A complex number z_(1) has a magnitude |z_(1)|=2 and an angle theta_(1)=49^(@). Express z_(1) in rectangular form, as z_(1)=a+bi. Round a and b to the nearest thousandth. z_(1)=◻+◻i

Convert angle 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 cosine and sine: First, we need to convert the angle from degrees to radians. The angle given is 49 degrees. To convert degrees to radians, we multiply by π/180. theta_(1) in radians = 49 * (π/180)Find a and b: Now we calculate the cosine and sine of theta_(1) in radians. cos(theta_(1)) = cos(49 * (π/180)) sin(theta_(1)) = sin(49 * (π/180))Calculate values of a and b: We then multiply the cosine and sine by the magnitude |z_(1)| to find a and b. a = |z_(1)| * cos(theta_(1)) = 2 * cos(49 * (π/180)) b = |z_(1)| * sin(theta_(1)) = 2 * sin(49 * (π/180))Determine rectangular form: Using a calculator, we find the values of a and b, rounding to the nearest thousandth. a ≈ 2 * cos(49 * (π/180)) ≈ 2 * 0.6561 ≈ 1.312 b ≈ 2 * sin(49 * (π/180)) ≈ 2 * 0.7547 ≈ 1.509The rectangular form of z_(1) is therefore approximately z_(1) = 1.312 + 1.509i.

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A complex number
z_(1) has a magnitude
|z_(1)|=2 and an angle
theta_(1)=49^(@).
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 $\theta_{1}=49^{\circ}$ . $\newline$ 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}=49^{\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 angle 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 cosine and sine: First, we need to convert the angle from degrees to radians. The angle given is $49$ degrees. To convert degrees to radians, we multiply by $\pi/180$ . $\newline$ $\theta_{1}$ in radians = $49 \times (\pi/180)$
Find $a$ and $b$ : Now we calculate the cosine and sine of $\theta_{1}$ in radians. $\newline$ $\cos(\theta_{1}) = \cos(49 \times (\pi/180))$ $\newline$ $\sin(\theta_{1}) = \sin(49 \times (\pi/180))$
Calculate values of a and b: We then multiply the cosine and sine by the magnitude $|z_{1}|$ to find a and b. $a = |z_{1}| \cdot \cos(\theta_{1}) = 2 \cdot \cos(49 \cdot (\frac{\pi}{180}))$ $b = |z_{1}| \cdot \sin(\theta_{1}) = 2 \cdot \sin(49 \cdot (\frac{\pi}{180}))$
Determine rectangular form: Using a calculator, we find the values of $a$ and $b$ , rounding to the nearest thousandth. $a \approx 2 \times \cos(49 \times (\frac{\pi}{180})) \approx 2 \times 0.6561 \approx 1.312$ $b \approx 2 \times \sin(49 \times (\frac{\pi}{180})) \approx 2 \times 0.7547 \approx 1.509$
Determine rectangular form: Using a calculator, we find the values of $a$ and $b$ , rounding to the nearest thousandth. $a \approx 2 \times \cos(49 \times (\pi/180)) \approx 2 \times 0.6561 \approx 1.312$ $b \approx 2 \times \sin(49 \times (\pi/180)) \approx 2 \times 0.7547 \approx 1.509$ The rectangular form of $z_{1}$ is therefore approximately $z_{1} = 1.312 + 1.509i$ .