Beispiel: Trigonometrische Identitäten

Funktionen > Trigonometrische, Logarithmus- und Exponentialfunktionen > Beispiel: Trigonometrische Identitäten

Beispiel: Trigonometrische Identitäten

Grundlegende Identitäten

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  <math resultRef="48" xml:lang="de">
    <apply xml:lang="de" xmlns="http://schemas.mathsoft.com/math50">
      <equal xml:lang="de" />
      <apply xml:lang="de">
        <div xml:lang="de" />
        <apply xml:lang="de">
          <id xml:space="preserve" xml:lang="de">sin</id>
          <id xml:space="preserve" xml:lang="de">x</id>
        </apply>
        <apply xml:lang="de">
          <id xml:space="preserve" xml:lang="de">cos</id>
          <id xml:space="preserve" xml:lang="de">x</id>
        </apply>
      </apply>
      <apply xml:lang="de">
        <id xml:space="preserve" xml:lang="de">tan</id>
        <id xml:space="preserve" xml:lang="de">x</id>
      </apply>
    </apply>
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      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" xml:lang="de" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="de" />
    </resultFormat>
  </math>
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  <math resultRef="50" xml:lang="de">
    <apply xml:lang="de" xmlns="http://schemas.mathsoft.com/math50">
      <equal xml:lang="de" />
      <apply xml:lang="de">
        <plus xml:lang="de" />
        <apply xml:lang="de">
          <pow xml:lang="de" />
          <apply xml:lang="de">
            <id xml:space="preserve" xml:lang="de">sin</id>
            <id xml:space="preserve" xml:lang="de">x</id>
          </apply>
          <real xml:lang="de">2</real>
        </apply>
        <apply xml:lang="de">
          <pow xml:lang="de" />
          <apply xml:lang="de">
            <id xml:space="preserve" xml:lang="de">cos</id>
            <id xml:space="preserve" xml:lang="de">x</id>
          </apply>
          <real xml:lang="de">2</real>
        </apply>
      </apply>
      <real xml:lang="de">1</real>
    </apply>
    <resultFormat xml:lang="de">
      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" xml:lang="de" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="de" />
    </resultFormat>
  </math>
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Komplexe Identitäten

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  <math resultRef="58" xml:lang="de">
    <apply xml:lang="de" xmlns="http://schemas.mathsoft.com/math50">
      <equal xml:lang="de" />
      <apply xml:lang="de">
        <id xml:space="preserve" xml:lang="de">sin</id>
        <id xml:space="preserve" xml:lang="de">z</id>
      </apply>
      <apply xml:lang="de">
        <div xml:lang="de" />
        <apply xml:lang="de">
          <minus xml:lang="de" />
          <apply xml:lang="de">
            <pow xml:lang="de" />
            <id xml:space="preserve" xml:lang="de">e</id>
            <apply xml:lang="de">
              <mult xml:lang="de" />
              <id xml:space="preserve" xml:lang="de">i</id>
              <id xml:space="preserve" xml:lang="de">z</id>
            </apply>
          </apply>
          <apply xml:lang="de">
            <pow xml:lang="de" />
            <id xml:space="preserve" xml:lang="de">e</id>
            <apply xml:lang="de">
              <mult xml:lang="de" />
              <apply xml:lang="de">
                <neg xml:lang="de" />
                <id xml:space="preserve" xml:lang="de">i</id>
              </apply>
              <id xml:space="preserve" xml:lang="de">z</id>
            </apply>
          </apply>
        </apply>
        <apply xml:lang="de">
          <mult xml:lang="de" />
          <real xml:lang="de">2</real>
          <id xml:space="preserve" xml:lang="de">i</id>
        </apply>
      </apply>
    </apply>
    <resultFormat xml:lang="de">
      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" xml:lang="de" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="de" />
    </resultFormat>
  </math>
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    <apply xml:lang="de" xmlns="http://schemas.mathsoft.com/math50">
      <equal xml:lang="de" />
      <apply xml:lang="de">
        <id xml:space="preserve" xml:lang="de">cos</id>
        <id xml:space="preserve" xml:lang="de">z</id>
      </apply>
      <apply xml:lang="de">
        <div xml:lang="de" />
        <apply xml:lang="de">
          <plus xml:lang="de" />
          <apply xml:lang="de">
            <pow xml:lang="de" />
            <id xml:space="preserve" xml:lang="de">e</id>
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              <id xml:space="preserve" xml:lang="de">i</id>
              <id xml:space="preserve" xml:lang="de">z</id>
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            <pow xml:lang="de" />
            <id xml:space="preserve" xml:lang="de">e</id>
            <apply xml:lang="de">
              <mult xml:lang="de" />
              <apply xml:lang="de">
                <neg xml:lang="de" />
                <id xml:space="preserve" xml:lang="de">i</id>
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              <id xml:space="preserve" xml:lang="de">z</id>
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        <real xml:lang="de">2</real>
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      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" xml:lang="de" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="de" />
    </resultFormat>
  </math>
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        <id xml:space="preserve" xml:lang="de">e</id>
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          <id xml:space="preserve" xml:lang="de">i</id>
          <id xml:space="preserve" xml:lang="de">z</id>
        </apply>
      </apply>
      <apply xml:lang="de">
        <plus xml:lang="de" />
        <apply xml:lang="de">
          <id xml:space="preserve" xml:lang="de">cos</id>
          <id xml:space="preserve" xml:lang="de">z</id>
        </apply>
        <apply xml:lang="de">
          <mult xml:lang="de" />
          <id xml:space="preserve" xml:lang="de">i</id>
          <apply xml:lang="de">
            <id xml:space="preserve" xml:lang="de">sin</id>
            <id xml:space="preserve" xml:lang="de">z</id>
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    </resultFormat>
  </math>
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          <id xml:space="preserve" xml:lang="de">e</id>
          <id xml:space="preserve" xml:lang="de">x</id>
        </apply>
        <parens xml:lang="de">
          <apply xml:lang="de">
            <plus xml:lang="de" />
            <apply xml:lang="de">
              <id xml:space="preserve" xml:lang="de">cos</id>
              <id xml:space="preserve" xml:lang="de">y</id>
            </apply>
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              <mult xml:lang="de" />
              <id xml:space="preserve" xml:lang="de">i</id>
              <apply xml:lang="de">
                <id xml:space="preserve" xml:lang="de">sin</id>
                <id xml:space="preserve" xml:lang="de">y</id>
              </apply>
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          </apply>
        </parens>
      </apply>
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      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" xml:lang="de" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="de" />
    </resultFormat>
  </math>
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    <apply xml:lang="de" xmlns="http://schemas.mathsoft.com/math50">
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      <apply xml:lang="de">
        <id xml:space="preserve" xml:lang="de">ln</id>
        <apply xml:lang="de">
          <plus xml:lang="de" />
          <id xml:space="preserve" xml:lang="de">x</id>
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        <plus xml:lang="de" />
        <apply xml:lang="de">
          <scale xml:lang="de" />
          <apply xml:lang="de">
            <div xml:lang="de" />
            <real xml:lang="de">1</real>
            <real xml:lang="de">2</real>
          </apply>
          <apply xml:lang="de">
            <id xml:space="preserve" xml:lang="de">ln</id>
            <apply xml:lang="de">
              <plus xml:lang="de" />
              <apply xml:lang="de">
                <pow xml:lang="de" />
                <id xml:space="preserve" xml:lang="de">x</id>
                <real xml:lang="de">2</real>
              </apply>
              <apply xml:lang="de">
                <pow xml:lang="de" />
                <id xml:space="preserve" xml:lang="de">y</id>
                <real xml:lang="de">2</real>
              </apply>
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          </apply>
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        <apply xml:lang="de">
          <mult xml:lang="de" />
          <id xml:space="preserve" xml:lang="de">i</id>
          <apply xml:lang="de">
            <id xml:space="preserve" xml:lang="de">atan</id>
            <apply xml:lang="de">
              <div xml:lang="de" />
              <id xml:space="preserve" xml:lang="de">y</id>
              <id xml:space="preserve" xml:lang="de">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" xml:lang="de" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="de" />
    </resultFormat>
  </math>
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Die Funktion ln0 ähnelt der Funktion ln, wird jedoch für null definiert.

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Hyperbolische Funktionen und Exponentiale

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        <id xml:space="preserve" xml:lang="de">sinh</id>
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      <apply xml:lang="de">
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            <id xml:space="preserve" xml:lang="de">x</id>
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            <pow xml:lang="de" />
            <id xml:space="preserve" xml:lang="de">e</id>
            <apply xml:lang="de">
              <neg xml:lang="de" />
              <id xml:space="preserve" xml:lang="de">x</id>
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        <real xml:lang="de">2</real>
      </apply>
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      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" xml:lang="de" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" xml:lang="de" />
    </resultFormat>
  </math>
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      <apply xml:lang="de">
        <div xml:lang="de" />
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          <plus xml:lang="de" />
          <apply xml:lang="de">
            <pow xml:lang="de" />
            <id xml:space="preserve" xml:lang="de">e</id>
            <id xml:space="preserve" xml:lang="de">x</id>
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            <pow xml:lang="de" />
            <id xml:space="preserve" xml:lang="de">e</id>
            <apply xml:lang="de">
              <neg xml:lang="de" />
              <id xml:space="preserve" xml:lang="de">x</id>
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    </resultFormat>
  </math>
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        <id xml:space="preserve" xml:lang="de">x</id>
      </apply>
      <apply xml:lang="de">
        <div xml:lang="de" />
        <apply xml:lang="de">
          <minus xml:lang="de" />
          <apply xml:lang="de">
            <pow xml:lang="de" />
            <id xml:space="preserve" xml:lang="de">e</id>
            <id xml:space="preserve" xml:lang="de">x</id>
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            <pow xml:lang="de" />
            <id xml:space="preserve" xml:lang="de">e</id>
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    </resultFormat>
  </math>
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        <real xml:lang="de">1</real>
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Hyperbolische Funktionen und trigonometrische Beziehungen

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