Exemple : Identités trigonométriques

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Exemple : Identités trigonométriques

Identités de base

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  <math resultRef="50">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <plus />
        <apply>
          <pow />
          <apply>
            <id xml:space="preserve">sin</id>
            <id xml:space="preserve">x</id>
          </apply>
          <real>2</real>
        </apply>
        <apply>
          <pow />
          <apply>
            <id xml:space="preserve">cos</id>
            <id xml:space="preserve">x</id>
          </apply>
          <real>2</real>
        </apply>
      </apply>
      <real>1</real>
    </apply>
    <resultFormat>
      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
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Identités complexes

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        <id xml:space="preserve">z</id>
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      <apply>
        <div />
        <apply>
          <minus />
          <apply>
            <pow />
            <id xml:space="preserve">e</id>
            <apply>
              <mult />
              <id xml:space="preserve">i</id>
              <id xml:space="preserve">z</id>
            </apply>
          </apply>
          <apply>
            <pow />
            <id xml:space="preserve">e</id>
            <apply>
              <mult />
              <apply>
                <neg />
                <id xml:space="preserve">i</id>
              </apply>
              <id xml:space="preserve">z</id>
            </apply>
          </apply>
        </apply>
        <apply>
          <mult />
          <real>2</real>
          <id xml:space="preserve">i</id>
        </apply>
      </apply>
    </apply>
    <resultFormat>
      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
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    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <id xml:space="preserve">cos</id>
        <id xml:space="preserve">z</id>
      </apply>
      <apply>
        <div />
        <apply>
          <plus />
          <apply>
            <pow />
            <id xml:space="preserve">e</id>
            <apply>
              <mult />
              <id xml:space="preserve">i</id>
              <id xml:space="preserve">z</id>
            </apply>
          </apply>
          <apply>
            <pow />
            <id xml:space="preserve">e</id>
            <apply>
              <mult />
              <apply>
                <neg />
                <id xml:space="preserve">i</id>
              </apply>
              <id xml:space="preserve">z</id>
            </apply>
          </apply>
        </apply>
        <real>2</real>
      </apply>
    </apply>
    <resultFormat>
      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
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        <id xml:space="preserve">e</id>
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          <mult />
          <id xml:space="preserve">i</id>
          <id xml:space="preserve">z</id>
        </apply>
      </apply>
      <apply>
        <plus />
        <apply>
          <id xml:space="preserve">cos</id>
          <id xml:space="preserve">z</id>
        </apply>
        <apply>
          <mult />
          <id xml:space="preserve">i</id>
          <apply>
            <id xml:space="preserve">sin</id>
            <id xml:space="preserve">z</id>
          </apply>
        </apply>
      </apply>
    </apply>
    <resultFormat>
      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
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      <equal />
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        <id xml:space="preserve">e</id>
        <apply>
          <plus />
          <id xml:space="preserve">x</id>
          <apply>
            <mult />
            <id xml:space="preserve">i</id>
            <id xml:space="preserve">y</id>
          </apply>
        </apply>
      </apply>
      <apply>
        <mult />
        <apply>
          <pow />
          <id xml:space="preserve">e</id>
          <id xml:space="preserve">x</id>
        </apply>
        <parens>
          <apply>
            <plus />
            <apply>
              <id xml:space="preserve">cos</id>
              <id xml:space="preserve">y</id>
            </apply>
            <apply>
              <mult />
              <id xml:space="preserve">i</id>
              <apply>
                <id xml:space="preserve">sin</id>
                <id xml:space="preserve">y</id>
              </apply>
            </apply>
          </apply>
        </parens>
      </apply>
    </apply>
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      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
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    </resultFormat>
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    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <id xml:space="preserve">ln</id>
        <apply>
          <plus />
          <id xml:space="preserve">x</id>
          <apply>
            <mult />
            <id xml:space="preserve">i</id>
            <id xml:space="preserve">y</id>
          </apply>
        </apply>
      </apply>
      <apply>
        <plus />
        <apply>
          <scale />
          <apply>
            <div />
            <real>1</real>
            <real>2</real>
          </apply>
          <apply>
            <id xml:space="preserve">ln</id>
            <apply>
              <plus />
              <apply>
                <pow />
                <id xml:space="preserve">x</id>
                <real>2</real>
              </apply>
              <apply>
                <pow />
                <id xml:space="preserve">y</id>
                <real>2</real>
              </apply>
            </apply>
          </apply>
        </apply>
        <apply>
          <mult />
          <id xml:space="preserve">i</id>
          <apply>
            <id xml:space="preserve">atan</id>
            <apply>
              <div />
              <id xml:space="preserve">y</id>
              <id xml:space="preserve">x</id>
            </apply>
          </apply>
        </apply>
      </apply>
    </apply>
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      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
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La fonction ln0 est similaire à ln, mais elle est définie pour zéro.

Fonctions hyperboliques et exponentielles

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        <id xml:space="preserve">sinh</id>
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        <div />
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          <minus />
          <apply>
            <pow />
            <id xml:space="preserve">e</id>
            <id xml:space="preserve">x</id>
          </apply>
          <apply>
            <pow />
            <id xml:space="preserve">e</id>
            <apply>
              <neg />
              <id xml:space="preserve">x</id>
            </apply>
          </apply>
        </apply>
        <real>2</real>
      </apply>
    </apply>
    <resultFormat>
      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
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        <apply>
          <plus />
          <apply>
            <pow />
            <id xml:space="preserve">e</id>
            <id xml:space="preserve">x</id>
          </apply>
          <apply>
            <pow />
            <id xml:space="preserve">e</id>
            <apply>
              <neg />
              <id xml:space="preserve">x</id>
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        <real>2</real>
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        <id xml:space="preserve">tanh</id>
        <id xml:space="preserve">x</id>
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      <apply>
        <div />
        <apply>
          <minus />
          <apply>
            <pow />
            <id xml:space="preserve">e</id>
            <id xml:space="preserve">x</id>
          </apply>
          <apply>
            <pow />
            <id xml:space="preserve">e</id>
            <apply>
              <neg />
              <id xml:space="preserve">x</id>
            </apply>
          </apply>
        </apply>
        <apply>
          <plus />
          <apply>
            <pow />
            <id xml:space="preserve">e</id>
            <id xml:space="preserve">x</id>
          </apply>
          <apply>
            <pow />
            <id xml:space="preserve">e</id>
            <apply>
              <neg />
              <id xml:space="preserve">x</id>
            </apply>
          </apply>
        </apply>
      </apply>
    </apply>
    <resultFormat>
      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

Fonctions hyperboliques et relations trigonométriques

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        <mult />
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          <neg />
          <id xml:space="preserve">i</id>
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        <apply>
          <id xml:space="preserve">tan</id>
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            <mult />
            <id xml:space="preserve">i</id>
            <id xml:space="preserve">z</id>
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      </apply>
    </apply>
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      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
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Rubriques associées

Fonctions hyperboliques

Fonctions trigonométriques

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