Mathematical special trigonometric function value
Application of trigonometric function in junior high school mathematics trigonometric function has a more important application in complex numbers. Trigonometric function is also a common tool in physics.
There are six basic functions: function names: sine, cosine, tangent, cotangent, secant and cotangent;
Symbols: sin, cos, tan, cot, sec, csc.
Sine function sin(A)=a/c
Cosine function cos(A)=b/c
Tangent function tan(A)=a/b
Cotangent function cot(A)=b/a
Where a is the opposite side, b is the adjacent side and c is the hypotenuse.
Formulas of trigonometric functions table product sum and difference formula
sinαcosβ=( 1/2)*[sin(α+β)+sin(α-β)]
cosαsinβ=( 1/2)*[sin(α+β)-sin(α-β)]
cosαcosβ=( 1/2)*[cos(α+β)+cos(α-β)]
sinαsinβ=-( 1/2)*[cos(α+β)-cos(α-β)]
Sum-difference product formula
sinα+sinβ= 2 sin[(α+β)/2]cos[(α-β)/2]
sinα-sinβ= 2cos[(α+β)/2]sin[(α-β)/2]
cosα+cosβ= 2cos[(α+β)/2]cos[(α-β)/2]
cosα-cosβ=-2 sin[(α+β)/2]sin[(α-β)/2]
Triple angle formula
sin3α=3sinα-4sin^3α；
cos3α=4cos^3α-3cosα
The trigonometric function relationship between the 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)
Sine double angle formula
sin2α=2cosαsinα
Deduction: SIN2A = SIN (a+a) = SINA COSA+COSA SINA = 2 SINA COSA.
Extended formula: sin2a = 2sinacosa = 2tanacosa = 2tana/[1+tan2a]
1+sin2A=(sinA+cosA)^2
Cosine double angle formula
The cosine double angle formula has three sets of expressions, which are equivalent:
1.Cos2a = Cos2a-Sin2a =[ 1-tan2a]/[ 1+tan2a]
2.Cos2a= 1-2Sin2a
3.Cos2a=2Cos2a- 1
Deduction: cos2a = cos (a+a) = cosacosa-sinasina = cos2a-sin2a = 2cos2a-1
= 1-2sin^2A
Tangent dihedral formula
tan2α=2tanα/[ 1-tan2α]
Deduction: tan2a = tan (a+a) = (tana+tana)/(1-tanatana) = 2tana/[1-tan2a]
Reduced power formula
cosA^2=[ 1+cos2A]/2
sinA^2=[ 1-cos2A]/2
tana^2=[ 1-cos2a]/[ 1+cos2a]
Variant: sin2α = sin2 (α+π/4)-cos2 (α+π/4) = 2sin2 (a+π/4)-1=1-2cos2 (α+π/4); cos2α=2sin(α+π/4)cos(α+π/4)
Cosine theorem:
a^2=b^2+c^2-2bc*cosA
b^2=c^2+a^2-2ca*cosB
c^2=a^2+b^2-2ab*cosC