Derivation of the triple angle formula, what is the derivation process of the doubling angle formula

sin3asin(2a+a)

sin2acosa+cos2asina

2sina(1-sin²a)+(1-2sin²a)sina3sina-4sin³a

cos3acos(2a+a)

cos2acosa-sin2asina

2cos²a-1)cosa-2(1-cos²a)cosa4cos³a-3cosa

sin3a=3sina-4sin³a

4sina(3/4-sin²a)

4sina[(√3/2)²-sin²a]

4sina(sin²60°-sin²a)

4sina(sin60°+sina)(sin60°-sina)4sina*2sin[(60+a)/2]cos[(60°-a)/2]*2sin[(60°-a)/2]cos[(60°-a)/2]

4sinasin(60°+a)sin(60°-a)cos3a=4cos³a-3cosa

4cosa(cos²a-3/4)

4cosa[cos²a-(√3/2)²]

4cosa(cos²a-cos²30°)

4cosa(cosa+cos30°)(cosa-cos30°)4cosa*2cos[(a+30°)/2]cos[(a-30°)/2]*

4cosasin(a+30°)sin(a-30°)-4cosasin[90°-(60°-a)]sin[-90°+(60°+a)]

4cosacos(60°-a)[-cos(60°+a)]4cosacos(60°-a)cos(60°+a)The above two equations can be compared.

tan3a=tanatan(60°-a)tan(60°+a)

Proof of the triple angle formula.

Idea: Trilogy: first decompose 3x into 2x+x, and use the sum angle formula; Then use the doubling angle formula to unify into a single angle x; Finally, it is simplified into a function, which is easy to remember and use.

The sinusoidal formula for the triple angle.

sin3x=3sinx-4sin^3 x

Proof: sin3x

sin(2x+x) (decomposed into 2x+x).

sin2xcosx + cos2xsinx (and angular sinusoidal formula).

2sinxcosxcosx+(1-2sin2 x)sinx (unified into a single angle x with the cosine formula).

2sinx(1-sin 2 x) + 1-2sin 2 x)sinx(reduced to a function).

3sinx-4sin^3 x

The cosine formula for the triple angle.

cos3x=4cos^3x-3cos x

Proof of: cos3x

cos(2x+x) (decomposed into 2x+x).

cos2xcosx-sin2xsinx (and angular cosine formula).

2cos 2 x-1) cosx-2sinxcosxsinx (unified into a single angle x with the cosine formula).

2cos 2 x-1)cosx-2cosx(1-cos 2 x) (reduced to a function).

4cos^3 x-3cosx

Tangent formula for triple angles.

tan3x=(3t-t 3) (1-3t 2), where t=tanx.

Proof : Let t = tanx, tan2x=2t (1-t 2).

tan3x=tan(2x+x) (decomposed into 2x+x).

tan2x+tanx) (1-tan2x tanx) (and tangent formula).

2t (1-t 2)+t] [1-2t (1-t 2) t] (unified into a single angle x by the doubling tangent formula).

3t-t 3) (1-3t 2), where t=tanx. (Simplify).

Application examples: Verification: tan3x=tan(60+x)tan(60-x)tanx

Proof: Let t = tanx

tan(60+x)=(√3+t)/(1-√3t)

tan(60-x) =(√3-t)/(1+√3t)

tan(60+x)tan(60-x)tanx

3-t^2)t/(1-3t^2)

tan3x (triple tangent formula).

2x Angle Formula:

1） sin2a=2sinacosa 。

2） cos2a=2(cosa)^2-1=1-2(sina)^2 。

3） tan2a=2tana/[1-(tana)^2]。

Derivation process: 1) sin2a=sin(a+a)=sinacosa+cosasina=2sinacosa.

2) cos2a=cos(a+a)=cosacosa-sinasina=(cosa)^2-(sina)^2=2(cosa)^2-1=1-2(sina)^2 。

3) tan2a=tan(a+a)=(tana+tana)/(1-tanatana)=2tana/[1-(tana)^2]。

2-fold angle transformation relation

Double angle formula.

Trigonometric value through the angle.

Some transformations of the Yuru Spike relation are used to represent the trigonometric value of its double angle 2, and the double angle formula includes the double angle formula of the rubber core string and the cosine.

The formula for the double angle as well as the formula for the tangent double angle.

In the calculation, it can be used to simplify the calculation formula and reduce the number of trigonometric functions, which is also widely used in engineering.

Sine double angle formula:

sin2α =2cosαsinα

Derivation: sin2a=sin(a+a)=sinacosa+cosinaina=2sinacosa

Cosine double angle formula:

The formula for the disadvantage of cosine double angle has three sets of representations, and the three sets of forms are equivalent:

1、cos2α =2(cosα)^2 − 1

2、cos2α =1 − 2(sinα)^2

3、cos2α =cosα)^2 − sinα)^2

Derivation: cos2a=cos(a+a)=cosacosa-sinasina=(cosa) 2-(sina) 2=2(cosa) 2-1=1-2(sina) 2

Tangent double angle formula:

tan2α=2tanα/[1-(tanα)^2]

Derivation: tan2a = tan(a+a) = (tana + tana) (1-tanatana) = 2tana [1-(tana) 2].

The formula of the double angle of the rock is expressed by some transformation of the trigonometric value of the angle of the angle, which includes the formula of the sine double angle, the formula of the cosine double angle and the tangent double angle formula.

It can be used to simplify the calculation formula and reduce the number of trigonometric functions in calculations, and it is also widely used in engineering.

The detailed process of deriving the induction formula is:

Since sin(- sin, sin( sin =sin(- let b= then b, substituting the two formulas into the above formula, we get sin(b)=sin( b). Rewrite b in the above equation to , that is, sin( shu) sin.

Derivation of the general formula:

sin2 2sin cos 2sin cos repentance, (because cos2( )sin2( )1) and divide the fraction up and down by cos 2 ( ) to get sin2 2tan and then replace it with 2. In the same way, the general formula of the front space conduction cosine can be deduced. The general formula for tangent can be obtained by sine ratio cosine.

The triple angle formula derives tan3 sin3 cos3 = (sin2 cos cos2 sin) cos2 sin ) cos2 sin2 sin )=cos3( )cos sin2.

2sin2( )cos is divided by cos3( ) to obtain: tan3 sin3 sin(2 sin2 cos cos2 sin = 2sin cos2( )sin.

2sinα－2sin3(α）sinα－2sin3(α）3sinα－4sin3(α）cos3α＝cos(2α＋αcos2αcosα－sin2αsinα=cosα－2cosαsin2(α）2cos3(α）cosα=4cos3(α）3cos。

The triple angle formula is an identity that expresses trigonometric functions of the shape sin(3x), cos(3x), etc., with the corresponding haplotype trigonometric functions. It is applied to mathematics, physics, astronomy and other subjects.

N-fold angle formula.

According to Euler's formula (cos + isin) n = cosn + isinn

The binomial theorem on the left is used to separate the real and imaginary parts to get the following two sets of formulas.

Triple angle formula: sin(3) = 3sin -4sin 3 = 4sin ·sin(60° + sin(60°-

cos(3α) = 4cos^3α-3cosα = 4cosα·cos(60°+αcos(60°-α

tan(3α) = (3tanα-tan^3α)/(1-3tan^2α) = tanαtan(π/3+α)tan(π/3-α)

The formula for triplicate angles in this paragraph is derived:

sin(2a+a)

sin2acosa+cos2asina

2sina(1-sin^2a)+(1-2sin^2a)sina

3sina-4sin^3a

cos(2a+a)

cos2acosa-sin2asina

2cos^2a-1)cosa-2(1-cos^2a)cosa

4cos^3a-3cosa

1）sin3a=3sina-4sin^3a

4sina(3/4-sin^2a)

4sina[(√3/2)^2-sin^2a]

4sina(sin^260°-sin^2a)

4sina(sin60°+sina)(sin60°-sina)

4sina*2sin[(60+a)/2]cos[(60°-a)/2]*2sin[(60°-a)/2]cos[(60°+a)/2]

4sinasin(60°+a)sin(60°-a)

2）cos3a=4cos^3a-3cosa

4cosa(cos^2a-3/4)

4cosa[cos^2a-(√3/2)^2]

4cosa(cos^2a-cos^230°)

4cosa(cosa+cos30°)(cosa-cos30°)

4cosa*2cos[(a+30°)/2]cos[(a-30°)/2]*

4cosasin(a+30°)sin(a-30°)

4cosasin[90°-(60°-a)]sin[-90°+(60°+a)]

4cosacos(60°-a)[-cos(60°+a)]

4cosacos(60°-a)cos(60°+a)

In summary, the above two formulas can be compared.

tan3a=tanatan(60°-a)tan(60°+a)

The detailed process of deriving the induction formula is:

Since sin(- sin, sin( sin =sin(- let b= then b, substituting the two formulas into the above formula, we get sin(b)=sin( b). Rewrite b in the above equation to , that is, sin( shu) sin.

Derivation of the general formula:

sin2 2sin cos 2sin cos repentance, (because cos2( )sin2( )1) and divide the fraction up and down by cos 2 ( ) to get sin2 2tan and then replace it with 2. In the same way, the general formula of the front space conduction cosine can be deduced. The general formula for tangent can be obtained by sine ratio cosine.

The triple angle formula derives tan3 sin3 cos3 = (sin2 cos cos2 sin) cos2 sin ) cos2 sin2 sin )=cos3( )cos sin2.

2sin2( )cos is divided by cos3( ) to obtain: tan3 sin3 sin(2 sin2 cos cos2 sin = 2sin cos2( )sin.

2sinα－2sin3(α）sinα－2sin3(α）3sinα－4sin3(α）cos3α＝cos(2α＋αcos2αcosα－sin2αsinα=cosα－2cosαsin2(α）2cos3(α）cosα=4cos3(α）3cos。