How to express e^{2-i} in the form a+ib?

Answered question

2022-01-17

How to express e2i in the form a+ib?

Answer & Explanation

nick1337

nick1337

Expert2022-01-18Added 777 answers

Step 1 I would write it as: e2×e1i Then I will use the conversion from expoential to trigonometric form of a complex number as: reiθ=r[cos(θ)isin(θ)] with θ=1 in radians: and convert it into rectangular form getting: e2[cos(1)isin(1)]46i
Vasquez

Vasquez

Expert2022-01-18Added 669 answers

Step 1 Given the expression of the exponential function 1) e2i We are asked to convert this exponential function given in the equation (1) in the form a complex number a+ib Firstly, we write the exponential function e2i as follows 2) e2i=e2×e1i Now we covert the exponential function e1i to trigonometric form of a complex number. This is done by using the Euler’s formula which is given by eiθ=cosθ+isinθ Where, e= base of the natural logarithmic function i= imaginary unit θ= angle in radians In the exponential function e1i, we have θ=1 Hence by Euler’s formula we get, 3) e1i=cos(1)isin(1) Now we substitute the expression of e1i given in the equation (3) in the equation (2), we get, 4) e2i=e2×[cos(1)isin(1)] Now we calculate the values of each term in the R.H.S. using the calculator. For e2 we obtain the value as, 7.38905627.4 For cos(1) we obtain the value as, 0.5403020.54 For sin(1) we obtain the value as, 0.841470.84 Substituting all this values in the equation (4) we get, e2i7.4×[0.84i(0.54)] e2i7.4×0.84i(7.4×0.54) e2i3.996i(6.21) e2i46i

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