The Relationship between sine, cosine and exponential functions
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`exp(+-itheta)=cos(theta)+-isin(theta)` `cos(theta)=1/2(exp(itheta)+exp(-itheta)` `sin(theta)=1/(2i)(exp(itheta)-exp(-itheta)` |
We can also see the relationship by considering the series expansions, viz.,
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`exp(x)=1+x/(1!)+x^2/(2!)+x^3/(3!)+...=sum_(n=0)^oo x^n/(n!)` `sin(x)=x-x^3/(3!)+x^5/(5!)-x^7/(7!)+...=sum_(n=0)^oo ((-1)^k)/((2k+1)!)x^(2k+1)` `cos(x)=1-x^2/(2!)+x^4/(4!)-...=sum_(n=0)^oo (-1)^k/((2k)!)x^(2k)` |