A complex number z_(1) has a magnitude |z_(1)|=20 and an angle theta_(1)=281^(@). Express z_(1) in rectangular form, as z_(1)=a+bi. Round a and b to the nearest thousandth. z_(1)=◻+◻i

Conversion equations: To convert a complex number from polar to rectangular form, we use the equations a = |z| * cos(θ) and b = |z| * sin(θ), where |z| is the magnitude and θ is the angle in radians.Convert angle to radians: First, we need to convert the angle from degrees to radians. The angle given is 281 degrees. To convert degrees to radians, we multiply by π/180. θ = 281 * (π/180)Calculate angle in radians: Now we calculate the angle in radians. θ = 281 * (π/180) ≈ 4.90874 radians (rounded to five decimal places for intermediate calculations)Calculate real part a: Next, we calculate the real part a using the cosine of the angle. a = |z_(1)| * cos(θ) a = 20 * cos(4.90874)Calculate value of a: Now we calculate the value of a. a ≈ 20 * cos(4.90874) ≈ 20 * (-0.85717) ≈ -17.1434 (rounded to four decimal places for intermediate calculations)Calculate imaginary part b: Next, we calculate the imaginary part b using the sine of the angle. b = |z_(1)| * sin(θ) b = 20 * sin(4.90874)Calculate value of b: Now we calculate the value of b. b ≈ 20 * sin(4.90874) ≈ 20 * (-0.51504) ≈ -10.3008 (rounded to four decimal places for intermediate calculations)Round a and b: Finally, we round a and b to the nearest thousandth. a ≈ -17.143 b ≈ -10.301

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A complex number
z_(1) has a magnitude
|z_(1)|=20 and an angle
theta_(1)=281^(@).
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 $\left|z_{1}\right|=20$ and an angle $\theta_{1}=281^{\circ}$ . $\newline$ 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|=20

and an angle

\theta_{1}=281^{\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

Conversion equations: 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.
Convert angle to radians: First, we need to convert the angle from degrees to radians. The angle given is $281$ degrees. To convert degrees to radians, we multiply by $\pi/180$ . $\newline$ $\theta = 281 \times (\pi/180)$
Calculate angle in radians: Now we calculate the angle in radians. $\theta = 281 \times (\pi/180) \approx 4.90874$ radians (rounded to five decimal places for intermediate calculations)
Calculate real part $a$ : Next, we calculate the real part $a$ using the cosine of the angle. $\newline$ $a = |z_{1}| \cdot \cos(\theta)$ $\newline$ $a = 20 \cdot \cos(4.90874)$
Calculate value of a: Now we calculate the value of $a$ . $a \approx 20 \times \cos(4.90874) \approx 20 \times (-0.85717) \approx -17.1434$ (rounded to four decimal places for intermediate calculations)
Calculate imaginary part $b$ : Next, we calculate the imaginary part $b$ using the sine of the angle. $b = |z_{1}| \cdot \sin(\theta)$ $b = 20 \cdot \sin(4.90874)$
Calculate value of b: Now we calculate the value of $b$ . $b \approx 20 \times \sin(4.90874) \approx 20 \times (-0.51504) \approx -10.3008$ (rounded to four decimal places for intermediate calculations)
Round $a$ and $b$ : Finally, we round $a$ and $b$ to the nearest thousandth. $a \approx -17.143$ $b \approx -10.301$