A complex number z_(1) has a magnitude |z_(1)|=10 and an angle theta_(1)=84^(@). Express z_(1) in rectangular form, as z_(1)=a+bi. Round a and b to the nearest thousandth. z_(1)=◻+◻i

Rectangular form formula: To express a complex number in rectangular form, we use the formula z = a + bi, where a is the real part and b is the imaginary part. We can find a and b using the polar form of the complex number, which is z = r(cos(θ) + i sin(θ)), where r is the magnitude and θ is the angle.Calculating real part: Given the magnitude |z_(1)| = 10 and the angle θ_(1) = 84 degrees, we can calculate the real part a as a = r * cos(θ) = 10 * cos(84 degrees).Calculating imaginary part: Using a calculator, we find that cos(84 degrees) ≈ 0.104528. Therefore, a ≈ 10 * 0.104528 ≈ 1.04528.Rounding to nearest thousandth: Now we calculate the imaginary part b as b = r * sin(θ) = 10 * sin(84 degrees).Final rectangular form: Using a calculator, we find that sin(84 degrees) ≈ 0.994522. Therefore, b ≈ 10 * 0.994522 ≈ 9.94522.Rounding a and b to the nearest thousandth, we get a ≈ 1.045 and b ≈ 9.945.Therefore, the complex number z_(1) in rectangular form is z_(1) = 1.045 + 9.945i.

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A complex number
z_(1) has a magnitude
|z_(1)|=10 and an angle
theta_(1)=84^(@).
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 $\left|z_{1}\right|=10$ and an angle $\theta_{1}=84^{\circ}$ . $\newline$ 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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Q. A complex number

z_{1}

has a magnitude

\left|z_{1}\right|=10

and an angle

\theta_{1}=84^{\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

Rectangular form formula: To express a complex number in rectangular form, we use the formula $z = a + bi$ , where $a$ is the real part and $b$ is the imaginary part. We can find $a$ and $b$ using the polar form of the complex number, which is $z = r(\cos(\theta) + i \sin(\theta))$ , where $r$ is the magnitude and $\theta$ is the angle.
Calculating real part: Given the magnitude $|z_{1}| = 10$ and the angle $\theta_{1} = 84$ degrees, we can calculate the real part $a$ as $a = r \cdot \cos(\theta) = 10 \cdot \cos(84$ degrees).
Calculating imaginary part: Using a calculator, we find that $\cos(84^\circ) \approx 0.104528$ . Therefore, $a \approx 10 \times 0.104528 \approx 1.04528$ .
Rounding to nearest thousandth: Now we calculate the imaginary part $b$ as $b = r \times \sin(\theta) = 10 \times \sin(84^\circ)$ .
Final rectangular form: Using a calculator, we find that $\sin(84^\circ) \approx 0.994522$ . Therefore, $b \approx 10 \times 0.994522 \approx 9.94522$ .
Final rectangular form: Using a calculator, we find that $\sin(84^\circ) \approx 0.994522$ . Therefore, $b \approx 10 \times 0.994522 \approx 9.94522$ .Rounding $a$ and $b$ to the nearest thousandth, we get $a \approx 1.045$ and $b \approx 9.945$ .
Final rectangular form: Using a calculator, we find that $\sin(84^\circ) \approx 0.994522$ . Therefore, $b \approx 10 \times 0.994522 \approx 9.94522$ . Rounding $a$ and $b$ to the nearest thousandth, we get $a \approx 1.045$ and $b \approx 9.945$ . Therefore, the complex number $z_{1}$ in rectangular form is $z_{1} = 1.045 + 9.945i$ .