How do you perform the operation in trigonometric form $\frac{cos (\frac{5 π}{3}) + i sin (\frac{5 π}{3})}{cos π + i sin π}$ $(cos((5pi)/3)+isin((5pi)/3))/(cospi+isinpi)$ ?

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How do you perform the operation in trigonometric form $\frac{cos (\frac{5 π}{3}) + i sin (\frac{5 π}{3})}{cos π + i sin π}$ $(cos((5pi)/3)+isin((5pi)/3))/(cospi+isinpi)$ ?

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Answer 1 · 2016-12-04T09:27:53

$\frac{cos (\frac{5 π}{3}) + i sin (\frac{5 π}{3})}{cos π + i sin π} = cos (\frac{2 π}{3}) + i sin (\frac{2 π}{3})$ $(cos((5pi)/3)+isin((5pi)/3))/(cospi+isinpi)=cos((2pi)/3)+isin((2pi)/3)$

= $- \frac{1}{2} + i \frac{\sqrt{3}}{2}$ $-1/2+isqrt3/2$

Explanation:

A complex number in polar form such as $(r cos θ + i r sin θ)$ $(rcostheta+irsintheta)$ can be written in exponential form as

$r e^{i θ}$ $re^(itheta)$

As such $cos (\frac{5 π}{3}) + i sin (\frac{5 π}{3}) = e^{\frac{5 π}{3} i}$ $cos((5pi)/3)+isin((5pi)/3)=e^((5pi)/3i)$

and $cos π + i sin π = e^{π i}$ $cospi+isinpi=e^(pii)$ , and hence

$\frac{cos (\frac{5 π}{3}) + i sin (\frac{5 π}{3})}{cos π + i sin π} = \frac{e^{\frac{5 π}{3} i}}{e^{π i}}$ $(cos((5pi)/3)+isin((5pi)/3))/(cospi+isinpi)=e^((5pi)/3i)/e^(pii)$

= $e^{\frac{5 π}{3} i - π i} = e^{\frac{2 π}{3} i}$ $e^((5pi)/3i-pii)=e^((2pi)/3i)$

= $cos (\frac{2 π}{3}) + i sin (\frac{2 π}{3})$ $cos((2pi)/3)+isin((2pi)/3)$

= $- \frac{1}{2} + i \frac{\sqrt{3}}{2}$ $-1/2+isqrt3/2$

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How do you perform the operation in trigonometric form $\frac{cos (\frac{5 π}{3}) + i sin (\frac{5 π}{3})}{cos π + i sin π}$ $(cos((5pi)/3)+isin((5pi)/3))/(cospi+isinpi)$ ?

1 Answer

Explanation:

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How do you perform the operation in trigonometric form $\frac{cos (\frac{5 π}{3}) + i sin (\frac{5 π}{3})}{cos π + i sin π}$ $(cos((5pi)/3)+isin((5pi)/3))/(cospi+isinpi)$ ?