三角関数の公式

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【一般の三角関数の公式】

sin(-x) = -sin(x), sin(π/2-x) = cos(x), sin(π-x) = sin(x), cos(-x) = cos(x), cos(π/2-x) = sin(x), cos(π-x) = -cos(x), sin(π/2+x) = cos(x), sin(π+x) = -sin(x), cos(π/2+x) = -sin(x), cos(π+x) = -cos(x),

sin(x)^2 + cos(x)^2 = 1

加算公式 sin(x+y) = sin(x)cos(y)+cos(x)sin(y), cos(x+y) = cos(x)cos(y)-sin(x)sin(y),

積和公式 2sin(x)cos(y) = sin(x+y) + sin(x-y), 2cos(x)sin(y) = sin(x+y) - sin(x-y), 2cos(x)cos(y) = cos(x+y) + cos(x-y), 2sin(x)sin(y) = -cos(x+y) + cos(x-y),

和積公式 sin(x)+sin(y) = 2sin( (x+y)/2 ) ・ cos( (x-y)/2 ), sin(x)-sin(y) = 2cos( (x+y)/2 ) ・ sin( (x-y)/2 ), cos(x)+cos(y) = 2cos( (x+y)/2 ) ・ cos( (x-y)/2 ), cos(x)-cos(y) = -2sin( (x+y)/2 ) ・ sin( (x-y)/2 ),

倍角公式 sin(2x) = 2sin(x)cos(x), cos(2x) = cos(x)^2 - sin(x)^2 = 2cos(x)^2 - 1 = 1 - 2sin(x)^2

半角公式 sin(x/2)^2 = ( 1-cos(x) )/2, cos(x/2)^2 = ( 1+cos(x) )/2

【球面三角法】
球面の三角形ABCの内角をa,b,c, 対辺をα,β,γとするとき、次のような関係が 成立する。

sin(a):sin(b):sin(c) = sin(α):sin(β):sin(γ)   正弦公式 cos(a) = cos(b)cos(c) + sin(b)sin(c)cos(α),etc.  余弦公式 cos(α) = -cos(β)cos(γ) + sin(β)sin(γ)cos(a),etc.   〃 sin(a)cos(β) = cos(b)sin(c) - sin(b)cos(c)cos(α),etc. 正弦余弦公式


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