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$e^\theta = cos\theta + isin\theta$
$cos\theta$ is not linear. $e^\theta$ provides linear-like property.
etc.
$cos(\alpha+\beta)$ is difficult to extract. With the formula, we have
$e^{\alpha+\beta} = cos(\alpha+\beta) + isin(\alpha+\beta)$
$e^{\alpha+\beta} = e^\alpha e^\beta = cos\alpha cos\beta - sin\alpha sin\beta + i(cos\alpha sin\beta + sin\alpha cos\beta)$
Two complex number is equal if and only if their real parts and imagine parts equal, respectively.
So we have
$cos(\alpha+\beta) = cos\alpha cos\beta - sin\alpha sin\beta$
$sin(\alpha+\beta) = cos\alpha sin\beta + sin\alpha cos\beta$
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<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-433-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mi>&#x03B8;</mi></msup><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>&#x03B8;</mi><mo>+</mo><mi>i</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>&#x03B8;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-8273" style="width: 9.402em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.503em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.302em 1007.5em 2.552em -999.998em); top: -2.298em; left: 0em;"><span class="mrow" id="MathJax-Span-8274"><span class="msubsup" id="MathJax-Span-8275"><span style="display: inline-block; position: relative; width: 0.853em; height: 0px;"><span style="position: absolute; clip: rect(3.402em 1000.45em 4.152em -999.998em); top: -3.998em; left: 0em;"><span class="mi" id="MathJax-Span-8276" style="font-family: MathJax_Math-italic;">e</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -4.348em; left: 0.453em;"><span class="mi" id="MathJax-Span-8277" style="font-size: 70.7%; font-family: MathJax_Math-italic;">θ</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-8278" style="font-family: MathJax_Main; padding-left: 0.302em;">=</span><span class="mi" id="MathJax-Span-8279" style="font-family: MathJax_Math-italic; padding-left: 0.302em;">c</span><span class="mi" id="MathJax-Span-8280" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-8281" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8282" style="font-family: MathJax_Math-italic;">θ</span><span class="mo" id="MathJax-Span-8283" style="font-family: MathJax_Main; padding-left: 0.202em;">+</span><span class="mi" id="MathJax-Span-8284" style="font-family: MathJax_Math-italic; padding-left: 0.202em;">i</span><span class="mi" id="MathJax-Span-8285" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8286" style="font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-8287" style="font-family: MathJax_Math-italic;">n</span><span class="mi" id="MathJax-Span-8288" style="font-family: MathJax_Math-italic;">θ</span></span><span style="display: inline-block; width: 0px; height: 2.302em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.184em; border-left: 0px solid; width: 0px; height: 1.316em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mi>θ</mi></msup><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>θ</mi><mo>+</mo><mi>i</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>θ</mi></math></span></span><script type="math/tex" id="MathJax-Element-433">e^\theta = cos\theta + isin\theta</script></p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-434-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>s</mi><mi>&#x03B8;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-8289" style="width: 2.353em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.852em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.453em 1001.85em 2.452em -999.998em); top: -2.298em; left: 0em;"><span class="mrow" id="MathJax-Span-8290"><span class="mi" id="MathJax-Span-8291" style="font-family: MathJax_Math-italic;">c</span><span class="mi" id="MathJax-Span-8292" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-8293" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8294" style="font-family: MathJax_Math-italic;">θ</span></span><span style="display: inline-block; width: 0px; height: 2.302em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>s</mi><mi>θ</mi></math></span></span><script type="math/tex" id="MathJax-Element-434">cos\theta</script> is not linear. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-435-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mi>&#x03B8;</mi></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-8295" style="width: 1.103em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.853em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.302em 1000.85em 2.452em -999.998em); top: -2.298em; left: 0em;"><span class="mrow" id="MathJax-Span-8296"><span class="msubsup" id="MathJax-Span-8297"><span style="display: inline-block; position: relative; width: 0.853em; height: 0px;"><span style="position: absolute; clip: rect(3.402em 1000.45em 4.152em -999.998em); top: -3.998em; left: 0em;"><span class="mi" id="MathJax-Span-8298" style="font-family: MathJax_Math-italic;">e</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -4.348em; left: 0.453em;"><span class="mi" id="MathJax-Span-8299" style="font-size: 70.7%; font-family: MathJax_Math-italic;">θ</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.302em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mi>θ</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-435">e^\theta</script> provides linear-like property.</p>
<p>etc.</p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-436-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>&#x03B1;</mi><mo>+</mo><mi>&#x03B2;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-8300" style="width: 5.702em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.553em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.403em 1004.45em 2.702em -999.998em); top: -2.298em; left: 0em;"><span class="mrow" id="MathJax-Span-8301"><span class="mi" id="MathJax-Span-8302" style="font-family: MathJax_Math-italic;">c</span><span class="mi" id="MathJax-Span-8303" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-8304" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-8305" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-8306" style="font-family: MathJax_Math-italic;">α</span><span class="mo" id="MathJax-Span-8307" style="font-family: MathJax_Main; padding-left: 0.202em;">+</span><span class="mi" id="MathJax-Span-8308" style="font-family: MathJax_Math-italic; padding-left: 0.202em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8309" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.302em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.372em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-436">cos(\alpha+\beta)</script> is difficult to extract. With the formula, we have </p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-437-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B1;</mi><mo>+</mo><mi>&#x03B2;</mi></mrow></msup><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>&#x03B1;</mi><mo>+</mo><mi>&#x03B2;</mi><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>&#x03B1;</mi><mo>+</mo><mi>&#x03B2;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-8310" style="width: 17.453em; display: inline-block;"><span style="display: inline-block; position: relative; width: 13.953em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.302em 1013.85em 2.702em -999.998em); top: -2.298em; left: 0em;"><span class="mrow" id="MathJax-Span-8311"><span class="msubsup" id="MathJax-Span-8312"><span style="display: inline-block; position: relative; width: 1.952em; height: 0px;"><span style="position: absolute; clip: rect(3.402em 1000.45em 4.152em -999.998em); top: -3.998em; left: 0em;"><span class="mi" id="MathJax-Span-8313" style="font-family: MathJax_Math-italic;">e</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -4.348em; left: 0.453em;"><span class="texatom" id="MathJax-Span-8314"><span class="mrow" id="MathJax-Span-8315"><span class="mi" id="MathJax-Span-8316" style="font-size: 70.7%; font-family: MathJax_Math-italic;">α</span><span class="mo" id="MathJax-Span-8317" style="font-size: 70.7%; font-family: MathJax_Main;">+</span><span class="mi" id="MathJax-Span-8318" style="font-size: 70.7%; font-family: MathJax_Math-italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-8319" style="font-family: MathJax_Main; padding-left: 0.302em;">=</span><span class="mi" id="MathJax-Span-8320" style="font-family: MathJax_Math-italic; padding-left: 0.302em;">c</span><span class="mi" id="MathJax-Span-8321" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-8322" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-8323" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-8324" style="font-family: MathJax_Math-italic;">α</span><span class="mo" id="MathJax-Span-8325" style="font-family: MathJax_Main; padding-left: 0.202em;">+</span><span class="mi" id="MathJax-Span-8326" style="font-family: MathJax_Math-italic; padding-left: 0.202em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8327" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-8328" style="font-family: MathJax_Main; padding-left: 0.202em;">+</span><span class="mi" id="MathJax-Span-8329" style="font-family: MathJax_Math-italic; padding-left: 0.202em;">i</span><span class="mi" id="MathJax-Span-8330" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8331" style="font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-8332" style="font-family: MathJax_Math-italic;">n</span><span class="mo" id="MathJax-Span-8333" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-8334" style="font-family: MathJax_Math-italic;">α</span><span class="mo" id="MathJax-Span-8335" style="font-family: MathJax_Main; padding-left: 0.202em;">+</span><span class="mi" id="MathJax-Span-8336" style="font-family: MathJax_Math-italic; padding-left: 0.202em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8337" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.302em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.372em; border-left: 0px solid; width: 0px; height: 1.503em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>α</mi><mo>+</mo><mi>β</mi></mrow></msup><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-437">e^{\alpha+\beta} = cos(\alpha+\beta) + isin(\alpha+\beta)</script></p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-438-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>&#x03B1;</mi><mo>+</mo><mi>&#x03B2;</mi></mrow></msup><mo>=</mo><msup><mi>e</mi><mi>&#x03B1;</mi></msup><msup><mi>e</mi><mi>&#x03B2;</mi></msup><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>&#x03B1;</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>&#x03B2;</mi><mo>&#x2212;</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>&#x03B1;</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>&#x03B2;</mi><mo>+</mo><mi>i</mi><mo stretchy="false">(</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>&#x03B1;</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>&#x03B2;</mi><mo>+</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>&#x03B1;</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>&#x03B2;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-8338" style="width: 34.203em; display: inline-block;"><span style="display: inline-block; position: relative; width: 27.352em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.302em 1027.25em 2.702em -999.998em); top: -2.298em; left: 0em;"><span class="mrow" id="MathJax-Span-8339"><span class="msubsup" id="MathJax-Span-8340"><span style="display: inline-block; position: relative; width: 1.952em; height: 0px;"><span style="position: absolute; clip: rect(3.402em 1000.45em 4.152em -999.998em); top: -3.998em; left: 0em;"><span class="mi" id="MathJax-Span-8341" style="font-family: MathJax_Math-italic;">e</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -4.348em; left: 0.453em;"><span class="texatom" id="MathJax-Span-8342"><span class="mrow" id="MathJax-Span-8343"><span class="mi" id="MathJax-Span-8344" style="font-size: 70.7%; font-family: MathJax_Math-italic;">α</span><span class="mo" id="MathJax-Span-8345" style="font-size: 70.7%; font-family: MathJax_Main;">+</span><span class="mi" id="MathJax-Span-8346" style="font-size: 70.7%; font-family: MathJax_Math-italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-8347" style="font-family: MathJax_Main; padding-left: 0.302em;">=</span><span class="msubsup" id="MathJax-Span-8348" style="padding-left: 0.302em;"><span style="display: inline-block; position: relative; width: 1.002em; height: 0px;"><span style="position: absolute; clip: rect(3.402em 1000.45em 4.152em -999.998em); top: -3.998em; left: 0em;"><span class="mi" id="MathJax-Span-8349" style="font-family: MathJax_Math-italic;">e</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -4.348em; left: 0.453em;"><span class="mi" id="MathJax-Span-8350" style="font-size: 70.7%; font-family: MathJax_Math-italic;">α</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-8351"><span style="display: inline-block; position: relative; width: 0.953em; height: 0px;"><span style="position: absolute; clip: rect(3.402em 1000.45em 4.152em -999.998em); top: -3.998em; left: 0em;"><span class="mi" id="MathJax-Span-8352" style="font-family: MathJax_Math-italic;">e</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -4.348em; left: 0.453em;"><span class="mi" id="MathJax-Span-8353" style="font-size: 70.7%; font-family: MathJax_Math-italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-8354" style="font-family: MathJax_Main; padding-left: 0.302em;">=</span><span class="mi" id="MathJax-Span-8355" style="font-family: MathJax_Math-italic; padding-left: 0.302em;">c</span><span class="mi" id="MathJax-Span-8356" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-8357" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8358" style="font-family: MathJax_Math-italic;">α</span><span class="mi" id="MathJax-Span-8359" style="font-family: MathJax_Math-italic;">c</span><span class="mi" id="MathJax-Span-8360" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-8361" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8362" style="font-family: MathJax_Math-italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8363" style="font-family: MathJax_Main; padding-left: 0.202em;">−</span><span class="mi" id="MathJax-Span-8364" style="font-family: MathJax_Math-italic; padding-left: 0.202em;">s</span><span class="mi" id="MathJax-Span-8365" style="font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-8366" style="font-family: MathJax_Math-italic;">n</span><span class="mi" id="MathJax-Span-8367" style="font-family: MathJax_Math-italic;">α</span><span class="mi" id="MathJax-Span-8368" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8369" style="font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-8370" style="font-family: MathJax_Math-italic;">n</span><span class="mi" id="MathJax-Span-8371" style="font-family: MathJax_Math-italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8372" style="font-family: MathJax_Main; padding-left: 0.202em;">+</span><span class="mi" id="MathJax-Span-8373" style="font-family: MathJax_Math-italic; padding-left: 0.202em;">i</span><span class="mo" id="MathJax-Span-8374" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-8375" style="font-family: MathJax_Math-italic;">c</span><span class="mi" id="MathJax-Span-8376" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-8377" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8378" style="font-family: MathJax_Math-italic;">α</span><span class="mi" id="MathJax-Span-8379" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8380" style="font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-8381" style="font-family: MathJax_Math-italic;">n</span><span class="mi" id="MathJax-Span-8382" style="font-family: MathJax_Math-italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8383" style="font-family: MathJax_Main; padding-left: 0.202em;">+</span><span class="mi" id="MathJax-Span-8384" style="font-family: MathJax_Math-italic; padding-left: 0.202em;">s</span><span class="mi" id="MathJax-Span-8385" style="font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-8386" style="font-family: MathJax_Math-italic;">n</span><span class="mi" id="MathJax-Span-8387" style="font-family: MathJax_Math-italic;">α</span><span class="mi" id="MathJax-Span-8388" style="font-family: MathJax_Math-italic;">c</span><span class="mi" id="MathJax-Span-8389" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-8390" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8391" style="font-family: MathJax_Math-italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8392" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.302em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.372em; border-left: 0px solid; width: 0px; height: 1.503em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mi>α</mi><mo>+</mo><mi>β</mi></mrow></msup><mo>=</mo><msup><mi>e</mi><mi>α</mi></msup><msup><mi>e</mi><mi>β</mi></msup><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>α</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>β</mi><mo>−</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>α</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>β</mi><mo>+</mo><mi>i</mi><mo stretchy="false">(</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>α</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>β</mi><mo>+</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>α</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>β</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-438">e^{\alpha+\beta} = e^\alpha e^\beta = cos\alpha cos\beta - sin\alpha sin\beta + i(cos\alpha sin\beta + sin\alpha cos\beta)</script></p>
<p>Two complex number is equal if and only if their real parts and imagine parts equal, respectively.</p>
<p>So we have</p>
<p><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-439-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>&#x03B1;</mi><mo>+</mo><mi>&#x03B2;</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>&#x03B1;</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>&#x03B2;</mi><mo>&#x2212;</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>&#x03B1;</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>&#x03B2;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-8393" style="width: 18.902em; display: inline-block;"><span style="display: inline-block; position: relative; width: 15.103em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.403em 1015.1em 2.702em -999.998em); top: -2.298em; left: 0em;"><span class="mrow" id="MathJax-Span-8394"><span class="mi" id="MathJax-Span-8395" style="font-family: MathJax_Math-italic;">c</span><span class="mi" id="MathJax-Span-8396" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-8397" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-8398" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-8399" style="font-family: MathJax_Math-italic;">α</span><span class="mo" id="MathJax-Span-8400" style="font-family: MathJax_Main; padding-left: 0.202em;">+</span><span class="mi" id="MathJax-Span-8401" style="font-family: MathJax_Math-italic; padding-left: 0.202em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8402" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-8403" style="font-family: MathJax_Main; padding-left: 0.302em;">=</span><span class="mi" id="MathJax-Span-8404" style="font-family: MathJax_Math-italic; padding-left: 0.302em;">c</span><span class="mi" id="MathJax-Span-8405" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-8406" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8407" style="font-family: MathJax_Math-italic;">α</span><span class="mi" id="MathJax-Span-8408" style="font-family: MathJax_Math-italic;">c</span><span class="mi" id="MathJax-Span-8409" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-8410" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8411" style="font-family: MathJax_Math-italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8412" style="font-family: MathJax_Main; padding-left: 0.202em;">−</span><span class="mi" id="MathJax-Span-8413" style="font-family: MathJax_Math-italic; padding-left: 0.202em;">s</span><span class="mi" id="MathJax-Span-8414" style="font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-8415" style="font-family: MathJax_Math-italic;">n</span><span class="mi" id="MathJax-Span-8416" style="font-family: MathJax_Math-italic;">α</span><span class="mi" id="MathJax-Span-8417" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8418" style="font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-8419" style="font-family: MathJax_Math-italic;">n</span><span class="mi" id="MathJax-Span-8420" style="font-family: MathJax_Math-italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.302em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.372em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>α</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>β</mi><mo>−</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>α</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>β</mi></math></span></span><script type="math/tex" id="MathJax-Element-439">cos(\alpha+\beta) = cos\alpha cos\beta - sin\alpha sin\beta</script>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-440-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>&#x03B1;</mi><mo>+</mo><mi>&#x03B2;</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>&#x03B1;</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>&#x03B2;</mi><mo>+</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>&#x03B1;</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>&#x03B2;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-8421" style="width: 18.953em; display: inline-block;"><span style="display: inline-block; position: relative; width: 15.152em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.403em 1015.15em 2.702em -999.998em); top: -2.298em; left: 0em;"><span class="mrow" id="MathJax-Span-8422"><span class="mi" id="MathJax-Span-8423" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8424" style="font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-8425" style="font-family: MathJax_Math-italic;">n</span><span class="mo" id="MathJax-Span-8426" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-8427" style="font-family: MathJax_Math-italic;">α</span><span class="mo" id="MathJax-Span-8428" style="font-family: MathJax_Main; padding-left: 0.202em;">+</span><span class="mi" id="MathJax-Span-8429" style="font-family: MathJax_Math-italic; padding-left: 0.202em;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8430" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-8431" style="font-family: MathJax_Main; padding-left: 0.302em;">=</span><span class="mi" id="MathJax-Span-8432" style="font-family: MathJax_Math-italic; padding-left: 0.302em;">c</span><span class="mi" id="MathJax-Span-8433" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-8434" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8435" style="font-family: MathJax_Math-italic;">α</span><span class="mi" id="MathJax-Span-8436" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8437" style="font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-8438" style="font-family: MathJax_Math-italic;">n</span><span class="mi" id="MathJax-Span-8439" style="font-family: MathJax_Math-italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-8440" style="font-family: MathJax_Main; padding-left: 0.202em;">+</span><span class="mi" id="MathJax-Span-8441" style="font-family: MathJax_Math-italic; padding-left: 0.202em;">s</span><span class="mi" id="MathJax-Span-8442" style="font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-8443" style="font-family: MathJax_Math-italic;">n</span><span class="mi" id="MathJax-Span-8444" style="font-family: MathJax_Math-italic;">α</span><span class="mi" id="MathJax-Span-8445" style="font-family: MathJax_Math-italic;">c</span><span class="mi" id="MathJax-Span-8446" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-8447" style="font-family: MathJax_Math-italic;">s</span><span class="mi" id="MathJax-Span-8448" style="font-family: MathJax_Math-italic;">β<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.302em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.372em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>α</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>β</mi><mo>+</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>α</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>β</mi></math></span></span><script type="math/tex" id="MathJax-Element-440">sin(\alpha+\beta) = cos\alpha sin\beta + sin\alpha cos\beta</script></p>
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