In ancient times, the "string" in "hook, three strands, four strings and five" was the hypotenuse of a right triangle, and "hook" and "strand" were the two right-angled sides of a right triangle.
Sine is the ratio of strands to chords, and cosine is the ratio of remaining right angles to chords.
Sine = unit head/chord length
Put the Pythagorean line in the circle. A chord is a line connecting two points on a circle. The largest chord is the diameter. If the chord of a right triangle is placed on the diameter, the chord is the inverse chord of ∠A, that is, sine, and the hook is the remaining chord-cosine.
According to modern theory, sine is the ratio of the opposite side to the hypotenuse of a right triangle.
The modern sine formula is
Sin= The opposite side of a right triangle is larger than the hypotenuse.
For example, the hypotenuse is R, the opposite side is Y, and the adjacent side is A. Sina = y/r of the included angle Ar between hypotenuse R and adjacent side A.
No matter what the values of a, y and r are, the sine value is always greater than or equal to 0 and less than or equal to 1, that is, 0≤sin≤ 1.
cosine
Cosine (cosine function), a kind of trigonometric function. In Rt△ABC (right triangle), ∠ C = 90, and ∠A's cosine is the hypotenuse of its adjacent side than the triangle, that is, cosA=b/c, which can also be written as cosa=AC/AB. Cosine function: f(x)=cosx(x∈R).
Trigonometric function is one of the basic elementary functions, which takes the angle (the most commonly used radian system in mathematics, the same below) as the independent variable, and the angle corresponds to the coordinates where the terminal edge of any angle intersects with the unit circle or its ratio as the dependent variable. It can also be equivalently defined as the lengths of various line segments related to the unit circle.
Trigonometric function plays an important role in studying the properties of geometric shapes such as triangles and circles, and is also a basic mathematical tool for studying periodic phenomena. In mathematical analysis, trigonometric function is also defined as the solution of infinite series or specific differential equation, which allows its value to be extended to any real value or even complex value.
Common trigonometric functions are sine function, cosine function and tangent function. Other trigonometric functions, such as cotangent function, secant function, cotangent function, dyadic function, cofactor function, semidyadic function and semifactorial function, are also used in other disciplines, such as navigation, surveying and engineering. The relationship between different trigonometric functions can be obtained by geometric intuition or calculation, which is called trigonometric identity.