Evaluate the following .?

Question

Update:

Edit : Plz Read the numerator as 4 ( cos 75 + i sin 75 )

Answer 1

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It's a simple question. You have to use simple formulae -

Answer 2

(4 cos 75° + i sin 75°) / (4(cos 30° + i sin 30°))

= (0.3449 + 0.07972 i)

Answer 3

360 = 2*2*2*3*3*5 and 75 = 3*5*5

The LCM of 75 and 360 = 5*360 = 1800.

Note 5*360=1800 and 24*75 = 1800.

1800º = 5*360º = 10π rad.  And since 1800/75 = 24:

75º = (10π)/24 = (5/12)π rad.

cos(75º) + isin(75º) =  (cos((5/12)π) + isin((5/12)π)

. . . . . . . . . . . . . . . .= e^((5/12)πi)

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Since 30º =  π/6 rad:

cos(30º) + isin(30º) = cos((π/6)i) + isin((π/6)i)

. . . . . . . . . . . . . . . . = e^((π/6)i)

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So your expression becomes:

4e^((5/12)πi)  / (0.4 e^((π/6)i)))

= 10 e^((3/12)πi (since e^x/e^y = e^(x-y) and 5/12-1/6 = 3/12)

= 10 e^(π/4)i)’

Since π/4 =45º you could write this as:

10(cos(45º) + isin(45º))