Complex numbers can be written in two forms:
[tex]\begin{gathered} z=a+b\cdot i \\ z=r\cdot e^{i\theta} \end{gathered}[/tex]Where a and b are known as the real and the imaginary part and r and theta are the magnitude and the angle of the number. In this case we are given these last two quantities and we have to find a and b. One way to do this is recalling an important property of the exponential expression above:
[tex]e^{i\theta}=\cos \theta+i\sin \theta[/tex]Then the exponential form of a number is equal to:
[tex]z=r\cdot e^{i\theta}=r\cdot(\cos \theta+i\sin \theta)=r\cos \theta+i\cdot r\sin \theta[/tex]And since we are talking about the same number then this expression must be equal to that given by a and b:
[tex]a+i\cdot b=r\cos \theta+i\cdot r\sin \theta[/tex]Equalizing terms without i and those with i we have two equations:
[tex]\begin{gathered} a=r\cos \theta \\ b=r\sin \theta \end{gathered}[/tex]Now let's use the data from the exercise:
[tex]\begin{gathered} r=\lvert z_1\rvert=2 \\ \theta=\theta_1=49^{\circ} \end{gathered}[/tex]Then we have:
[tex]\begin{gathered} a=2\cdot\cos 49^{\circ} \\ b=2\cdot\sin 49^{\circ} \end{gathered}[/tex]Using a calculator we can find a and b:
[tex]\begin{gathered} a=1.312 \\ b=1.509 \end{gathered}[/tex]Then the answers for the two boxes are 1.312 and 1.509
