Beispiel: Trigonometrische Identitäten

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

Beispiel: Trigonometrische Identitäten

Grundlegende Identitäten

<region id="ID0ETO5DB" top="211.20000000000005" left="19.200000000000003" height="34" width="150.05666666666664" xmlns="http://schemas.mathsoft.com/worksheet50">
  <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" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

Komplexe Identitäten

<region id="ID0EUQ5DB" top="441.60000000000008" left="19.200000000000003" height="49.352000000000004" width="137.53000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="58">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <id xml:space="preserve">sin</id>
        <id xml:space="preserve">z</id>
      </apply>
      <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" />
    </resultFormat>
  </math>
</region>

<region id="ID0ECR5DB" top="480.00000000000006" left="19.200000000000003" height="49.352000000000004" width="137.83666666666667" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="60">
    <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" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

<region id="ID0EQR5DB" top="537.60000000000014" left="19.200000000000003" height="27.752000000000002" width="158.92000000000002" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="62">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <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>
        <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" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

<region id="ID0E5R5DB" top="576.00000000000011" left="19.200000000000003" height="27.752000000000002" width="215.66666666666669" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="64">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <pow />
        <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>
    <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>

<region id="ID0EMS5DB" top="633.60000000000014" left="19.200000000000003" height="47.2" width="279.2166666666667" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="66">
    <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>
    <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>

Die Funktion ln0 ähnelt der Funktion ln, wird jedoch für null definiert.

Hyperbolische Funktionen und Exponentiale

<region id="ID0E5V5DB" top="844.80000000000018" left="38.400000000000006" height="49.352000000000004" width="131.08333333333334" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="72">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <id xml:space="preserve">sinh</id>
        <id xml:space="preserve">x</id>
      </apply>
      <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>
        <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>
</region>

<region id="ID0EMW5DB" top="883.20000000000016" left="38.400000000000006" height="49.352000000000004" width="131.39000000000002" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="74">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <id xml:space="preserve">cosh</id>
        <id xml:space="preserve">x</id>
      </apply>
      <apply>
        <div />
        <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>
        <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>
</region>

<region id="ID0E1W5DB" top="940.80000000000018" left="38.400000000000006" height="51.504000000000005" width="133.66666666666666" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="76">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <id xml:space="preserve">tanh</id>
        <id xml:space="preserve">x</id>
      </apply>
      <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>

Hyperbolische Funktionen und trigonometrische Beziehungen

<region id="ID0EXZ5DB" top="1286.4" left="19.200000000000003" height="25.6" width="159.23666666666668" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="88">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <id xml:space="preserve">tanh</id>
        <id xml:space="preserve">z</id>
      </apply>
      <apply>
        <mult />
        <apply>
          <neg />
          <id xml:space="preserve">i</id>
        </apply>
        <apply>
          <id xml:space="preserve">tan</id>
          <apply>
            <mult />
            <id xml:space="preserve">i</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" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

Zugehörige Themen

Hyperbolische Funktionen

Trigonometrische Funktionen

War dies hilfreich?