# the amplitude of 1- cos theta - i sin theta

dear student

$amplitude\phantom{\rule{0ex}{0ex}}=\left|z\right|\phantom{\rule{0ex}{0ex}}=\left|1-\mathrm{cos}\theta -i\mathrm{sin}\theta \right|\phantom{\rule{0ex}{0ex}}=\sqrt{\left(1-\mathrm{cos}\theta {\right)}^{2}+\left(-\mathrm{sin}\theta {\right)}^{2}}\phantom{\rule{0ex}{0ex}}=\sqrt{1+{\mathrm{cos}}^{2}\theta -2\mathrm{cos}\theta +{\mathrm{sin}}^{2}\theta }\phantom{\rule{0ex}{0ex}}=\sqrt{1+1-2\mathrm{cos}\theta }\phantom{\rule{0ex}{0ex}}=\sqrt{2\left(1-\mathrm{cos}\theta \right)}\phantom{\rule{0ex}{0ex}}=\sqrt{4{\mathrm{sin}}^{2}\left(\theta /2\right)}\phantom{\rule{0ex}{0ex}}=2\mathrm{sin}\left(\theta /2\right)$

regards

• -8
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