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

Answer 1

Josiah Anderson

The answer is #=-(sqrt2+sqrt6)/4+(sqrt2-sqrt6)/4*i#

Apply Euler's relation

#e^(itheta)=costheta+isintheta#

Therefore,

#e^(i7/12pi)=cos(7/12pi)+isin(7/12pi)#

#cos(7/12pi)=cos(1/4pi+1/3pi)=cos(1/4pi)cos(1/3pi)-sin(1/4pi)sin(1/3pi)#

#=(sqrt2/2).(1/2)-(sqrt2/2)(sqrt3/2)#

#=(sqrt2-sqrt6)/4#

#sin(7/12pi)=sin(1/4pi+1/3pi)=sin(1/4pi)cos(1/3pi)+cos(1/4pi)sin(1/3pi)#

#=(sqrt2/2).(1/2)+(sqrt2/2)(sqrt3/2)#

#=(sqrt2+sqrt6)/4#

#e^(ipi/2)=cos(pi/2)+isin(pi/2)=0+i.1=i#

#i^2=-1#

Therefore,

#e^(i7/12pi).e^(ipi/2)=(((sqrt2-sqrt6)+(sqrt2+sqrt6)i)/4)(i)#

#=-(sqrt2+sqrt6)/4+(sqrt2-sqrt6)/4*i#

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Answer 2

Madelyn Adams

To multiply ( e^{(7\pi/12)} ) by ( e^{(\pi/2)i} ) in trigonometric form, you can add the exponents and express the result as a single exponential function.

( e^{(7\pi/12)} \times e^{(\pi/2)i} = e^{(7\pi/12 + \pi/2)i} )

Now, combine the exponents:

( 7\pi/12 + \pi/2 = (7\pi + 6\pi)/12 = (13\pi)/12 )

So, the result in trigonometric form is ( e^{(13\pi/12)i} ).

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Answer from HIX Tutor

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