Formulas for Trigonometric Functions

三角関数公式

一般的な関係式

• sin(-x) = -sin(x)
• cos(-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(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

加法定理

• sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
• cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
• 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

複素

• e^{i x} = cos(x) + i sin(x)
• e^{i (x+y)} = e^{i x} e^{i y}

リンク

この式集は 三角関数資料 を参考に整理した(殆どそのままである)。感謝

• 三角関数資料 URL:http://hp.vector.co.jp/authors/VA003604/sincos.htm

letters

• π · ± ⋅ ² ³ × ÷ ℑ ℘ ℜ ℵ ∞

rdソース

tri.rd.txt

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