Properties and images of sine function

The image and property of sine function is sine function y=sinx.

Sine function is a function in the form of y=Asin(ωx+φ)+k, where a, ω, φ and k are constants and ω≠0. Function y=Asin(ωx+φ), (a >；; 0，ω& gt； 0), the image of x∈R can be regarded as obtained by the following methods: firstly, all points on the image of y=sinx are left (φ >; 0) or to the right (Φ < 0) parallel movement | Φ | unit.

Then shorten the abscissa of each point (ω >); 1) or elongation (0; 1) or shortened (0; 0，ω& gt； 0), when x ∈ [0, +∞] represents a vibration quantity.

A refers to the maximum distance from the equilibrium position when the quantity vibrates, which is usually called vibration amplitude; The time required for reciprocating vibration once is T=2π/ω, which is called the period of vibration. The frequency of reciprocating vibration in unit time f= 1/T=ω/2π, which is called the frequency of vibration, and ωx+φ is called the phase.

φ is called the initial phase (that is, the phase when x=0). The geometric drawing of sine function y=Asin(ωx+φ) is as follows: Take any point C on the horizontal axis Ox as the center, make a circle with radius, and intersect with the X axis at A0 and A6. With A0 as the starting point, randomly divide the circle (1 2 in figure1).

Take OA ′ 0 =-φ/ω on the X-axis, and then take A ′ 0 as A ′ i (I = 0, 1 2, …, 12) to make A ′ ia ′ i+1= π/6ω, that is, the period is 2π. The image of sine function is also called sine curve or sine wave.