Cosine
Cosine is a fundamental trigonometric function that establishes a relationship between the angles and sides of a triangle, particularly in the context of right triangles. It is defined as the ratio of the length of the adjacent side to the length of the hypotenuse, and is commonly denoted as "cos" in mathematical equations. To aid in remembering this and other trigonometric functions, the mnemonic SOHCAHTOA is often used, covering Sine, Cosine, and Tangent. The cosine function not only serves practical purposes in geometry and engineering but also plays a crucial role in more advanced mathematical concepts, such as the law of cosines. This law generalizes the Pythagorean theorem, applicable to all types of triangles, and it was developed through contributions from various mathematicians over time, including ancient Greeks and Renaissance figures like Francois Viete. Additionally, the concept of cosine extends beyond traditional Euclidean geometry into areas such as spherical geometry, highlighting its versatility in mathematical applications. Understanding cosine is essential for solving problems related to triangles and can have broader implications in fields like physics and engineering.
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Cosine
Cosine is one of the trigonometric functions, all of which relate the angles of a triangle to the lengths of its sides. The cosine (represented in equations as cos) of an angle of a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse, sometimes written as b/h. The mnemonic SOHCAHTOA (pronounced "soak a toe") is sometimes recommended for the three main trigonometric functions: Sine, Opposite/Hypotenuse, Cosine, Adjacent/Hypotenuse, Tangent, Opposite/Adjacent. The hypotenuse is the side opposite the right angle, and is always the longest side of a right triangle. The adjacent side is the one that includes both the right angle and the angle in question. The third side is the opposite side.
The trigonometric functions can also be described in terms of the rise, run, and slope (rise/run) of a line segment relative to the horizontal, and because cosine is traditionally introduced to students after sine, the mnemonic "cosine is second, run is second" reminds that cosine takes the angle of the line segment and tells its horizontal run, when the length of the line is 1.
Overview
Cosine and other trigonometric functions allow the angles of a triangle to be determined if the sides are known, or vice versa, which is useful in numerous engineering applications and other areas. The law of cosines states that:
Where a, b, and c are the sides opposite points A, B, and C of a triangle, and y is the angle between a and b. This is actually a generalization of the Pythagorean theorem, which applies to only right triangles and states that c2 = a2 + b2, since in a right triangle, the cosine of y will equal 0. The law of cosines was discovered long after the Pythagorean theorem, though many mathematicians understood that some sort of generalization of Pythagoras must exist. Special cases were contributed early on. For example, the ancient Greek Euclid generalized the Pythagorean theorem to an obtuse triangle by dividing it into two right triangles. Francois Viete, a French mathematician who served the royal court as codebreaker, contributed the law of cosines in the late sixteenth century, using the angles of an oblique triangle in his proof. Viete was one of the innovators of western algebra—the synthesis of western mathematical knowledge with algebraic tools contributed by the Arab world—and self-published a number of treatises on mathematical laws and formulas.
Versions of the law of cosines have also been defined for non-Euclidean geometries. In spherical geometry, a triangle is defined as the space enclosed by the three arcs of great circles connecting the points u, v, and w on the unit sphere. For angles A, B, and C with opposite sides a, b, and c, the spherical law of cosines says that cos a = cos b cos c + sin b sin c cos A.
Bibliography
Aigner, Martin, and Gunter M. Ziegler. Proofs from the Book. New York: Springer, 2014.
Grunbaum, Branko, and G. C. Shephard. Tilings and Patterns. New York: Dover, 2015.
Hanna, Gila. Explanation and Proof in Mathematics. New York: Springer, 2014.
Kahn, David. Attacking Trigonometry Problems. New York: Dover, 2015.
Larson, Ron. Algebra and Trigonometry. Boston: Cengage, 2015.
Millman, Richard, Peter Shiue, and Eric Brendan Kahn. Problems and Proofs in Numbers and Algebra. New York: Springer, 2015.