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Fig. 7a – Proof of the law of cosines for acute angle γ by "cutting and pasting". Fig. 7b – Proof of the law of cosines for obtuse angle γ by "cutting and pasting". One can also prove the law of cosines by calculating areas. The change of sign as the angle γ becomes obtuse makes a case distinction necessary. Recall that
Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle. For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse.
The spherical cosine formulae were originally proved by elementary geometry and the planar cosine rule (Todhunter, [1] Art.37). He also gives a derivation using simple coordinate geometry and the planar cosine rule (Art.60). The approach outlined here uses simpler vector methods. (These methods are also discussed at Spherical law of cosines.)
The original proof is based on the Taylor series expansions of the exponential function e z (where z is a complex number) and of sin x and cos x for real numbers x . In fact, the same proof shows that Euler's formula is even valid for all complex numbers x.
The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles: [85] = + ...
The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion.
Morrie's law is a special trigonometric identity. Its name is due to the physicist Richard Feynman , who used to refer to the identity under that name. Feynman picked that name because he learned it during his childhood from a boy with the name Morrie Jacobs and afterwards remembered it for all of his life.
is the imaginary unit, which by definition satisfies =, and π {\displaystyle \pi } is pi , the ratio of the circumference of a circle to its diameter . Euler's identity is named after the Swiss mathematician Leonhard Euler .