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 an odd number, what do you 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, with the symbol "-".
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, you can also remember the formula "a full pair; Two sine (cotangent); Cut in twos and threes; Four cosines (secant) ".
The meaning of this 12 formula is:
The four trigonometric functions at any angle in the first quadrant are "+";
In the second quadrant, only the sine is "+",and the rest are "-";
The tangent function of the third quadrant is+and the chord function is-.
In the fourth quadrant, only cosine is "+",others are "-".
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:
Tancott. = 1
sincsc？ = 1
cossec？ = 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 memory method: the structure is "winding, middle cutting and lower 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？ )
And tan (? /2)=( 1-cos？ )/sin? = sin? /( 1+cos？ )
General formula of trigonometric function
Sin? =2tan(？ /2)/[ 1+tan^2(？ /2)]
Because? =[ 1-tan^2(？ /2)]/[ 1+tan^2(？ /2)]
Tan? =2tan(？ /2)/[ 1-tan^2(？ /2)]
Derivation of universal formula
Additional derivation:
sin2？ =2sin？ Because? =2sin？ Because? /(cos^2(？ )+sin^2(？ ))......*,
(because cos^2 (? )+sin^2(？ )= 1)
Then divide the * score up and down by cos^2 (? ), can you get sin2? =2tan？ /( 1+tan^2(？ ))
And use it? /2 instead? Do it.
Similarly, the universal formula of cosine can be derived. By comparing sine and cosine, a general formula of tangent can be obtained.
Triple angle formula
Sine, cosine and tangent formulas of triple angle
sin3？ =3sin？ -4sin^3(？ )
cos3？ =4cos^3(？ )-3cos？
tan3？ =【3 tan？ -tan^3(？ )]/[ 1-3tan^2(？ )]
Derivation of triple angle formula
Additional derivation:
tan3？ =sin3？ /cos3？
=(sin2？ Because? +cos2？ Sin? )/(cos2？ Because? -sin2？ Sin? )
=(2sin？ cos^2(？ )+cos^2(？ ) sin? -sin^3(？ ))/(cos^3(？ )-cos？ sin^2(？ )-2sin^2(？ )cos？ )
Up and down divided by cos^3 (? ), have to:
tan3？ =(3tan？ -tan^3(？ ))/( 1-3tan^2(？ ))
sin3？ =sin(2？ +? )=sin2？ Because? +cos2？ Sin?
=2sin？ cos^2(？ )+( 1-2sin^2(？ )) sin?
=2sin？ -2sin^3(？ )+sin? -2sin^3(？ )
=3sin？ -4sin^3(？ )
cos3？ =cos(2？ +? )=cos2？ Because? -sin2？ Sin?
=(2cos^2(？ )- 1)cos？ -2cos？ sin^2(？ )
=2cos^3(？ )-cos？ +(2cos？ -2cos^3(？ ))
=4cos^3(？ )-3cos？
that is
sin3？ =3sin？ -4sin^3(？ )
cos3？ =4cos^3(？ )-3cos？
Associative memory of triangle formula
★ Memory method: homophonic association.
Sine Triangle: 3 yuan minus 4 yuan Triangle (debt (minus negative number), so "making money" (sounds like "sine").
Cosine triple angle: 4 yuan minus 3 yuan (there is a "remainder" after subtraction).
☆☆ Pay attention to the name of the function, that is, all three angles of sine are represented by sine, and all three angles of cosine are represented by cosine.
★ Another memory method:
Sine triangle: the crime of "three times" when there is no commander in the mountain (homophonic for three without four stands) , no sign, four refers to "four times", vertical refers to sin? cube
Cosine triple angle: commander without mountain is the same as above.
Sum-difference product formula
Sum and difference product formula of trigonometric function
Sin? +sin? =2sin[(？ +? )/2]? cos[(？ -? )/2]
Sin? Sin? =2cos[(？ +? )/2]? Sin [(? -? )/2]
Because? +cos？ =2cos[(？ +? )/2]? cos[(？ -? )/2]
Because? Because? =-2sin[(？ +? )/2]? Sin [(? -? )/2]
Product sum and difference formula
Formula of product and difference of trigonometric function
Sinkos? =0.5[sin(？ +? )+sin(？ -? )]
Cousin? =0.5[sin(？ +? )-sin (? -? )]
coscos？ =0.5[cos(？ +? )+cos(？ -? )]
Xin Xin? =-0.5[cos(？ +? )-cos(？ -? )]
Derivation of sum-difference product formula
Additional derivation:
First of all, we know that sin(a+b)=sina*cosb+cosa*sinb, sin(a-b)=sina*cosb-cosa*sinb.
We add these two expressions to get sin(a+b)+sin(a-b)=2sina*cosb.
So sin a * cosb = (sin (a+b)+sin (a-b))/2.
Similarly, if you subtract the two expressions, you get COSA * SINB = (SIN (A+B)-SIN (A-B))/2.
Similarly, we also know that cos(a+b)=cosa*cosb-sina*sinb, cos(a-b)=cosa*cosb+sina*sinb.
Therefore, by adding the two expressions, we can get cos(a+b)+cos(a-b)=2cosa*cosb.
So we get, COSA * COSB = (COS (A+B)+COS (A-B))/2.
Similarly, by subtracting two expressions, Sina * sinb =-(cos (a+b)-cos (a-b))/2 can be obtained.
In this way, we get the formulas of the sum and difference of four products:
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
With the four formulas of sum and difference, we only need one deformation to get the four formulas of sum and difference product.
Let a+b be X and a-b be Y in the above four formulas, then a=(x+y)/2 and b=(x-y)/2.
If a and b are represented by x and y respectively, we can get four sum-difference product formulas:
sinx+siny = 2 sin((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 =-2 sin((x+y)/2)* sin((x-y)/2)