A complex number z_(1) has a magnitude |z_(1)|=7 and an angle theta_(1)=158^(@). 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 formulas a = r * cos(theta) and b = r * sin(theta), where r is the magnitude and theta is the angle in radians.Calculate Real Part: First, we need to convert the angle from degrees to radians because the trigonometric functions in most calculators use radians. The conversion is done by multiplying the angle in degrees by pi/180. theta_(1) in radians = 158 * (pi/180).Calculate Imaginary Part: Now we calculate the real part a of the complex number z_(1) using the cosine function. a = |z_(1)| * cos(theta_(1)) = 7 * cos(158 * (pi/180)).Perform Calculations: Next, we calculate the imaginary part b of the complex number z_(1) using the sine function. b = |z_(1)| * sin(theta_(1)) = 7 * sin(158 * (pi/180)).Round to Nearest Thousandth: Perform the calculations for a and b using a calculator. a ≈ 7 * cos(158 * (pi/180)) ≈ 7 * cos(2.7596) ≈ -6.848. b ≈ 7 * sin(158 * (pi/180)) ≈ 7 * sin(2.7596) ≈ 1.913.Round a and b to the nearest thousandth. a ≈ -6.848 rounded to -6.848. b ≈ 1.913 rounded to 1.913.

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A complex number
z_(1) has a magnitude
|z_(1)|=7 and an angle
theta_(1)=158^(@).
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 $\left|z_{1}\right|=7$ and an angle $\theta_{1}=158^{\circ}$ . $\newline$ 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|=7

and an angle

\theta_{1}=158^{\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 formulas $a = r \cdot \cos(\theta)$ and $b = r \cdot \sin(\theta)$ , where $r$ is the magnitude and $\theta$ is the angle in radians.
Calculate Real Part: First, we need to convert the angle from degrees to radians because the trigonometric functions in most calculators use radians. The conversion is done by multiplying the angle in degrees by $\pi/180$ . $\newline$ $\theta_{1}$ in radians = $158 \times (\pi/180)$ .
Calculate Imaginary Part: Now we calculate the real part $a$ of the complex number $z_{1}$ using the cosine function. $a = |z_{1}| \cdot \cos(\theta_{1}) = 7 \cdot \cos(158 \cdot (\pi/180)).$
Perform Calculations: Next, we calculate the imaginary part $b$ of the complex number $z_{1}$ using the sine function. $b = |z_{1}| \cdot \sin(\theta_{1}) = 7 \cdot \sin(158 \cdot (\pi/180)).$
Round to Nearest Thousandth: Perform the calculations for $a$ and $b$ using a calculator. $\newline$ $a \approx 7 \times \cos(158 \times (\pi/180)) \approx 7 \times \cos(2.7596) \approx -6.848.$ $\newline$ $b \approx 7 \times \sin(158 \times (\pi/180)) \approx 7 \times \sin(2.7596) \approx 1.913.$
Round to Nearest Thousandth: Perform the calculations for $a$ and $b$ using a calculator. $\newline$ $a \approx 7 \times \cos(158 \times (\pi/180)) \approx 7 \times \cos(2.7596) \approx -6.848.$ $\newline$ $b \approx 7 \times \sin(158 \times (\pi/180)) \approx 7 \times \sin(2.7596) \approx 1.913.$ Round $a$ and $b$ to the nearest thousandth. $\newline$ $a \approx -6.848$ rounded to $-6.848.$ $\newline$ $b \approx 1.913$ rounded to $1.913.$