Complete works of four inductive formulas in senior one mathematics.
Formula 1:
Settings? For any angle, the values of the same trigonometric function with the same angle of the terminal edge are equal:
sin(2k？ +? ) = sin? (k？ z)
cos(2k？ +? )=cos？ (k？ z)
Tan (2k? +? ) = Tan? (k？ z)
cot(2k？ +? )=cot？ (k？ z)
Equation 2:
Settings? For any angle, +? What is the trigonometric function value of? The relationship between trigonometric function values is:
Sin (? +? ) =-sin?
cos(？ +? )=-cos？
Tan (? +? ) = Tan?
cot(？ +? )=cot？
Formula 3:
Any angle? Use-? The relationship between trigonometric function values is:
Sin (-? ) =-sin?
cos(-？ )=cos？
Tan (- ) =-Tan?
cot(-？ )=-cot？
Equation 4:
Can be obtained by Formula 2 and Formula 3? -? With what? The relationship between trigonometric function values is:
Sin (? -? ) = sin?
cos(？ -? )=-cos？
Tan (? -? ) =-Tan?
cot(？ -? )=-cot？
Formula 5:
Using the first-order equation and the third-order equation, we can get 2? -? With what? The relationship between trigonometric function values is:
Sin (2? -? ) =-sin?
cos(2？ -? )=cos？
Tan (2? -? ) =-Tan?
cot(2？ -? )=-cot？
Equation 6:
? /2 and 3? /2 and? The relationship between trigonometric function values is:
Sin (? /2+? )=cos？
cos(？ /2+? ) =-sin?
Tan (? /2+? )=-cot？
cot(？ /2+? ) =-Tan?
Sin (? /2-? )=cos？
cos(？ /2-? ) = sin?
Tan (? /2-? )=cot？
cot(？ /2-? ) = Tan?
Sin (3? /2+? )=-cos？
cos(3？ /2+? ) = sin?
Tan (3? /2+? )=-cot？
cot(3？ /2+? ) =-Tan?
Sin (3? /2-? )=-cos？
cos(3？ /2-? ) =-sin?
Tan (3? /2-? )=cot？
cot(3？ /2-? ) = Tan?
(above k? z)
Note: When doing the problem, it is best to regard A as an acute angle.
Inductive formula memory formula
Summary of the law. ※。
The above inductive formula can be summarized as follows:
For what? /2*k (k？ Z) trigonometric function value,
(1) When k is even, get? The function value of the same name, that is, the function name remains unchanged;
(2) When k is odd, get? The corresponding complementary function value, namely sin? Cos because? Sin; Tan? Kurt. Kurt. Tan.
(Odd and even numbers remain the same)
Put a handle in the front? The sign of the original function value when regarded as an acute angle.
(Symbols look at quadrants)
For example:
Sin (2? -? )=sin(4/2-？ ), k=4 is an even number, so take the crime? .
What time? When it is an acute angle, 2? -(270? ,360? ), sin (2? -? )<0, the symbol is? -? .
So sin (2? -? ) =-sin?
The above memory formula is:
Odd couples, symbols look at quadrants.
The symbol on the right side of the formula is Ba? As an acute angle, angle k? 360? +? (k？ z)，-？ 、 180? ,360? -?
The sign of the original trigonometric function value in the quadrant can be remembered.
The name of horizontal induction remains unchanged; Symbols look at quadrants.
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How to judge the symbols of various trigonometric functions in four quadrants and remember the formulas? One is no problem; Two sine (cotangent); Cut in twos and threes; Four cosines (secant)? .
The meaning of this 12 formula is:
What are the four trigonometric functions at any angle in the first quadrant? +? ;
The second quadrant is only sine. +? , the rest are all? -? ;
What is the tangent function of the third quadrant? +? What is the function of chords? -? ;
Only cosine in the fourth quadrant? +? , the rest are all? -? .
The above memory formulas are all positive, sine, inscribed and cosine.
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There is another way to define positive and negative according to the function type:
Function Type First Quadrant Second Quadrant Third Quadrant Fourth Quadrant
Sine ...........+............+............? ............？ ........
Cosine ...........+............? ............？ ............+ ........
Tangent ...........+............? ............+............？ ........
I cut ...........+............? ............+............？ ........
Basic relations of trigonometric functions with the same angle
Basic relations of trigonometric functions with the same angle
Reciprocal relationship:
Brown canvas bed? = 1
sin csc？ = 1
Cos seconds? = 1
Relationship between businesses:
Sin? /cos？ = Tan? = seconds? /csc？
Because? /sin? =cot？ =csc？ /sec?
Square relation:
sin^2(？ )+cos^2(？ )= 1
1+tan^2(？ )=sec^2(？ )
1+cot^2(？ )=csc^2(？ )
Hexagon memory method of equilateral trigonometric function relationship
Hexagonal mnemonics: (see pictures or links to resources)
The structure is "winding, cutting and cutting; Zuo Zheng, the right remainder and the regular hexagon of the middle 1 "are models.
(1) Reciprocal relation: The two functions on the diagonal are reciprocal;
(2) Quotient relation: the function value at any vertex of a hexagon is equal to the product of the function values at two adjacent vertices.
(Mainly the product of trigonometric function values at both ends of two dotted lines). From this, the quotient relation can be obtained.
(3) Square relation: In a triangle with hatched lines, the sum of squares of trigonometric function values on the top two vertices is equal to the square of trigonometric function values on the bottom vertex.
Two-angle sum and difference formula
Formulas of trigonometric functions of sum and difference of two angles.
Sin (? +? ) = sin? Because? +cos？ Sin?
Sin (? -? ) = sin? Because? Because? Sin?
cos(？ +? )=cos？ Because? Sin? Sin?
cos(？ -? )=cos？ Because? +sin? Sin?
Tan (? +? ) = (Tan? +Tan? )/( 1-tan？ Tan? )
Tan (? -? ) = (Tan? Tan? ) /( 1+ Tantan? )
Double angle formula
Sine, Cosine and Tangent Formulas of Double Angles (Ascending Power and Shrinking Angle Formula)
sin2？ =2sin？ Because?
cos2？ =cos^2(？ )-sin^2(？ )=2cos^2(？ )- 1= 1-2sin^2(？ )
tan2？ =2tan？ /[ 1-tan^2(？ )]
half-angle formula
Sine, cosine and tangent formulas of half angle (power decreasing and angle expanding formulas)
sin^2(？ /2)=( 1-cos？ )/2
cos^2(？ /2)=( 1+cos？ )/2
tan^2(？ /2)=( 1-cos？ )/( 1+cos？ )