A complex number z_(1) has a magnitude |z_(1)|=8 and an angle theta_(1)=67^(@). 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 = r * cos(theta) and b = r * sin(theta), where r 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 67 degrees. To convert degrees to radians, we multiply by π/180. theta_(1) in radians = 67 * (π/180)Calculate theta in radians: Now we calculate the value of theta_(1) in radians. theta_(1) in radians = 67 * (π/180) ≈ 1.16937 radiansCalculate real part a: Next, we calculate the real part a of the complex number using the magnitude and the cosine of the angle. a = |z_(1)| * cos(theta_(1)) = 8 * cos(1.16937)Calculate value of a: Now we calculate the value of a. a ≈ 8 * cos(1.16937) ≈ 8 * 0.39073 ≈ 3.12584Calculate imaginary part b: Next, we calculate the imaginary part b of the complex number using the magnitude and the sine of the angle. b = |z_(1)| * sin(theta_(1)) = 8 * sin(1.16937)Calculate value of b: Now we calculate the value of b. b ≈ 8 * sin(1.16937) ≈ 8 * 0.92050 ≈ 7.364Round a and b: We round a and b to the nearest thousandth as requested. a ≈ 3.126 b ≈ 7.364Complex number in rectangular form: The complex number z_(1) in rectangular form is z_(1) = a + bi, so we have: z_(1) = 3.126 + 7.364i

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A complex number
z_(1) has a magnitude
|z_(1)|=8 and an angle
theta_(1)=67^(@).
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 $\left|z_{1}\right|=8$ and an angle $\theta_{1}=67^{\circ}$ . $\newline$ 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|=8

and an angle

\theta_{1}=67^{\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 $a = r \cdot \cos(\theta)$ and $b = r \cdot \sin(\theta)$ , where $r$ 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 $67$ degrees. To convert degrees to radians, we multiply by $\pi/180$ . $\newline$ $\theta_{1}$ in radians = $67 \times (\pi/180)$
Calculate theta in radians: Now we calculate the value of $\theta_{1}$ in radians. $\theta_{1}$ in radians = $67 \times (\pi/180) \approx 1.16937$ radians
Calculate real part $a$ : Next, we calculate the real part $a$ of the complex number using the magnitude and the cosine of the angle. $a = |z_{1}| \cdot \cos(\theta_{1}) = 8 \cdot \cos(1.16937)$
Calculate value of a: Now we calculate the value of $a$ . $a \approx 8 \times \cos(1.16937) \approx 8 \times 0.39073 \approx 3.12584$
Calculate imaginary part $b$ : Next, we calculate the imaginary part $b$ of the complex number using the magnitude and the sine of the angle. $b = |z_{1}| \cdot \sin(\theta_{1}) = 8 \cdot \sin(1.16937)$
Calculate value of b: Now we calculate the value of $b$ . $b \approx 8 \times \sin(1.16937) \approx 8 \times 0.92050 \approx 7.364$
Round $a$ and $b$ : We round $a$ and $b$ to the nearest thousandth as requested. $a \approx 3.126$ $b \approx 7.364$
Complex number in rectangular form: The complex number $z_{1}$ in rectangular form is $z_{1} = a + bi$ , so we have: $\newline$ $z_{1} = 3.126 + 7.364i$