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A complex number zı has a magnitude z1] = 2 and an angle 6, = 49°.=Express zy in rectangular form, as 21 == a +bi.Round a and b to the nearest thousandth.21 =+iShow Calculator

A Complex Number Zı Has A Magnitude Z1 2 And An Angle 6 49Express Zy In Rectangular Form As 21 A BiRound A And B To The Nearest Thousandth21 IShow Calculator class=

Sagot :

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