EM@trigonometric function induction formula@trigonometric function formula simplification

induction formula

• The trigonometric functions of acute angles are simple and easy to find (easy to express)
• For any angle, there are certain relationships between its various trigonometric functions that need to be discussed
• The most basic and commonly used formula for inducing trigonometric functions

Same end angles

• In the Cartesian coordinate system, $a,a+2k\pi$,$k\in If\; the\; terminal\; sides\; of\; K$ are the same, they are defined by trigonometric functions. It is easy to know that the two trigonometric functions are equal.
  • $\cos (a+2kp)=\cos a$
  • $\sin (a+2kp)=\sin a$
  • $\tan (a+2kp)=\tan a$

Internalization of trigonometric functions for any angle

• According to the relationship between the trigonometric functions of the same terminal angles, all absolute values ​​exceed one cycle ( $2\pi$ or$-2\pi$ ) can be converted into an absolute value less than$2\pi$ angle to calculate

Opposite angle

• About xxAngle that is symmetrical about$the\; x-\; axis$

• $The\; opposite\; angle\; of\; \alpha$ is$-a$

• Obviously, the terminal sides of opposite angles are about $The\; x-$ axis is symmetrical. According to the definition of trigonometric functions, we have

  • $\cos (-\alpha )$ =$\cos a$

  • $\sin (-\alpha )$ =$-\sin a$

  • $\tan (-\alpha )$ =$\sin (-\alpha )/\cos (-\alpha )$ =$-\tan a$

• summary

  • $\cos \alpha$ is an even function, and$\sin a,\tan \alpha$ is an odd function

Negative angle normalization of any angle

• From the conclusion of opposite angles, any negative angle can be converted into a positive angle to calculate and express
• For example $\cos (-\frac{}{4})$ =$\cos \frac{}{4}$; $\sin (-\frac{7}{3})$ =$-\sin \frac{7}{3}$, $\tan (-\frac{}{3})$ =$-\tan \frac{}{3}$

Origin symmetry angle

• Angle α \alphaThe starting side of$\alpha$$In\; the\; rectangular\; coordinate\; system\; of\; the\; positive\; x$ -axis,α \alphaThe angle corresponding to the terminal side of$\alpha$$a+(2k\_+1)\pi \psi \alpha$ + ( 2 k − 1 ) π \alpha+(2k-1)\pi$a+(2k\_-1)\pi$,$k\in Z$ ; Let’s call this type of angleα \alphaThe origin symmetry angleof$\alpha$

  • In $[0,$α \alphawithin$2$$\pi$$)$$The\; terminal\; side\; of\; \alpha$ is symmetrical about the origin and is expressed as$a\pm Pi$
  • Then according to the formula for generating angles with the same terminal side, we get the expression for the symmetrical angle at the origin.
  • The coordinate signs of the points on the two terminal edges that are symmetrical about the origin are inverted.

• $The\; symmetry\; angle\; between\; \alpha$ and its origin$a+(2k\_+1)\; Trigonometric\; function\; relationship\; of\pi$ :

  • $\cos (a+(2k\_+1\left)\pi \right)$ =$-\cos a$

  • $\sin (a+(2k\_+1\left)\pi \right)$ =$-\sin a$

  • $\tan (a+(2k\_+1\left)\pi \right)$ =$\tan a$

Any angle sharpening

• Let the set of odd numbers be N 1 = { 2 k ∣ k ∈ Z } N_1=\set{2k|k\in{\mathbb{Z}}}N1​={ 2k _∣k∈Z} , the even number set is$N_2=\{2k\_+1|k\in Z\}$

  • $\sin (a+k\pi )$ =$\{\begin{array}{ll}-\sin a& k\in N_1\\ \sin & k\in N_2\end{array}$

  • $\cos (a+k\pi )$ =$\{\begin{array}{ll}-\cos a& k\in N_1\\ \cos & k\in N_2\end{array}$

  • $\tan (a+k\pi )$ =$\tan \alpha$,$k\in \mathbb{Z}$

• After the above analysis and discussion, it can be seen that any angle can be transformed into $a+k\pi$ ,$(|\alpha |⩽\frac{}{2})$ form

• Then according to $a,-\; The\; trigonometric\; function\; relationship\; of\alpha$ is further transformed into$[0,\frac{}{2}]$ to represent and calculate acute angle trigonometric functions

summary

• The first three sets of formulas above: (three terminal edge relationships correspond to three sets of formulas)
  1. Same end angles
  2. Opposite angle
  3. Origin symmetry angle
• Collectively called induced formulas , the formulas can be reasoned and memorized with the help of the terminal edge of any angle.
• The induction formula can be used to find the value of a trigonometric function or simplify a trigonometric function.

complementary angle

• Two angles are complementary $(a,Pi-\alpha )$ , then their terminal sides are about$y-$ axis symmetry

  • $Pi-\alpha$ can be viewed as$-a+\pi$ , that is, first aboutxxDraw− α -\alpha$symmetrically\; about\; the\; x-\; axis$$-\alpha$ final edge, and then make$-\alpha$ is symmetric about the origin$-a+Pi$
  • Press α \alpha respectivelyWhen the terminal side of$\alpha$$a,Pi-\alpha About$ yy_$y-$ axis symmetry

• Known $\alpha$ ,$Pi-\alpha$ are complementary angles, then$\sin (p-\alpha )$ =$\sin a$ ? $\cos (p-\alpha )$ =$-\cos a$

  • whereπ $Pi-\alpha$ is equivalent to$\alpha About$ xx_After symmetry on the$x-\; axis,\; then\; symmetry\; about\; the\; origin$
  • $\sin (p-\alpha )$ =$-\sin (-\alpha )$ =$-(-\sin \alpha )$ =$\sin a$
  • Similarly, $\cos (p-\alpha )$ =$-\cos (-\alpha )$ =$-\cos a$

• In short, the sine values ​​of two complementary angles are equal, and the cosine values ​​are opposite to each other.

与$\frac{}{2}$Trigonometric functions related to angle expressions

• Formula (the latter two can convert α \alpha in the first two$\alpha$ is replaced by$-\alpha$ is obtained,$5\sim 8$ can be directly obtained from the ratio relationship between each and the first two formulas)
  1. $\cos (a+\frac{}{2})$ =$-\sin a$ ?
  2. $\sin (a+\frac{}{2})$ =$\cos a$
  3. $\cos (-a+\frac{}{2})$ =$\sin a$ ?
  4. $\sin (-a+\frac{}{2})$ =$\cos a$
  5. $\tan (a+\frac{}{2})$=$-\cot a$
  6. $\cot (a+\frac{}{2})$ =$-\tan a$
  7. $\tan (-a+\frac{}{2})=\cot a$
  8. $\cot (-a+\frac{}{2})$ =$\tan a$

$a,a+\frac{}{2}$

• Discuss $\alpha$和$a+\frac{}{2}$trigonometric function relationship, we use
  • The coordinates of the intersection point of the terminal side and the unit circle (the horizontal and vertical coordinates respectively reflect the angle α \alphasine and cosine of$\alpha \; )$
  • And the straight line y = ± xy=\pm x on the Cartesian coordinate system$y=\pm xauxiliary$ (transition)
    • $P(x,y)$和$Q(y,x)$ with respect to$y=xsymmetry$ _
      • For example $(2,1)$,$(1,2)$ ; also as$(1,-2)$,$(-2,1)$
    • $P(x,y)$和$Q(-y,-x)$ about$y=-xsymmetry\_$ _
      • For example $(2,1),(-1,-2)$ ; also as$(1,-2)$,$(2,-1)$
  • Coordinate relationships of points symmetrical about coordinate axes
    • $P(x,y),Q(x,-y)$ about$x-$ axis symmetry
    • $P(x,y),Q(-x,y)$ About$y-$ axis symmetry
  • In fact, $\alpha$ can be transformed toα + π 2 \alpha+\frac{\pi}{2}through two appropriateaxial symmetry transformations$a+\frac{}{2}$
• Let $The\; terminal\; side\; \alpha$ and the unit circle intersect at point$P(\cos a,\sin a)$
• Take α \alphaDiscussion of the first quadrant angle of $\alpha \; formula\; as\; an\; example$
  • The first axial symmetry transformation is about the straight line $y=x$ , the new coordinates obtained are denoted as$M$ , from symmetry we can know$M(\sin a,\cos \alpha )$ , terminal edgeOM OMThe corresponding angle of$OM$$(\frac{}{2}-a)+2kp,k\in \mathbb{Z}$
  • The second axial symmetry transformation is about $x=0$ , the new coordinates obtained are$N$ , by$N$ and$M$ with respect to$x=0$ symmetry, so$N(-\sin a,\cos a)$ ;角$a+\frac{}{2}$The terminal edge is $ON$ ,$N\left(\cos \right(a+\frac{}{2}),\sin (a+\frac{}{2}))$
  • So $\cos (a+\frac{}{2})$ =$-\sin a$ ? $\sin (a+\frac{}{2})$ =$\cos a$
• Using similar techniques, α \alpha can be completely summarizedThe same conclusion (formula) is established when$\alpha \; is\; in\; the\; four\; quadrants.$

$a,a-\frac{}{2}$

• $a-\frac{}{2}$= $-(-a+\frac{}{2})$
• It can be directly deduced from the formulas in the previous group, for example
  • $\cos (a-\frac{}{2})$ =$\cos (-(-a+\frac{}{2}))$ =$\cos (-a+\frac{}{2})$ =$\sin a$
  • $\sin (a-\frac{}{2})$ =$\sin (-(-a+\frac{}{2}))$ =$-\sin (-a+\frac{}{2})$ =$-\cos a$
  • $\cdots$

Summary@mantra

• Through research and induction on the induced formulas of trigonometric functions, people have summarized a set of formulas to quickly complete the conversion of the following form

  • $U(a+k\cdot \frac{}{2})$,$k\in \mathbb{Z}$到$V\left(a\right)$
  • Inside $U,V$ represents$\sin ,$A function name in$\cos$$U,V$ may take the same function name

• Here is the most commonly used formula, mainly used for $\sin ,\cos$

• " Odd changes to even, unchanged, symbols look at the quadrants "

  • Changes from odd to even remain unchanged:
    • ifkk $k$ is an even number, then$U,The\; function\; names\; of\; V$ are the same, for example, they are all$\sin$ or both$\cos$ , that is,the function name remains unchanged
    • ifkk $k$ is an odd number, the function name changes ($\sin \to \cos ;\cos \to \sin$)
  • Symbols look at the quadrants:
    • The sign refers to the plus or minus sign
    • Change $\alpha$ is regarded as an acute angle, and then judge$a+k\cdot \frac{}{2}$The quadrant where the terminal edge is located
    • The sign of the terminal side (corresponding angle) under the trigonometric function U is $The\; symbol\; of\; V$ is simplythe same sign
  • 例$\cos (-\frac{19}{4})$ =$\cos (\frac{19}{4}\pi )$ =$\cos (\frac{3}{4}+4\pi )$ =$\cos \frac{3}{4}\pi$ =$\cos (\frac{}{2}+\frac{}{4})$ =$-\sin \frac{}{4}$=$-\frac{\sqrt{2}}{2}$

• Other trigonometric functions can be converted into $\sin ,\cos$ is calculated, so it is not necessary to remember

• If necessary, you can refer to other information for other formulas.

refs

• EM@Trigonometric function induced formula CSDN blog