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A complex number
z_(1) has a magnitude
|z_(1)|=22 and an angle
theta_(1)=56^(@).
Express
z_(1) in rectangular form, as
z_(1)=a+bi.
Round
a and
b to the nearest thousandth.

z_(1)=◻+◻i

A complex number $z_{1}$ has a magnitude $\left|z_{1}\right|=22$ and an angle $\theta_{1}=56^{\circ}$ . $\newline$ Express $z_{1}$ in rectangular form, as $z_{1}=a+b i$ . $\newline$ Round $a$ and $b$ to the nearest thousandth. $\newline$ $z_{1}=\square+\square i$

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Compare linear and exponential growth

Full solution

Q. A complex number

z_{1}

has a magnitude

\left|z_{1}\right|=22

and an angle

\theta_{1}=56^{\circ}

.

\newline

Express

z_{1}

in rectangular form, as

z_{1}=a+b i

.

\newline

Round

a

and

b

to the nearest thousandth.

\newline

z_{1}=\square+\square i

Formulas for conversion: To convert a complex number from polar to rectangular form, we use the formulas $a = r \cdot \cos(\theta)$ and $b = r \cdot \sin(\theta)$ , where $r$ is the magnitude and $\theta$ is the angle in degrees.
Calculating the real part: First, we calculate the real part $a$ of the complex number $z_{1}$ . We have $r = 22$ and $\theta = 56$ degrees. So, $a = 22 \times \cos(56^\circ)$ .
Finding the value of $a$ : Using a calculator, we find that $\cos(56^\circ)$ is approximately $0.559192$ . Therefore, $a = 22 \times 0.559192$ .
Calculating the imaginary part: Performing the multiplication, we get $a \approx 22 \times 0.559192 \approx 12.302224$ . We round this to the nearest thousandth to get $a \approx 12.302$ .
Finding the value of $b$ : Next, we calculate the imaginary part $b$ of the complex number $z_{1}$ . We have $r = 22$ and $\theta = 56$ degrees. So, $b = 22 \times \sin(56$ degrees).
Finding the value of $b$ : Next, we calculate the imaginary part $b$ of the complex number $z_{1}$ . We have $r = 22$ and $\theta = 56$ degrees. So, $b = 22 \times \sin(56$ degrees). Using a calculator, we find that $\sin(56$ degrees) is approximately $0.829038$ . Therefore, $b = 22 \times 0.829038$ .
Finding the value of $b$ : Next, we calculate the imaginary part $b$ of the complex number $z_{1}$ . We have $r = 22$ and $\theta = 56$ degrees. So, $b = 22 \times \sin(56^\circ)$ .Using a calculator, we find that $\sin(56^\circ)$ is approximately $0.829038$ . Therefore, $b = 22 \times 0.829038$ .Performing the multiplication, we get $b \approx 22 \times 0.829038 \approx 18.238836$ . We round this to the nearest thousandth to get $b$ $0$ .

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