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A complex number
z_(1) has a magnitude
|z_(1)|=12 and an angle
theta_(1)=163^(@).
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 $\left|z_{1}\right|=12$ and an angle $\theta_{1}=163^{\circ}$ . $\newline$ 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|=12

and an angle

\theta_{1}=163^{\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

Convert angle to radians: To convert a complex number from polar to rectangular form, we use the equations $a = |z| \cdot \cos(\theta)$ and $b = |z| \cdot \sin(\theta)$ , where $|z|$ is the magnitude and $\theta$ is the angle in radians.
Calculate theta in radians: First, we need to convert the angle from degrees to radians. The angle given is $163$ degrees. To convert degrees to radians, we multiply by $\pi/180$ . $\newline$ $\theta_{1}$ in radians = $163 \times (\pi/180)$
Calculate real part $a$ : Now we calculate the value of $\theta_1$ in radians. $\theta_1$ in radians $\approx 163 \times (\pi/180) \approx 2.844886680750757$
Calculate value of a: Next, we calculate the real part $a$ of the complex number using the cosine of the angle. $a = |z_{1}| \cdot \cos(\theta_{1}) = 12 \cdot \cos(2.844886680750757)$
Calculate imaginary part b: Now we calculate the value of $a$ . $a \approx 12 \times \cos(2.844886680750757) \approx 12 \times (-0.961261695938319) \approx -11.535140351259428$
Calculate value of b: Next, we calculate the imaginary part $b$ of the complex number using the sine of the angle. $b = |z_{1}| \cdot \sin(\theta_{1}) = 12 \cdot \sin(2.844886680750757)$
Round to nearest thousandth: Now we calculate the value of $b$ . $\newline$ $b \approx 12 \times \sin(2.844886680750757) \approx 12 \times 0.27563735581699916 \approx 3.30764787060399$
Round to nearest thousandth: Now we calculate the value of $b$ . $b \approx 12 \times \sin(2.844886680750757) \approx 12 \times 0.27563735581699916 \approx 3.30764787060399$ Finally, we round $a$ and $b$ to the nearest thousandth. $a \approx -11.535$ (rounded to the nearest thousandth) $b \approx 3.308$ (rounded to the nearest thousandth)

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