Commonly used revulsive formula has the following groups:
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
(Z of above K ∈ )
Attention: When becoming a problem, regard A as it is better that acute angle will do do.
※ rule sums up ※
These revulsive formula can generalize above for:
To π / α of 2*k ± (the trigonometric function value of K ∈ Z) ,
① is become K is even when, the homonymic letter that receives α is numeric, namely function name is not changed;
② is become K is odd when, get the complementary function with corresponding α is worth, namely Cot of → of Sin;tan of → of Sin → Cos;cos, cot → Tan.
(strange change occasionally changeless)
Add a α to regard as in front next the symbol that numerical value cases formerly when acute angle.
(the symbol sees quadrant)
For example:
Sin(2 π - α ) π of ＝ Sin(4 · / 2 - α ) , k ＝ 4 for even number, take Sin α so.
When α it is acute angle when, 2 π - α ∈ (270 ° , 360 ° ) , sin(2 π - α ) ＜ 0, the symbol is " - " .
So Sin(2 π - α ) ＝ - Sin α
Afore-mentioned memory a pithy formula is:
Strange change occasionally changeless, the symbol sees quadrant.
When the symbol of formulary right is a α to regard acute angle as, 360 ° of horny K · + α (K ∈ Z) , - α of α , 180 ° ± , 360 ° - α
The symbol that is in value of quadrantal primary trigonometric function can be remembered
The level is revulsive the name is changeless; The symbol sees quadrant.
How are all sorts of trigonometric function judged in 4 quadrantal symbols, also can remember a pithy formula " one complete; 2 sine (cosecant) ; 29 cut; More than 4 bowstring (secant) " .
The meaning that is to say of a pithy formula of this 12 words:
The value of 4 kinds of trigonometric function of horn of the any inside the first quadrant is " ＋ " ;
Only sine is inside the 2nd quadrant " ＋ " , the others is entirely " - " ;
Function is being cut inside the 3rd quadrant is " ＋ " , bowstring function is " - " ;
Only Yu Xian is inside the 4th quadrant " ＋ " , the others is entirely " - " .
Afore-mentioned memorial a pithy formula, one complete, 2 sine, 3 inside cut, more than 4 bowstring
Still one kind divides quadrant to losing surely according to function type:
Function type the first quadrant the 2nd quadrant the 3rd quadrant the 4th quadrant
Sine. . . . . . . . . . . ＋ . . . . . . . . . . . . ＋ . . . . . . . . . . . . , . . . . . . . . . . . . , . . . . . . . .
Cosine. . . . . . . . . . . ＋ . . . . . . . . . . . . , . . . . . . . . . . . . , . . . . . . . . . . . . ＋ . . . . . . . .
Tangential. . . . . . . . . . . ＋ . . . . . . . . . . . . , . . . . . . . . . . . . ＋ . . . . . . . . . . . . , . . . . . . . .
Cotangent. . . . . . . . . . . ＋ . . . . . . . . . . . . , . . . . . . . . . . . . ＋ . . . . . . . . . . . . , . . . . . . . .
The trigonometric function that be the same as horn concerns basically
The basic relation of the trigonometric function that be the same as horn
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 )
