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A complex number
z_(1) has a magnitude
|z_(1)|=3 and an angle
theta_(1)=20^(@).
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 $\left|z_{1}\right|=3$ and an angle $\theta_{1}=20^{\circ}$ . $\newline$ 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|=3

and an angle

\theta_{1}=20^{\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 to rectangular form: Convert the polar form of the complex number to rectangular form using the formula $z = r(\cos(\theta) + i\sin(\theta))$ , where $r$ is the magnitude and $\theta$ is the angle. $\newline$ Given: $|z_{1}| = 3$ and $\theta_{1} = 20$ degrees. $\newline$ We need to calculate the cosine and sine of $20$ degrees.
Calculate cosine and sine: Calculate the cosine and sine of $20$ degrees. $\newline$ Using a calculator or trigonometric tables: $\newline$ $\cos(20^\circ) \approx 0.9397$ $\newline$ $\sin(20^\circ) \approx 0.3420$
Multiply by magnitude: Multiply the cosine and sine values by the magnitude to get the real and imaginary parts of the complex number. $\newline$ $a = r \cdot \cos(\theta) = 3 \cdot 0.9397 \approx 2.8191$ $\newline$ $b = r \cdot \sin(\theta) = 3 \cdot 0.3420 \approx 1.0260$
Round to nearest thousandth: Round the real and imaginary parts to the nearest thousandth. $\newline$ $a \approx 2.819$ $\newline$ $b \approx 1.026$
Express in rectangular form: Express $z_{1}$ in rectangular form using the rounded values of $a$ and $b$ . $z_{1} = a + bi = 2.819 + 1.026i$

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