A complex number z_(1) has a magnitude |z_(1)|=2 and an angle theta_(1)=39^(@). Express z_(1) in rectangular form, as z_(1)=a+bi. Round a and b to the nearest thousandth. z_(1)=◻+◻i

Convert to Radians: To convert a complex number from polar to rectangular form, we use the equations a = |z| * cos(theta) and b = |z| * 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 39 degrees. To convert degrees to radians, we multiply by π/180. theta_(1) in radians = 39 * (π/180)Calculate Real Part: Now we calculate the value of theta_(1) in radians. theta_(1) in radians = 39 * (π/180) ≈ 0.6807 radians (rounded to four decimal places for intermediate calculation)Calculate Value of A: Next, we calculate the real part a of the complex number z_(1) using the magnitude and the cosine of the angle. a = |z_(1)| * cos(theta_(1)) = 2 * cos(0.6807)Calculate Imaginary Part: Now we calculate the value of a. a ≈ 2 * cos(0.6807) ≈ 2 * 0.7771 ≈ 1.5542 (rounded to four decimal places for intermediate calculation)Calculate Value of B: Next, we calculate the imaginary part b of the complex number z_(1) using the magnitude and the sine of the angle. b = |z_(1)| * sin(theta_(1)) = 2 * sin(0.6807)Round to Nearest Thousandth: Now we calculate the value of b. b ≈ 2 * sin(0.6807) ≈ 2 * 0.6293 ≈ 1.2586 (rounded to four decimal places for intermediate calculation)Finally, we round a and b to the nearest thousandth as requested. a ≈ 1.554 (rounded to the nearest thousandth) b ≈ 1.259 (rounded to the nearest thousandth)

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A complex number
z_(1) has a magnitude
|z_(1)|=2 and an angle
theta_(1)=39^(@).
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 $\left|z_{1}\right|=2$ and an angle $\theta_{1}=39^{\circ}$ . $\newline$ 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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Algebra 2

Write equations of cosine functions using properties

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Q. A complex number

z_{1}

has a magnitude

\left|z_{1}\right|=2

and an angle

\theta_{1}=39^{\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 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 $39$ degrees. To convert degrees to radians, we multiply by $\pi/180$ . $\newline$ $\theta_{1}$ in radians = $39 \times (\pi/180)$
Calculate Real Part: Now we calculate the value of $\theta_{1}$ in radians. $\newline$ $\theta_{1}$ in radians $= 39 \times (\pi/180) \approx 0.6807$ radians (rounded to four decimal places for intermediate calculation)
Calculate Value of A: Next, we calculate the real part $a$ of the complex number $z_{1}$ using the magnitude and the cosine of the angle. $a = |z_{1}| \cdot \cos(\theta_{1}) = 2 \cdot \cos(0.6807)$
Calculate Imaginary Part: Now we calculate the value of $a$ . $a \approx 2 \times \cos(0.6807) \approx 2 \times 0.7771 \approx 1.5542$ (rounded to four decimal places for intermediate calculation)
Calculate Value of B: Next, we calculate the imaginary part $b$ of the complex number $z_{1}$ using the magnitude and the sine of the angle. $b = |z_{1}| \cdot \sin(\theta_{1}) = 2 \cdot \sin(0.6807)$
Round to Nearest Thousandth: Now we calculate the value of $b$ . $b \approx 2 \times \sin(0.6807) \approx 2 \times 0.6293 \approx 1.2586$ (rounded to four decimal places for intermediate calculation)
Round to Nearest Thousandth: Now we calculate the value of $b$ . $b \approx 2 \times \sin(0.6807) \approx 2 \times 0.6293 \approx 1.2586$ (rounded to four decimal places for intermediate calculation)Finally, we round $a$ and $b$ to the nearest thousandth as requested. $a \approx 1.554$ (rounded to the nearest thousandth) $b \approx 1.259$ (rounded to the nearest thousandth)