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Solve the equation for \\(\\theta\\) and \\(x\\)
\n\\(\\theta\\) = [[0]]
\n\\(x\\) = [[1]]
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\n\\(\\var{a}(cos(\\theta)+jsin(\\theta))+\\var{b}(cos(-\\theta)+jsin(-\\theta))=\\var{c}+xj\\)
\n\\(\\var{a}(cos(\\theta)+jsin(\\theta))+\\var{b}(cos(\\theta)-jsin(\\theta))=\\var{c}+xj\\)
\n\\(\\var{a}cos(\\theta)+j\\var{a}sin(\\theta))+\\var{b}cos(\\theta)-j\\var{b}sin(\\theta))=\\var{c}+xj\\)
\n\\(\\simplify{{a}+{b}}cos(\\theta)+j\\simplify{{a}-{b}}sin(\\theta)=\\var{c}+xj\\)
\nThis equation can be split into two parts. Real = Real and Imaginary = Imaginary.
\n\\(\\simplify{{a}+{b}}cos(\\theta)=\\var{c}\\) and \\(\\simplify{{a}-{b}}sin(\\theta)=x\\)
\n\\(cos(\\theta)=\\simplify{{c}/({a}+{b})}\\)
\n\\(\\theta = \\var{theta}\\)
\nInserting this into the second part gives
\n\\(\\simplify{{a}-{b}}sin(\\var{theta})=x\\)
\n\\(x=\\simplify{({a}-{b})}\\simplify{sin({theta})}\\)
\n\\(x=\\var{x}\\)
", "extensions": [], "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "name": "Euler's identity", "metadata": {"description": "Euler's identity \\(e^{j\\theta}=cos(\\theta)+jsin(\\theta)\\)
", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["a", "b", "d", "c", "theta", "x"], "rulesets": {}, "statement": "Given the equation \\(\\var{a}e^{j\\theta}+\\var{b}e^{-j\\theta}=\\var{c}+xj\\)
", "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}