How do you multiply $e^{\frac{7 π}{12}} \cdot e^{\frac{π}{2} i}$ $e^(( 7 pi )/ 12 ) * e^( pi/2 i )$ in trigonometric form?

Question

How do you multiply $e^{\frac{7 π}{12}} \cdot e^{\frac{π}{2} i}$ $e^(( 7 pi )/ 12 ) * e^( pi/2 i )$ in trigonometric form?

1 Answer

Impact of this question

1372 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-11-22T14:43:55

The answer is $= - \frac{\sqrt{2} + \sqrt{6}}{4} + \frac{\sqrt{2} - \sqrt{6}}{4} \cdot i$ $=-(sqrt2+sqrt6)/4+(sqrt2-sqrt6)/4*i$

Explanation:

Apply Euler's relation

$e^{i θ} = cos θ + i sin θ$ $e^(itheta)=costheta+isintheta$

Therefore,

$e^{i \frac{7}{12} π} = cos (\frac{7}{12} π) + i sin (\frac{7}{12} π)$ $e^(i7/12pi)=cos(7/12pi)+isin(7/12pi)$

$cos (\frac{7}{12} π) = cos (\frac{1}{4} π + \frac{1}{3} π) = cos (\frac{1}{4} π) cos (\frac{1}{3} π) - sin (\frac{1}{4} π) sin (\frac{1}{3} π)$ $cos(7/12pi)=cos(1/4pi+1/3pi)=cos(1/4pi)cos(1/3pi)-sin(1/4pi)sin(1/3pi)$

$= (\frac{\sqrt{2}}{2}) . (\frac{1}{2}) - (\frac{\sqrt{2}}{2}) (\frac{\sqrt{3}}{2})$ $=(sqrt2/2).(1/2)-(sqrt2/2)(sqrt3/2)$

$= \frac{\sqrt{2} - \sqrt{6}}{4}$ $=(sqrt2-sqrt6)/4$

$sin (\frac{7}{12} π) = sin (\frac{1}{4} π + \frac{1}{3} π) = sin (\frac{1}{4} π) cos (\frac{1}{3} π) + cos (\frac{1}{4} π) sin (\frac{1}{3} π)$ $sin(7/12pi)=sin(1/4pi+1/3pi)=sin(1/4pi)cos(1/3pi)+cos(1/4pi)sin(1/3pi)$

$= (\frac{\sqrt{2}}{2}) . (\frac{1}{2}) + (\frac{\sqrt{2}}{2}) (\frac{\sqrt{3}}{2})$ $=(sqrt2/2).(1/2)+(sqrt2/2)(sqrt3/2)$

$= \frac{\sqrt{2} + \sqrt{6}}{4}$ $=(sqrt2+sqrt6)/4$

$e^{i \frac{π}{2}} = cos (\frac{π}{2}) + i sin (\frac{π}{2}) = 0 + i .1 = i$ $e^(ipi/2)=cos(pi/2)+isin(pi/2)=0+i.1=i$

$i^{2} = - 1$ $i^2=-1$

Therefore,

$e^{i \frac{7}{12} π} . e^{i \frac{π}{2}} = ⎛ ⎜ ⎝ \frac{(\sqrt{2} - \sqrt{6}) + (\sqrt{2} + \sqrt{6}) i}{4} ⎞ ⎟ ⎠ (i)$ $e^(i7/12pi).e^(ipi/2)=(((sqrt2-sqrt6)+(sqrt2+sqrt6)i)/4)(i)$

$= - \frac{\sqrt{2} + \sqrt{6}}{4} + \frac{\sqrt{2} - \sqrt{6}}{4} \cdot i$ $=-(sqrt2+sqrt6)/4+(sqrt2-sqrt6)/4*i$

Science

Math

Humanities

... and beyond

How do you multiply $e^{\frac{7 π}{12}} \cdot e^{\frac{π}{2} i}$ $e^(( 7 pi )/ 12 ) * e^( pi/2 i )$ in trigonometric form?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you multiply e7π12⋅eπ2ie^(( 7 pi )/ 12 ) * e^( pi/2 i ) in trigonometric form?

1 Answer

Explanation:

Related questions

Impact of this question

How do you multiply $e^{\frac{7 π}{12}} \cdot e^{\frac{π}{2} i}$ $e^(( 7 pi )/ 12 ) * e^( pi/2 i )$ in trigonometric form?