Sine, mathematical term, istrigonometric functionAny one of the right trianglesAcute angle∠ AOpposite sideAndhypotenuseThe ratio of, called the sine of ∠ A, is recorded as sinA (abbreviated from the English word sine), that is, sinA=the opposite side/bevel of ∠ A.
In ancient times, sine is the combination ofstringScale of.
The "string" in the ancient "Gou San Gu Si Xian Wu" isright triangle"Gou" and "Gu" are the two right sides of a right triangle.
Figure 1
Sine is the ratio of strands to strings,cosineIs the ratio of the remaining right angle side to the chord.
Sine=strand length/chord length
Put the hook, strand and chord into the circle.The string iscircumferenceThe last two points are connected.The largest chord is the diameter.holdright angleThe chord of the triangle is placed on the diameter, and the strand is the chord of ∠ A, that is, the sine, and the hook is the remaining chord - cosine.
In modern parlance, sine is the ratio of the opposite side to the hypotenuse of a right triangle.
As shown in Figure 1, the hypotenuse is r, the opposite edge is y, and the adjacent edge is a.Sinusoidal sine A=y/r of the angle Ar between the hypotenuse r and the adjacent edge a
Regardless of a, y, r, the sine value is always greater than or equal to 0 and less than or equal to 1, that is, 0 ≤ sin ≤ 1.
Trigonometric functions have important applications in complex numbers.In physics,trigonometric functionIt is also a commonly used tool.
In RT △ ABC, if acute angle A is determined, thenOpposite sideThe ratio to the adjacent edge is then determined. This ratio is called the tangent of angle A and recorded as tanA
That is, tanA=opposite side of angle A/adjacent side of angle A
Similarly, in RT △ ABC, if acute angle A is determined, then the opposite side of angle A ishypotenuseThe ratio of is then determined. This ratio is called the sine of angle A and is recorded as sinA
That is, sinA=opposite side of angle A/bevel side of angle A
Similarly, in RT △ ABC, if the acute angle A is determined, then the ratio of the adjacent side of angle A to the bevel will be determined accordingly. This ratio is called the ratio of angle Acosine, recorded as cosA
That is, cosA=the adjacent side of angle A/the bevel of angle A
Sine function
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Generally, in rectangular coordinate system, givenUnit circleFor any angle α, make the vertex of angle α coincide with the origin, the starting edge coincides with the non negative half axis of x-axis, and the ending edge intersects the unit circle at point P (u, v), then the ordinate v of point P is called angle αSine function, recorded as v=sin α.Generally, x represents the independent variable, that is, x represents the size of the angle, and y represents the value of the function. Thus, the trigonometric function y=sin x of any angle is definedDefine FieldsbyAll real numbers, the value field is [- 1,1].
The proof that the perimeter of an ellipse is equal to the length of a specific sine curve in a period:
The cylinder with radius r intersects with an inclined plane to obtain an ellipse. The angle between the inclined plane and the horizontal plane is α, and a circle passing through the short diameter of the ellipse is intercepted.Turn a θ angle starting from a certain intersection point of the circle and ellipse.Then the height of the point on the ellipse and the point perpendicular to the circle can be obtained
f(c)=r tanα sin(c/r)
r: Cylinder radius
α: The angle between the ellipse and the horizontal plane
c: Corresponding arc length (moving from a certain intersection point to a certain direction)
The above is to prove the simple process, then the circumference of the ellipse (x * cos α) ^ 2+y ^ 2=r ^ 2 is equal to the length of the sine curve of f (c)=r tan α sin (c/r) in a period, and a period T=2 π r, which is exactly the circumference of a circle.
triangle
The Law of Sines is a basic theorem in trigonometry, which points out that "in any plane triangle, the ratio of each side to the sine value on its opposite corner is equal and equal to the diameter of the circumscribed circle", that is
(r is the radius of the circumscribed circle, D is the diameter).
As early as the 2nd century AD, the sine theorem was thought to be an ancient Greek astronomerPtolemy (C. Ptolemy).The famous Arab astronomer al Birunj (973-1048) in the Middle Ages also knew the theorem.However, Nasir al Din, an Arab mathematician and astronomer in the 13th century, was the first to clearly express and prove the theorem.In Europe, Jewish mathematician Germson stated the theorem in his Sine, String and Arc: "In all triangles, the ratio of one side to the other is equal to the ratio of the sine of its diagonal", but he did not give a clear proof.German mathematician in the 15th centuryRegmontanusOn《On various triangles》The sine theorem is given in, but the proof of Nasirdin is simplified.French mathematician in 1571Weida(F. Viete, 1540-1603) proved the sine theorem with a new method in his Mathematical Rules.Later, the German mathematician B. Pitiscus (1561-1613) in his "Trigonometry" followed Weida's method to prove the sine theorem[2]。
Unit circle
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The image shows a common angle measured in radians.The counter clockwise measurement is positive angle and the clockwise measurement isNegative angle。Set a line crossing the origin, the same asxThe positive half of the axis gets an angle θ and intersects the unit circle.Of this intersectionyThe coordinates are equal to sin θ.The triangle in this figure ensures the formula;The radius is equal to the hypotenuse and has length 1, so we have sin θ=y/1。The unit circle can be considered as a way to view an infinite number of triangles by changing the lengths of adjacent and opposite sides and keeping the hypotenuse equal to 1.
For angles greater than 2 π or less than − 2 π, simply continue to rotate around the unit circle.In this way, the sine becomes 2 πPeriodic function：
