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Commonly used and revulsive formula
Formulary one:
Set α to be aleatoric horn, eventually the value of the same trigonometric function of the role with same limit is equal:
Sin (α of 2k π ＋ ) ＝ Sin α (K ∈ Z)
Cos (α of 2k π ＋ ) ＝ Cos α (K ∈ Z)
Tan (α of 2k π ＋ ) ＝ Tan α (K ∈ Z)
Cot (α of 2k π ＋ ) ＝ Cot α (K ∈ Z)
Formulary 2:
Set α to be aleatoric horn, π + the relation between the trigonometric function value of the trigonometric function value of α and α :
Sin (π ＋ α ) ＝ - Sin α
Cos (π ＋ α ) ＝ - Cos α
Tan (π ＋ α ) ＝ Tan α
Cot (π ＋ α ) ＝ Cot α
Formulary 3:
Aleatoric horn α and - the relation between the trigonometric function value of α :
Sin (- α ) ＝ - Sin α
Cos (- α ) ＝ Cos α
Tan (- α ) ＝ - Tan α
Cot (- α ) ＝ - Cot α
Formulary 4:
Use formula 2 with formula 3 can get π - the relation between the trigonometric function value of α and α :
Sin (π - α ) ＝ Sin α
Cos (π - α ) ＝ - Cos α
Tan (π - α ) ＝ - Tan α
Cot (π - α ) ＝ - Cot α
Formulary 5:
Use formula to mix formulary 3 can get 2 π - the relation between the trigonometric function value of α and α :
Sin (2 π - α ) ＝ - Sin α
Cos (2 π - α ) ＝ Cos α
Tan (2 π - α ) ＝ - Tan α
Cot (2 π - α ) ＝ - Cot α
Formulary 6:
π / 2 ± α reachs 3 π / the relation between the trigonometric function value of 2 ± α and 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 (π / 2 - α ) ＝ Cos α
Cos (π / 2 - α ) ＝ Sin α
Tan (π / 2 - α ) ＝ Cot α
Cot (π / 2 - α ) ＝ Tan α
Sin (3 π / 2 ＋ α ) ＝ - Cos α
Cos (3 π / 2 ＋ α ) ＝ Sin α
Tan (3 π / 2 ＋ α ) ＝ - Cot α
Cot (3 π / 2 ＋ α ) ＝ - Tan α
Sin (3 π / 2 - α ) ＝ - Cos α
Cos (3 π / 2 - α ) ＝ - Sin α
Tan (3 π / 2 - α ) ＝ Cot α
Cot (3 π / 2 - α ) ＝ Tan α
The trigonometric function that be the same as horn concerns
Reciprocal concerns:
＝ of α of Cot of Tan α · 1
＝ of α of Csc of Sin α · 1
＝ of α of Sec of Cos α · 1
The relation of business:
Sin α / α of Sec of ＝ of α of Tan of Cos α ＝ / Csc α
Cos α / α of Csc of ＝ of α of Cot of Sin α ＝ / Sec α
Square concerns:
Sin^2(α ) ＋ Cos^2(α ) ＝ 1
1 ＋ Tan^2(α ) ＝ Sec^2(α )
1 ＋ Cot^2(α ) ＝ Csc^2(α )
Relation of the trigonometric function that be the same as horn is hexagonal memorial law:
Hexagonal memory law:
Tectonic with " first quarter, medium cut, next cutting; Zun Zheng, right beyond, of the 1" intermediate hexagon is a model.
(1) reciprocal relation: Each other of on diagonal two function is reciprocal;
(2) quotient concerns: The case numerical value on one acme is equal to hexagon random as numeric as the case on two acme of its photograph adjacent product. (basically be the product that the trigonometric function of two end is worth two dotted line) . From this, can get quotient relation.
(3) square concerns: In the triangle that contains shadow line, above the sum of squares that the trigonometric function on two acme is worth is equal to below the square that the trigonometric function on acme is worth.
Two horn and difference are formulary:
Two horn and the trigonometric function with difference are formulary:
Sin (α ＋ β ) β of Sin of α of Cos of ＋ of β of Cos of ＝ Sin α
Sin (α - β ) β of Cos of ＝ Sin α - β of Cos α Sin
Cos (α ＋ β ) β of Cos of ＝ Cos α - β of Sin α Sin
Cos (α - β ) β of Sin of α of Sin of ＋ of β of Cos of ＝ Cos α
Tan (α ＋ β ) ＝ (Tan α + Tan β ) ／ (β of 1-tan α Tan)
Tan (α - β ) ＝ (Tan α - Tan β ) ／ (β of Tan of · of α of 1 ＋ Tan)
Duple horn is formulary:
The sine of duple horn, Yu Xian and tangential formula (ascending power shrinks horn is formulary)
α of Cos of α of 2sin of Sin2 α ＝
Cos^2(α of Cos2 α ＝ ) - Sin^2(α ) ＝ 2cos^2(α ) - 1 ＝ 1 - 2sin^2(α )
α of 2tan of Tan2 α ＝ / [1 - Tan^2(α ) ]
Half horn is formulary:
The sine of half horn, Yu Xian and tangential formula (horn of Jiang Mi enlarge is formulary)
Sin^2(α / 2) ＝ (1 - Cos α ) ／ 2
Cos^2(α / 2) ＝ (α of 1 ＋ Cos) ／ 2
Tan^2(α / 2) ＝ (1 - Cos α ) ／ (α of 1 ＋ Cos)
Also have Tan(α additionally / 2)=(1 - Cos α ) / α of Sin α =sin / (1+cos α )
All-purpose and formulary:
Sin α =2tan(α / 2)/[1+tan^2(α / 2) ]
α of Cos α =[1-tan^2(/ 2)]/[1+tan^2(α / 2) ]
Tan α =2tan(α / 2)/[1-tan^2(α / 2) ]
All-purpose derivation of equation:
Add derivation:
α of Cos of α of =2sin of α of Cos of α of Sin2 α =2sin / (Cos^2(α ) + Sin^2(α ) ) . . . . . . * ,
(because of Cos^2(α ) + Sin^2(α ) =1)
Fluctuate * cent type to divide Cos^2(α ) together again, can get α of 2tan of Sin2 α ＝ / (α ) of 1 ＋ Tan^2()
Use α next / 2 replace α can.
Manage together but all-purpose formula of derivation cosine. Tangential all-purpose formula can compare Yu Xian to get through sine.
3 times horn is formulary:
The sine of 3 times horn, Yu Xian and tangent are formulary:
α of 3sin of Sin3 α ＝ - 4sin^3(α )
α of 4cos^3(of Cos3 α ＝ ) - 3cos α
Tan3 α ＝ [3tan α - Tan^3(α ) ] ／ [1 - 3tan^2(α ) ]
3 times horn derivation of equation:
Add derivation:
α of Sin3 of Tan3 α ＝ / Cos3 α
＝ (α of Sin of α of Cos2 of ＋ of α of Sin2 α Cos) / (α of Cos2 α Cos - α of Sin2 α Sin)
＝ (2sin α Cos^2(α ) ＋ Cos^2(α ) Sin α - Sin^3(α ) ) / (Cos^3(α ) - Cos α Sin^2(α ) - 2sin^2(α ) Cos α )
Divide together up and down with Cos^3(α ) , :
Tan3 α ＝ (3tan α - Tan^3(α ) ) / (1-3tan^2(α ) )
α of ＋ of α of Sin(2 of Sin3 α ＝ ) α of Sin of α of Cos2 of ＋ of α of Cos of ＝ Sin2 α
Cos^2(α of ＝ 2sin α ) ＋ (1 - 2sin^2(α ) ) Sin α
＝ 2sin α - 2sin^3(α ) ＋ Sin α - 2sin^3(α )
＝ 3sin α - 4sin^3(α )
α of ＋ of α of Cos(2 of Cos3 α ＝ ) α of Cos of ＝ Cos2 α - α of Sin2 α Sin
＝ (2cos^2(α ) - 1)cos α - 2cos α Sin^2(α )
＝ 2cos^3(α ) - Cos α ＋ (2cos α - 2cos^3(α ) )
＝ 4cos^3(α ) - 3cos α
Namely
α of 3sin of Sin3 α ＝ - 4sin^3(α )
α of 4cos^3(of Cos3 α ＝ ) - 3cos α
Formula of 3 times horn associates memory:
Memorial method: Homophonic, associate
Sine 3 times horn: 3 yuan reduce 4 yuan of 3 role (was in debt (be decreased negative number) , want so " earn money " (sound be like " sine " ) )
Cosine 3 times horn: 4 yuan of 3 horn decrease 3 yuan (after be being decreased, still have " beyond " )
Notice function name, namely the 3 times horn of sine expresses with sine, the 3 times horn of cosine expresses with Yu Xian.
Another memorial methods:
Sine 3 times horn: Hill does not have commander (homophonic is 3 without 4 stand) 3 those who point to is "3 times "sin α , without what point to it is minus sign, 4 those who point to is "4 times " , stand those who point to is Sin α cubic metre
Cosine 3 times horn: Commander is not had hill and go up to manage together
Change with difference accumulate formula
Of trigonometric function change with difference accumulate formula
β of ＋ of 2sin[(α of ＝ of β of Sin of Sin α ＋ ) / α of 2] · Cos[(- β ) / 2]
Sin α - β of ＋ of 2cos[(α of Sin β ＝ ) / α of 2] · Sin[(- β ) / 2]
β of ＋ of 2cos[(α of ＝ of β of Cos of Cos α ＋ ) / α of 2] · Cos[(- β ) / 2]
Cos α - Cos β ＝ - β of 2sin[(α ＋ ) / α of 2] · Sin[(- β ) / 2]
Of trigonometric function indigestion change and need formula:
β of ＋ of 0.5[sin(α of ＝ of β of Cos of Sin α · ) ＋ Sin(α - β ) ]
β of ＋ of 0.5[sin(α of ＝ of β of Sin of Cos α · ) - Sin(α - β ) ]
β of ＋ of 0.5[cos(α of ＝ of β of Cos of Cos α · ) ＋ Cos(α - β ) ]
＝ of β of Sin of Sin α · - β of 0.5[cos(α ＋ ) - Cos(α - β ) ]
Change with difference accumulate derivation of equation:
Add derivation:
Above all, we know Sin(a+b)=sina*cosb+cosa*sinb, sin(a-b)=sina*cosb-cosa*sinb
We receive two type addition Sin(a+b)+sin(a-b)=2sina*cosb
So, sina*cosb=(sin(a+b)+sin(a-b))/2
Manage together, if decrease two type photograph, get Cosa*sinb=(sin(a+b)-sin(a-b))/2
We still know Cos(a+b)=cosa*cosb-sina*sinb, cos(a-b)=cosa*cosb+sina*sinb
So, two type addition, we can get Cos(a+b)+cos(a-b)=2cosa*cosb
So we have to arrive, cosa*cosb=(cos(a+b)+cos(a-b))/2
Manage together, two type photograph decreases us to get Sina*sinb=-(cos(a+b)-cos(a-b))/2
Such, we got accumulating 4 formula that change and need:
Sina*cosb=(sin(a+b)+sin(a-b))/2
Cosa*sinb=(sin(a+b)-sin(a-b))/2
Cosa*cosb=(cos(a+b)+cos(a-b))/2
Sina*sinb=-(cos(a+b)-cos(a-b))/2
Had after accumulating 4 formula that change and need, we need to be out of shape only, with respect to 4 formula that can get mixing differring changing accumulating.
We set the A+b in 4 afore-mentioned formula for X, a-b is set for Y, so A=(x+y)/2, b=(x-y)/2
A, b uses X respectively, y expresses to be able to get mixing differring changing 4 formula that accumulate:
Sinx+siny=2sin((x+y)/2)*cos((x-y)/2)
Sinx-siny=2cos((x+y)/2)*sin((x-y)/2)
Cosx+cosy=2cos((x+y)/2)*cos((x-y)/2)
Cosx-cosy=-2sin((x+y)/2)*sin((x-y)/2)
