EM@Graphic properties of commonly used trigonometric functions (middle school part)

abstract

• Discuss $\sin ,\cos ,The\; image\; properties\; of\; tan$ , these properties can be analyzed with the help of the unit circle
  • $y=\cos x$ : cosine curve
  • $y=\sin x$ : sinusoidal curve
  • $y=\tan x$ :tangent curve

	Sinusoids and sinusoids
	cosine curve
	Tangent lines and tangent curves

sine function

• $y=\sin x$,$x\in R$ is a sine function, where the independent variable$x$ is the radian value
  • Domain: $\mathbb{R}$
  • Value range: $[-1,1]$
    • 当$x=-\frac{}{2}+2k\pi$,$k\in$Obtain the minimum value at$Z$$-1$
    • 当$x=\frac{}{2}+2k\pi$,$k\in$Get the maximum value 1 when$Z$
  • Boundedness: $|\sin x|⩽1$
  • Parity: odd function
  • Periodicity: The minimum positive period is $2p.m$
  • Solution: $[-\frac{}{2}+2kp,\frac{}{2}+2k\pi ]$ equivalent; $[\frac{}{2}+2kp,\frac{3}{2}+2k\pi ]$ is a decreasing interval;$k\in \mathbb{Z}$
• The sine function describes radians xxThe corresponding sine value of$x$$\sin x$

sine function

• $y=A\sin (\omega x+\varphi )$ , called a sinusoidal function
• Compared to the sine function $y=\sin x$ ,$y=A\sin (\omega x+\varphi )$ adds two parameters$A,oh\_\varphi$ (they are not variables, but constants)
• This function is very common in physical applications and has obvious physical meaning.
• The sum of sinusoidal functions can still be described by circular motion:
  • Assume a certain point P (x, y) P(x,y) in the rectangular coordinate system$P(x,y)$ around the origin$O$ takes radius asRRThe angular velocityof the circular trajectory of$R$ is$Circular\; motion\; of\; \omega$ rad/s
  • Let PP before rotation$The\; position\; of\; P$ is$P_0$, 且$OP$ isϕ \phiterminal edge of$\varphi$
  • After $After\; t$ seconds, click$P$ has arrived at a new location, and$OP$ is$\varphi +$Terminal edge of$\omega$$t$
  • Easy to calculate PPCoordinates of$P$$(x,y)$ about timettFunctional relationship of $t$
    • $x=R\cos (\omega t+)\_$
    • $y=R\sin (\omega t+)\_$
    • The derivation method is as follows:
      • In the Cartesian coordinate system $On\; xOy$ , let the angle$The\; vertex\; of\; \varphi$ is$The\; origins\; of\; the\; O$ coordinates coincide with each other,$The\; initial\; side\; of\; \varphi$ and$Positive\; x-$ axis coincidence
      • And with $O$ is the center of the circle to construct the unit circle,ϕ \phiThe coordinates of the intersection of$\varphi$$E(\cos \varphi ,\sin )\_$
      • And $P_0$Also $\varphi The$ point on the terminal edge, and$O{P}_0=R$ ; then$P_0$The coordinates of $(P_{0x},P_0)$ isEEThe coordinates of$E$$(\sin a,\cos$RRof$\alpha$$)$$Rfold$ :$P_{0x}=R\cos \varphi$ ,$P_0=R\sin \varphi$
      • For the terminal side $\omega t+$PPon$\varphi$$The\; coordinates\; of\; point\; P$ are$(R\cos (\omega t+\varphi ),R\sin (\omega t+\varphi ))$
  • This gives us the sinusoidal function $y=A\sin (\omega x+)\_$

Rotation related concepts

Rotation angular velocity

• Point P ( x , y ) P(x,y) in the coordinate system$P(x,y)$ point$The\; number\; of\; radians\; of\; the\; angle\; that\; O$ rotates in unit time$oh$

rotation period

• $y=R\sin (\omega t+\varphi )$ , pointPPThe time required for$P$$T=\frac{2}{oh}$, this time is also called the rotation period

  • 令$a=\omega t+\varphi$ , let the function$The\; minimum\; positive\; period\; of\; y$ is$T_0$,则$y(t+T_0)$=$y(t)$
  • 即$R\sin \left(\omega \right(t+T_0)+\varphi )$ =$R\sin (\omega t+\varphi )$ , 即$\sin ((\omega t+)\_+\omega T_0)$ =$\sin (\omega t+)\_$
  • So $\sin (a+T_0)=\sin \left(a\right)$
  • $\sin The\; period\; of\; \alpha$ is$2\pi$ , then$\omega T_0$= $2\pi$ , so$T_0=\frac{2}{oh}$

• In addition, the rotation period calculation formula can also be obtained from the expansion and contraction angle of the coordinates.

• Example: $\sin (kx)$,$k=1,2,3,\cdots ,When\; n$ , at$[0,2\pi ]$ to obtain the extreme point of the maximum value 1 (satisfying$kx=\frac{}{2}$) are: $\frac{}{2}$, $\frac{}{4}$, $\frac{}{6}$,$\cdots$ ,$\frac{}{2n}$

Rotation frequency

• Within one second, click PPThe number of times$P$$f=\frac{1}{T}$= $\frac{}{2p.m}$, called the frequency of rotation

first appearance

• Angle $\varphi$ is also calledthe first phase

summary

• The rotation period (rotation frequency) only sums up with $\omega$ is related to$\varphi irrelevant$ _
• $The\; larger\; \omega$ is, the more times the curve fluctuates within a certain range, and vice versa.

Graph and properties of cosine function

• We can induce the formula $y=\sin (\frac{}{2}+x)$=$\cos x$得知,$y=\cos Graph\; of\; x$ and$\sin (\frac{}{2}+$The image of$x$$)\; is\; the\; same$
• Therefore, we can study the cosine function by studying the sine function (the image of the cosine function can be obtained by shifting the sine function 2 units to the left)
• In addition, the cosine function $y=A\cos (\omega x+\varphi )$ can be converted to$y=A\sin (\omega x+\varphi +\frac{}{2})$

nature

• The domain, range and period are the same as the sine function
  • If and only if $x=Pi+2k\pi$,$k\in When\; Z$ , obtain the minimum value$-1$
  • If and only if $x=2k\pi$,$k\in When\; Z$ , the cosine function takes the maximum value 1
• Parity: even function
• Form: $[2kp,Pi+2k\pi ]$,$k\in$The function at$Z$$[p+2kp,2p.m+2k\pi ]$,$k\in The\; Z$ function is monotonically increasing

Graph and properties of tangent function

• Form: { x ∣ x ≠ k π + π 2 , k ∈ Z } \set{x|x\neq{k\pi+\frac{\pi}{2}},k\in\mathbb{Z}}{ x∣x=kπ+2p,k∈Z},
• Range: $\mathbb{R}$
  • In the interval $(-\frac{}{2},\frac{}{2})$ , when$x<\frac{}{2}$And infinitely close to $\frac{}{2}$time, $\tan x$ tends to infinity, recorded as$\tan x\to +\infty$
  • The other side has $x\to -\frac{}{2}$time $\tan x\to -\infty$
• Period: $Pi$
  • Yutan $\tan (\pi +x)$ =$\tan x$ , so$\pi$ is$\tan$a period of$x$
  • And combined with the sinusoidal line in the unit circle, it is easy to show that the minimum positive period is $Pi$
• Parity: odd function
• Solution: $(-\frac{}{2}+k\pi ,\frac{}{2}+k\pi )$,$k\in The\; function\; in\; the\; Z$ interval increases monotonically

Find angles from known trigonometric function values

Within any angle range

• The unit circle can often be used to solve for angles in radians with given values ​​of trigonometric functions.
• For example: It is known that $\sin x=\frac{\sqrt{2}}{2}$,beg
  • x x Possible values ​​of$x$
  • $x\in [-\frac{}{2},\frac{}{2}]$ Conditionxxvalue of$x$
• untie:
  • It can be seen from the unit circle that $\frac{}{4}+2k\pi$ ,$(p-\frac{}{4})+2k\pi$,$(k\in Z)$ all satisfy$\sin x=\frac{\sqrt{2}}{2}$,
    • Expressed as a set: { x ∣ x = 2 k π + π 4 ( k ∈ Z ) } \set{x|x=2k\pi+\frac{\pi}{4}(k\in{\mathbb{Z }})}{ x∣x=2 kp+4p(k∈Z)} $\bigcup$ {   x ∣ x = 2 k π + 3 π 4 ( k ∈ Z )   } \set{x|x=2k\pi+\frac{3\pi}{4}(k\in\mathbb{Z})} { x∣x=2 kp+43 p.m(k∈Z)}
  • 若$x\in [-\frac{}{2},\frac{}{2}]$ , from the unit circle,$x=\frac{}{4}$

Inverse trigonometric function (within limited range)

• The function established by the process of finding angles with known trigonometric function values ​​is called inverse trigonometric function
• Since only injective functions have inverse functions, inverse trigonometric functions are defined according to a monotonic interval within the restricted trigonometric function.

arcsine

• Generally, for the sine function $y=\sin x$ , if the known function value is$and(and\in [-1,1])$ , then$[-\frac{}{2},\frac{}{2}]$ by the one and only$The\; x$ value corresponds to it, recorded as$x=\arcsin y$ ,(其中$-1⩽y⩽1$ ,$-\frac{}{2}⩽x⩽\frac{}{2}$)
• For example: $\sin x=\frac{1}{2}$, $x\in [-\frac{}{2},\frac{}{2}]$ , please$The\; problem\; of\; x$ can be expressed as$\arcsin \frac{1}{2}$

Arc cosine

• In the interval $[0,\pi ]$ meets the condition$\cos x=y$,$(y\in [-1,1])$ to find angle$x$ , recorded as$x=\arccos y$
• example
  1. $\arccos \frac{1}{2}=\frac{}{3}$
  2. It is known that $\cos x=-\frac{\sqrt{2}}{2}$,且$x\in [0,2\pi ]$ , findxxThe value set of $x$
     • It can be seen from the unit circle that the solution set is { 3 π 4 , 5 π 4 } \set{\frac{3\pi}{4},\frac{5\pi}{4}}{ 43 p.m,45 p.m}
     • Expressed by the inverse cosine function: { arccos ⁡ ( − 2 2 ) , π + arccos ⁡ 2 2 } \set{\arccos{(-\frac{\sqrt{2}}{2})},\pi+\arccos\ frac{\sqrt{2}}{2}}{ arccos(−22 ​​),Pi+arccos22 ​​}

arctangent

• For example, write $\tan x=and(and\in R)$ , 且$x\in (-\frac{}{2},\frac{}{2})$ , then for each tangent value$y$ , in the open interval$(-\frac{}{2},\frac{}{2})$ , there is and is only one angle$x$ full$\tan x=y$ ,definition$x=\arctan y$ ,$x\in (-\frac{}{2},\frac{}{2})$
• For example $\arctan \frac{\sqrt{3}}{3}$= $\frac{}{6}$