Example: Trigonometric Identities

Functions > Trigonometric, Log, and Exponential > Example: Trigonometric Identities

Example: Trigonometric Identities

Basic Identities

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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" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

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Complex Identities

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  <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>

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  <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>

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  <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>

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  <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>

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  <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>

The ln0 function is similar to ln, but it is defined for zero.

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Hyperbolic Functions and Exponentials

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  <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>

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    <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>

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  <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>

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Hyperbolic Functions and Trigonometric Relationships

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    <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>

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