1. Please provide complete correct detailed solution. Preferentially done by han
ID: 2871680 • Letter: 1
Question
1. Please provide complete correct detailed solution. Preferentially done by hand or on paper.
Problem #1 This problem is a review of complex numbers. It must be clone by hand, without the aid of math software like Wolfram Alpha. (a) Euler^'s formula states that e^itheta = cos(theta) + i sin(theta). Use this expression to write sin(theta) and cos(theta) in terms of e^itheta and e^-itheta. (b) Use your results from part (a) to prove the identity cos(theta1 + theta2) = cos(theta1) cos(theta2) - sin(theta1) sin(theta2) (e) Find the simplest expression for i^378. (cl) Convert the complex number 1 + i to the form re^itheta. Using this result, calculate (1 + i)^30, then write it in the form a + bi where a and b are real numbers. Also find both complex square roots of 1 + i and plot them in the complex plane along with 1 + i itself.Explanation / Answer
a)
ei=cos+isin
e-i=cos-isin
ei+e-i =cos+isin+cos-isin
2cos=ei+e-i
cos=(1/2)[ei+e-i]
similarly sin=(1/2)[ei-e-i]
b)cos(1+2)=(1/2)[ei(1+2)+e-(1+2)]
=(1/2)[cos(1+2)+isin(1+2) +cos(1+2)-isin(1+2)
=(1/2)[2cos(1+2)]
=cos(1+2)
=cos1cos2 -sin1sin2
c)i378
=[ei/2]378
=[ei378/2]
=[ei189]
=[ei189]
=[ei(188+1)]
=[ei(1)]
=cos+isin
=-1 +i*0
=-1
d)(1+i)
((sqrt2)(1/sqrt2 +i/sqrt2))
(sqrt2)(ei/4)
(1+i)30
((sqrt2)(1/sqrt2 +i/sqrt2))30
=215(ei/4)30
=215(ei30/4)
=215(ei15/2)
=215(ei6 + 3/2)
=215(ei3/2)
=215(cos3/2 +isin3/2)
=215(0 -i)
=-i215
(1+i)1/2
=((sqrt2)(1/sqrt2 +i/sqrt2))1/2
=21/4 (ei/4)1/2
=21/4ei/8
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