm table, in order to compute the square root. In modern times the law of tangents may have better numerical properties than the law of cosines: If γ is small, and a and b are almost equal, then an application of the law of cosines leads to a subtraction of almost equal values, incurring catastrophic cancellation. On a sphere of unit radius, the sides of the triangle are arcs of great circles. Accordingly, their lengths can be expressed in radians or any other units of angular measure. Let A, B, C be the angles at the three vertices of the triangle and let a, b, c be the respective lengths of the opposite sides.

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Law of cosines

In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides and opposite respective angles and (see Fig. 1), the law of cosines states: The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if is a right angle then and the law of cosines reduces to The law of cosines is useful for solving a triangle when all three sides or two sides and their included angle are given.

Trigonometry

Trigonometry () is a branch of mathematics concerned with relationships between angles and ratios of lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.

Law of tangents