恒等式








恒等式（こうとうしき、英: identity）は、恒真な等式、すなわち等号 (=) を含む数式であって、そこに現れるあらゆる変数がどのような値にあっても、常に等号で結ばれた左右二つの数式の "値" が等しいもののことを言う。変数の動く範囲は、文脈によって異なる。恒等式であることを明示するとき、= の代わりに ≡ が使われる。


重要な恒等式の中には、公式、定理、法則などと呼ばれて知られているものも多く存在する。オイラーの公式、三角関数の加法定理、指数法則などはその例である。



例



  • 次の式は実数 x, y について恒等式である。
    x2+2xy+y2=(x+y)2.{displaystyle x^{2}+2xy+y^{2}=(x+y)^{2}.}{displaystyle x^{2}+2xy+y^{2}=(x+y)^{2}.}


  • (1) が実変数 x について恒等式であるとき、(2) が成立する


    ax2+bx+c=0{displaystyle ax^{2}+bx+c=0}{displaystyle ax^{2}+bx+c=0} … (1),


    a=b=c=0{displaystyle a=b=c=0}{displaystyle a=b=c=0} … (2).




  • 三角関数は次のような恒等式で結ばれている。

    sin2⁡x+cos2⁡x=1,{displaystyle sin ^{2}x+cos ^{2}x=1,}{displaystyle sin ^{2}x+cos ^{2}x=1,}

    tan⁡x=sin⁡xcos⁡x.{displaystyle tan x={frac {sin x}{cos x}}.}{displaystyle tan x={frac {sin x}{cos x}}.}



  • 1 = 1 はあらゆる変数に関する恒等式である。



関連項目





ウィキブックスに恒等式関連の解説書・教科書があります。


  • 恒真式

  • 数式

  • 等式

  • 方程式

  • 不等式



外部リンク


  • Weisstein, Eric W. "Identity". MathWorld（英語）..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}



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