Question

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

Answer 1

Hint: In this question, we need to express an exponential function in terms of a complex number. Firstly, we will split the exponential function ${e^{2 - i}}$ as ${e^2} \cdot {e^{ - 1i}}$. Then we find the expression for ${e^{ - 1i}}$ using the Euler’s formula given by ${e^{i\theta }} = \cos \theta + i\sin \theta $. We then multiply the obtained expression from this formula by the exponent ${e^2}$. Then using the calculator we find the values for exponential function ${e^2}$, cosine and sine of the angle $\theta $and obtain the expression in the form of the complex number $a + ib$.

Complete step by step solution:
Given the expression of the exponential function ${e^{2 - i}}$ …… (1)
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 ${e^{2 - i}}$ as follows.
${e^{2 - i}} = {e^2} \cdot {e^{ - 1i}}$ …… (2)
Now we covert the exponential function ${e^{ - 1i}}$ to trigonometric form of a complex number.
This is done by using the Euler’s formula which is given by,
${e^{i\theta }} = \cos \theta + i\sin \theta $
Where, $e = $ base of the natural logarithmic function
               $i = $ imaginary unit
              $\theta = $ angle in radians
In the exponential function ${e^{ - 1i}}$, we have $\theta = 1$.
Hence by Euler’s formula we get,
$ \Rightarrow {e^{ - 1i}} = \cos (1) - i\sin (1)$ …… (3)
Now we substitute the expression of ${e^{ - 1i}}$ given in the equation (3) in the equation (2), we get,
$ \Rightarrow {e^{2 - i}} = {e^2} \cdot [\cos (1) - i\sin (1)]$ …… (4)
Now we calculate the values of each term in the R.H.S. using the calculator.
For ${e^2}$ we obtain the value as, $7.3890561 \approx 7.4$
For $\cos (1)$ we obtain the value as, $0.540302 \approx 0.54$
For $\sin (1)$ we obtain the value as, $0.84147 \approx 0.84$
Substituting all this values in the equation (4) we get,
$ \Rightarrow {e^{2 - i}} \cong 7.4 \times [0.84 - i(0.54)]$
$ \Rightarrow {e^{2 - i}} \cong 7.4 \times 0.84 - i(7.4 \times 0.54)$
$ \Rightarrow {e^{2 - i}} \cong 3.996 - i(6.21)$
$ \Rightarrow {e^{2 - i}} \cong 4 - 6i$

Hence the expression for ${e^{2 - i}}$ in the form of complex number is given by, ${e^{2 - i}} \cong 4 - 6i$.

Note: Complex numbers are expressions in the form $x + iy$, where $x$ is the real part and $y$ is the imaginary part. These numbers cannot be marked on the real number line.
(Here note that the imaginary part is $y$, and not $iy$)
Students note that the backbone of this new number system is the number $i$, also known as the imaginary unit.
Students must know the Euler’s formula which is important to convert the exponential function to any complex number. The Euler’s formula is given by,
${e^{i\theta }} = \cos \theta + i\sin \theta $
Where, $e = $ base of the natural logarithmic function
               $i = $ imaginary unit
              $\theta = $ angle in radians