When did you learn the auxiliary angle formula?
Auxiliary angle formula is the content of compulsory four in senior high school mathematics. The auxiliary angle formula is as follows:
1. Sum and difference formula of two angles (remember everything written)
sin(A+B)= Sina cosb+cosa sinb；
sin(A-B)= Sina cosb-sinBcosA；
cos(A+B)= cosa cosb-Sina sinb；
cos(A-B)= cosa cosb+Sina sinb；
tan(A+B)=(tanA+tanB)/( 1-tanA tanB)；
tan(A-B)=(tanA-tanB)/( 1+tanA tanB).
2. With the above formula, the following double-angle formula can be derived.
tan2a=2tana/[ 1-(tana)^2]；
cos2a=(cosa)^2-(sina)^2=2(cosa)^2 - 1= 1-2(sina)^2；
(The cosine above is very important)
sin2A=2sinA*cosA .
3. Half-angle ones just need to remember this.
tan(A/2)=( 1-cosA)/sinA = sinA/( 1+cosA)。
4. The power decreasing formula can be derived from the cosine of double angle.
(sina)^2=( 1-cos2a)/2；
(cosA)^2=( 1+cos2A)/2。
5. Using the power drop formula above, the following commonly used simplified formulas can be derived.
1-cosa=sin^(a/2)*2；
1-sinA=cos^(A/2)*2。
Further reading: What else is the formula of trigonometric function?
Acute angle formula of trigonometric function:
Opposite side/hypotenuse of sinα=∞α;
Adjacent side/hypotenuse of cosα=∞α;
Opposite side of tan α = adjacent side of ∠ α/∠α;
Adjacent side of cot α = opposite side of ∠ α/∠α.
Double angle formula:
Sin2A = 2SinACosA
cos2a=cosa^2-sina^2= 1-2sina^2=2cosa^2- 1；
tan2A=(2tanA)/( 1-tanA^2)。
(Note: Sina 2 is the square of Sina 2 (a))
Triple angle formula:
sin 3α= 4 sinαsin(π/3+α)sin(π/3-α)；
cos 3α= 4 cosαcos(π/3+α)cos(π/3-α)；
tan3a = tan a tan(π/3+a) tan(π/3-a).
Derivation of triple angle formula;
sin3a = sin(2a+a)= sin 2 acosa+cos 2 asina .
Power consumption reduction formula:
sin^2(α)=( 1-cos(2α))/2=versin(2α)/2；
cos^2(α)=( 1+cos(2α))/2=covers(2α)/2；
tan^2(α)=( 1-cos(2α))/( 1+cos(2α))。