Basic relations of trigonometric functions with the same angle
Reciprocal relation: tan α cotα =1sin α CSC α =1cos α secα =1.
The relationship of quotient: sin α/cos α = tan α = sec α/CSC α cos α/sin α = cot α = CSC α/sec α.
Square relation: sin 2 (α)+cos 2 (α) =1+tan 2 (α) = sec2 (α)1+cot2 (α) = CSC 2 (α).
Two commonly used formulas under different conditions
Sin 2 (α)+cos 2 (α) =1tan α *cot α= 1
Special formula
(Sina+sinθ)*(Sina-sinθ)= sin(a+θ)* sin(a-θ)
It is proved that (sina+sinθ) * (sina-sinθ) = 2sin [(θ+a)/2] cos [(a-θ)/2] * 2cos [(θ+a)/2] sin [(a-θ)/2] = sin (a+).
slope formula
We usually refer to the ratio of the vertical height h to the horizontal height l of the half slope as slope (also called slope ratio), which is expressed by the letter I, that is, I = h/L. The general form of slope is written as l: m, such as i= 1:5. If the included angle between the inclined plane and the horizontal plane is recorded as
A (called inclination angle), then I = h/l = tan a.
Acute angle formula of trigonometric function
Sine: the hypotenuse of the opposite side of sine α = ∠ α/α cosine: the hypotenuse of the adjacent side of cosine α = ∠ α/α.
Tangent: the opposite side of tan α = ∠ α/∠ α's adjacent cotangent: the adjacent side of cot α = ∠α/∠α's opposite side.
Double angle formula
Sin2A=2sinA cosA Cosa cosine 1. Cos2a = Cos2 (a)-Sin2 (a) 2。 Cos2a = 1-2sin 2 (a) 3。 Cos2a = 2cos 2 (a)- 1。 -1 = 1-2 sin 2 (a) tangent tan2a = (2 tana)/( 1-tan 2 (a))
Triple angle formula
sin 3α= 4 siinαsin(π/3+α)sin(π/3-α)cos 3α= 4 cosαcos(π/3+α)cos(π/3-α)tan3a = tana tan(π/3+a)tan(π/3)。 a)+( 1-2sin？ A) Sina = 3sina-4s in 3a cos3a = cos (2a+a) = cos2acosa-sin2asina = (2cos? A-1) COSA-2 (1-cos a) COSA = 4cos 3a-3cosasin3a = 3sina-4sin 3a = 4sina (3/4- sin? a) =4sina[(√3/2)？ Sin? a] =4sina(sin？ 60- sin? a)= 4 Sina(sin 60+Sina)(sin 60-Sina)= 4 Sina * 2 sin[(60+a)/2]cos[(60-a)/2]* 2 sin[(60-a)/2]cos[(60-a)/2]= 4 Sina sin(60+a)sin(60-a)cos3a=4cos^3a-3cosa = 4 cosa(cos？ a-3/4) =4cosa[cos？ a-(√3/2)^2] =4cosa(cos？ a-cos？ 30)= 4 cosa(cosa+cos 30)(cosa-cos 30)= 4 cosa * 2 cos[(a+30)/2]cos[(a-30)/2]* {-2 sin[(a+30)/2]sin[(a-30)/2]} =-4 cosa sin(a+30)sin(a-30) =-4cosasin [90-(60-a)] sin [-90+(60+a)] =-4cosacos (60-a) List the following formulas: sin2alpha = 2sinα cos α tan2alpha = 2tanα/(1-tan2 (α). Including some image problems and functional problems.
Triple angle formula
sin3α=3sinα-4sin^3(α)=4sinαsin(π/3+α)sin(π/3-α)cos3α=4cos^3(α)-3cosα=4cosαcos(π/3+α)cos(π/3-α)tan3α=tan(α)*(-3+tan(α)^2)/(- 1+3*tan(α)^2)=tan a tan(π/3+a)tan(π/3-a)
half-angle formula
sin^2(α/2)=( 1-cosα)/2 cos^2(α/2)=( 1+cosα)/2tan^2(α/2)=( 1-cosα)/( 1+cosα)tan(α/2)= sinα/( 1+cosα)=( 1-cosα)/sinα
General formula of trigonometric function
sinα=2tan(α/2)/[ 1+tan^2(α/2)]cosα=[ 1-tan^2(α/2)]/[ 1+tan^2(α/2)]tanα=2tan(α/2)/[ 1-tan^2(α/2)]
other
sinα+sin(α+2π/n)+sin(α+2π* 2/n)+sin(α+2π* 3/n)+……+sin[α+2π*(n- 1)/n]= 0 cosα+cos(α+2π/n)+cos(α+2π* 2/n)+……+cos[α+2π*(n- 1)/n
four times the angle formula
sin4a=-4*(cosa*sina*(2*sina^2- 1 cos4a= 1+(-8*cosa^2+8*cosa^4 tan4a=(4*tana-4*tana^3)/( 1-6*tana^2+tana^4)
Five-fold angle formula
sin5a= 16sina^5-20sina^3+5sina cos5a= 16cosa^5-20cosa^3+5cosa tan5a=tana*(5- 10*tana^2+tana^4)/( 1- 10*tana^2+5*tana^4)
Hexagonal formula
sin6a=2*(cosa*sina*(2*sina+ 1)*(2*sina- 1)*(-3+4*sina^2))cos6a=((- 1+2*cosa^2)*( 16*cosa^4- 16*cosa^2+ 1)) tan6a =(-6 * tana+20*tana^3-6*tana^5)/(- 1+ 15*tana^2- 15*tana^4+tana^6)
Seven-angle formula
sin7a=-(sina*(56*sina^2- 1 12*sina^4-7+64*sina^6))cos7a=(cosa*(56*cosa^2- 1 12*cosa^4+64*cosa^6-7))tan7a = tana *(7+35 * tana^2-2 1*tana^4+tana^6)/(- 1+2 1*tana^2-35*tana^4+7*tana^6)
Octagonal formula
sin8a=-8*(cosa*sina*(2*sina^2- 1)*(-8*sina^2+8*sina^4+ 1))cos8a= 1+( 160*cosa^4-256*cosa^6+ 128*cosa^8-32*cosa^2) tan8a =-8 * tana*(- 1+7*tana^2-7*tana^4+tana^6)/( 1-28*tana^2+70*tana^4-28*tana^6+tana^8)
Nine-angle formula
sin9a=(sina*(-3+4*sina^2)*(64*sina^6-96*sina^4+36*sina^2-3))cos9a=(cosa*(-3+4*cosa^2)*(64*cosa^6-96*cosa^4+36*cosa^2-3))tan9a = tana *(9-84 * tana^2+ 126*tana^4-36*tana^6+tana^8)/( 1-36*tana^2+ 126*tana^4-84*tana^6+9*tana^8)
Ten-fold angle formula
sin 10a=2*(cosa*sina*(4*sina^2+2*sina- 1)*(4*sina^2-2*sina- 1)*(-20*sina^2+5+ 16*sina^4 cos 10a=((- 1+2* cosa^2)*(256* cosa^8-5 12*cosa^6+304*cosa^4-48*cosa^2+ 1 tan 10a=-2*tana*(5-60*tana^2+ 126*tana^4-60*tana^6+5*tana^8)/(- / kloc-0/+45* tana^2-2 10*tana^4+2 10*tana^6-45*tana^8+tana^ 10)
N times angle formula
According to Demeifu's theorem, (cos θ+I sin θ) n = cos (n θ)+I sin (n θ) For convenience of description, let sinθ=s and cosθ=c consider the case that n is a positive integer: cos (n θ)+I sin (n θ) = (c+I s) n = c (n, 2) *. 4)*c^(n-4)*(i s)^4+...+c(n, 1)*c^(n- 1)*(i s)^ 1+c(n,3)*c^(n-3)*(i s)^3+c(n,5)*c^(n-5)*(i s)^5+。 ... = > compare the real and imaginary parts: cos (nθ) = c (n, 0) * c n+c (n, 2) * c (n-2) * (I s) 2+c (n, 4) * c (n-4) * (I s). 1)* c(n- 1)*(I s) 1+c(n，3) * c (n-3) * (I s) 3+c (n，5) * c (n-。 2.SIN (nθ): (1) When n is odd: in the formula, c is an even power, while C 2 = 1-S 2 (square relation), so both can be expressed as s (that is, sinθ). (2) When n is even, all C in the formula are odd powers, and C 2 = 1-S 2 (square relation), so even if it is changed to S, at least the first power of C (that is, cosθ) will be retained. (For example, c 3 = c * c 2 = c * (1-s 2), c 5 = c * (c 2) 2 = c * (1-s 2) 2)
half-angle formula
tan(A/2)=( 1-cosA)/sinA = sinA/( 1+cosA)； cot(A/2)= sinA/( 1-cosA)=( 1+cosA)/sinA。 sin^2(a/2)=( 1-cos(a))/2 cos^2(a/2)=( 1+cos(a))/2 tan(a/2)=( 1-cos(a))/sin(a)= sin(a)/( 1+cos(a))
Sum difference product
sinθ+sinφ= 2 sin[(θ+φ)/2]cos[(θ-φ)/2]
sinθ-sinφ= 2 cos[(θ+φ)/2]sin[(θ-φ)/2]cosθ+cosφ= 2 cos[(θ+φ)/2]cosθ-cosφ=-2 sin[(θ+φ)/2]sin[(θ+φ)/2]tanA+tanB = sin(A+B)/cosa cosb = tan(A+B)( 1-tanA tanB)tanA-tanB = sin(A-B)/cosa cosb = tan(
Two-angle sum formula
tan(α+β)=(tanα+tanβ)/( 1-tanαtanβ)tan(α-β)=(tanα-tanβ)/( 1+tanαtanβ)cos(α+β)= cosαcosβ-sinαsinβcos(α-β)= cosαcosβ+sinαsinβsin(α+β)= sinαcosβ+cosαsinβsin(α-β)= sinαcosβ
Sum and difference of products
sinαsinβ=-[cos(α+β)-cos(α-β)]/2 cosαcosβ=[cos(α+β)+cos(α-β)]/2 sinαcosβ=[sin(α+β)+sin(α-β)]/2 cosαsinβ=[sin(α+β)-sin(α-β)]/2
Hyperbolic function
Sh a = [e a-e (-a)]/2cha = [e a+e (-a)]/2tha = Xin h(a)/ Kosh h(a).
Formula 1: Let α be any angle and the values of the same trigonometric function of the same terminal angle are equal: sin (2kπ+α) = sinα cos (2kπ+α) = cos α tan (2kπ+α) = tan α cot (2kπ+α) = cot α.
Equation 2: Let α be an arbitrary angle, and the trigonometric function relationship between π+α and α is as follows: sin (π+α) =-sin α cos (π+α) =-cos α tan (π+α) = tan α cot (π+α) = cot α.
Equation 3: the relationship between any angle α and trigonometric function value of-α: sin (-α) =-sin α cos (-α) = cos α tan (-α) =-tan α cot (-α) =-cot α.
Equation 4: Using Equation 2 and Equation 3, we can get the relationship between the trigonometric function values of π-α and α: sin (π-α) = sin α cos (π-α) =-cos α tan (π-α) =-tan α cot (π-α) =-cot α.
Formula 5: The relationship between the trigonometric function values of 2π-α and α can be obtained by Formula-and Formula 3: sin (2π-α) =-sin α cos (2π-α) = cos α tan (2π-α) =-tan α cot (2π-α) =-cot α.
Equation 6: The relationship between π/2α and 3 π/2 α and the trigonometric function value of α: sin (π/2+α) = cos α cos (π/2+α) =-sin α tan (π/2+α) =-cot α cot (π/2+α) =-tan α sin. 2-α) = cot α cot (π/2-α) = tan α sin (3π/2+α) =-cos α cos (3π/2+α) = sin α tan (3π/2+α) =-cot α cot (3π/2+α) =-tan α. +2 abcos(θ-φ)} sin {ωt+arcsin[(a sinθ+b sinφ)/√{a^2 +b^2； +2abcos (θ-φ)} √ indicates the radical sign, including {...}.
Inductive formulas of trigonometric functions (six formulas)
Formula1sin (-α) =-sin α cos (-α) = cos α tan (-α) =-tan α formula 2 sin (π/2-α) = cos α cos (π/2-α) = sin α formula 3 sin (π/2+α) = cos α cos (. = -sinα cos(π+α) = -cosα Formula Six tana = Sina/Cosatan (π/2+α) =-Cot
General formula of trigonometric function
sinα= 2tan(α/2)/[ 1+(tan(α/2))？ ] cosα=[ 1-(tan(α/2))？ ]/[ 1+(tan(α/2))？ ]tanα= 2tan(α/2)/[ 1-(tan(α/2))？ ]
Other formulas
(1) (sin α) 2+(cos α) 2 =1(sum of squares formula) (2)1+(tan α) 2 = (sec α) 2 (3)1+(cot α). The second can be divided by (cos α) 2. (4) For any non-right triangle, there is always a syndrome of Tana+Tanb+Tanc = Tanatan Tanc: a+b = π-c tan (a+b) = tan (π-c) (tana+tanb)/(1-tanatanb) = (tanπ-). You can also obtain a certificate of Tana+Tanb+Tanc = Tanatanbntanc. When x+y+z=nπ(n∈Z), tana+tanb+tanc = tanatantbanc (5) cotacotb+cotacotc+cotbctc =1(6) cot (a/2)+cot (b/2). +(cosb) 2+(cosc) 2 =1-2 Cosacosbosco (8) (Sina) 2+(sinb) 2+(sinc) 2 = 2+2 Cosacosbosco Other nonessential trigonometric functions CSC (A) = 65438+. (SECA) 2+(CSCA) 2 = (SECA) 2 (CSCA) 2 Power Series Expansion Sinx = X-X 3/3! +x^5/5！ -……+(- 1)^(k- 1)*(x^(2k- 1))/(2k- 1)！ +………(-∞& lt； x & lt∞) cos x = 1-x^2/2！ +x^4/4！ -……+(- 1)k*(x^(2k))/(2k)！ +……(-∞& lt； x & lt∞)arcsin x = x+ 1/2*x^3/3+ 1*3/(2*4)*x^5/5+……(| x | & lt； 1)arccos x =π-(x+ 1/2*x^3/3+ 1*3/(2*4)*x^5/5+……)(| x | & lt； 1) arctan x = x-x 3/3+x 5/5-... (x ≤1) Infinite formula sinx = x (1-x2/π 2) (1-x2) (/kloc-0) ……tanx=8x[ 1/(π^2-4x^2)+ 1/(9π^2-4x^2)+ 1/(25π^2-4x^2)+……)secx=4π[ 1/(π^2-4x^2)- 1/(9π^2-4x^2)+ 1/ (25π^2-4x^2)-+……)(sinx)x = cosx/2 cosx/4 cosx/8……( 1/4)tanπ/4+( 1/8)tanπ/8+( 1/ 16)tanπ/65438...(x ≤ 1) And the formula sinx+sin2x+sin3x+...+sinnx = [sin (NX/2) sin ((n+1)) is related to the sum of independent variables. cosx+cos2x+cos3x+……+cosnx =[cos((n+ 1)x/2 sin(NX/2)]/sin(x/2)tan((n+ 1)x/2)=(sinx+sin2x+sin3x+……+sinnx)/(cosx+cos2x+cos3x+……+cosnx) sinx+sin3x+sin5x+……+sin(2n- 1)x=(sinnx)^2/sinx cosx+cos3x+cos5x+……+cos(2n- 1)x = sin(22
Edit the content rules of this paragraph.
Trigonometric functions seem to be many and complicated, but as long as we master the essence and internal laws of trigonometric functions, we will find that there is a strong connection between the formulas of trigonometric functions. And mastering the inherent law and essence of trigonometric function is also the key to learn trigonometric function well. The essence of 1. trigonometric function:
[1] According to the picture on the right, there is sin θ = y/r; cosθ= x/r； tanθ= y/x； Cot θ = x/y. I have a deep understanding of this point, and all the following trigonometric formulas can be derived from this point. For example, take the derivative of sin(A+B) = sinAcosB+cosAsinB as an example: Derivation: Draw a unit circle with the X axis at C and D, and there are any points A and B on the unit circle. The angle AOD is α and BOD is β. Rotating AOB makes OB and OD overlap to form a new A'OD. A(cosα, sinα), B(cosβ, sinβ), A'(cos(α-β), sin(α-β)) OA'=OA=OB=OD= 1, 0)d( 1, 0) ? But the definition of the unit circle does allow trigonometric functions to define all positive and negative radians, not just the angle between 0 and π/2 radians. It also provides images containing all the important trigonometric functions. According to Pythagorean theorem, the equation of unit circle is: some commonly used angles measured in radians are given in the image. The counterclockwise measurement is a positive angle, while the clockwise measurement is a negative angle. Let a straight line passing through the origin make an angle θ with the positive half of the X axis and intersect the unit circle. The x and y coordinates of this intersection point are equal to cos θ and sin θ respectively. The triangle in the image ensures this formula; The radius is equal to the hypotenuse and the length is 1, so there are sin θ = y/ 1 and cos θ = x/ 1. The unit circle can be regarded as a way to view an infinite number of triangles by changing the lengths of adjacent sides and opposite sides, but keeping the hypotenuse equal to 1. Two-angle sum formula
sin(A+B)= Sina cosb+cosa sinb sin(A-B)= Sina cosb-cosa sinb cos(A+B)= cosa cosb-Sina sinb cos(A-B)= cosa cosb+Sina sinb tan(A+B)=(tanA+tanB)/( / kloc-0/-tanA tanB)tan(A-B)=(tanA-tanA tanB)/( 1+tanA tanB)cot(A+B)=(cotA cotB- 1)/(cot