1, formulas of trigonometric functions of sum and difference of two angles:
sin(α+β)=sinαcosβ+cosαsinβ
sin(α-β)=sinαcosβ-cosαsinβ
cos(α+β)=cosαcosβ-sinαsinβ
cos(α-β)=cosαcosβ+sinαsinβ
tan(α+β)=(tanα+tanβ)/( 1-tanαtanβ)
tan(α-β)=(tanα-tanβ)/( 1+tanαtanβ)
2, double angle formula:
Sine, Cosine and Tangent Formulas of Double Angles (Ascending Power and Shrinking Angle Formula)
sin2α=2sinαcosα
cos2α=cos^2(α)-sin^2(α)=2cos^2(α)- 1= 1-2sin^2(α)
tan2α=2tanα/[ 1-tan^2(α)]
3. Half-angle formula:
Sine, cosine and tangent formulas of half angle (power decreasing and angle expanding formulas)
sin^2(α/2)=( 1-cosα)/2
cos^2(α/2)=( 1+cosα)/2
tan^2(α/2)=( 1-cosα)/( 1+cosα)
And tan (α/2) = (1-cos α)/sin α = sin α/(1+cos α).
4. General formula:
sinα=2tan(α/2)/[ 1+tan^2(α/2)]
cosα=[ 1-tan^2(α/2)]/[ 1+tan^2(α/2)]
tanα=2tan(α/2)/[ 1-tan^2(α/2)]
Derivation of general formula;
Attached derivation: sin 2 α = 2 sin α cos α = 2 sin α cos α/(cos 2 (α)+sin 2 (α)) ...
(because cos 2 (α)+sin 2 (α) = 1)
Divide the * fraction up and down by COS 2 (α) to get SIN 2 α = 2 tan α/( 1+tan 2 (α)).
Then replace α with α/2.
Similarly, the universal formula of cosine can be derived. By comparing sine and cosine, a general formula of tangent can be obtained.
5, triple angle formula:
Sine, cosine and tangent formulas of triple angle;
sin3α=3sinα-4sin^3(α)
cos3α=4cos^3(α)-3cosα
tan3α=[3tanα-tan^3(α)]/[ 1-3tan^2(α)]