A complex number z_(1) has a magnitude |z_(1)|=10 and an angle theta_(1)=210^(@). Express z_(1) in rectangular form, as z_(1)=a+bi. Express a+bi in exact terms. z_(1)=◻

Use formula for conversion: To convert a complex number from polar to rectangular form, use the formula z = r(cos(θ) + i*sin(θ)), where r is the magnitude and θ is the angle.Plug in values: Given |z_(1)|=10 and theta_(1)=210 degrees, plug these values into the formula. z_(1) = 10(cos(210^(@)) + i*sin(210^(@)))Calculate cosine and sine: Calculate the cosine and sine of 210 degrees. Cos(210^(@)) = -cos(30^(@)) and sin(210^(@)) = -sin(30^(@)) because 210 degrees is in the third quadrant where both sine and cosine are negative.Use exact values: Use the exact values for cos(30^(@)) and sin(30^(@)), which are (√3)/2 and 1/2, respectively. z_(1) = 10(-√3/2 + i*(-1/2))Multiply magnitude: Multiply the magnitude 10 by the cosine and sine values to get the rectangular form. z_(1) = 10 * -√3/2 + 10 * i * -1/2Simplify expression: Simplify the expression to get the final rectangular form. z_(1) = -5√3 + i*(-5)

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A complex number
z_(1) has a magnitude
|z_(1)|=10 and an angle
theta_(1)=210^(@).
Express
z_(1) in rectangular form, as
z_(1)=a+bi.
Express
a+bi in exact terms.

z_(1)=◻

A complex number $z_{1}$ has a magnitude $\left|z_{1}\right|=10$ and an angle $\theta_{1}=210^{\circ}$ . $\newline$ Express $z_{1}$ in rectangular form, as $z_{1}=a+b i$ . $\newline$ Express $a+b i$ in exact terms. $\newline$ $z_{1}=\square$

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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|=10

and an angle

\theta_{1}=210^{\circ}

\newline

Express

z_{1}

in rectangular form, as

z_{1}=a+b i

\newline

Express

a+b i

in exact terms.

\newline

z_{1}=\square

Use formula for conversion: To convert a complex number from polar to rectangular form, use the formula $z = r(\cos(\theta) + i\sin(\theta))$ , where $r$ is the magnitude and $\theta$ is the angle.
Plug in values: Given $\lvert z_{1} \rvert=10$ and $\theta_{1}=210$ degrees, plug these values into the formula. $z_{1} = 10(\cos(210^{\circ}) + i\sin(210^{\circ}))$
Calculate cosine and sine: Calculate the cosine and sine of $210$ degrees. $\cos(210^{\circ}) = -\cos(30^{\circ})$ and $\sin(210^{\circ}) = -\sin(30^{\circ})$ because $210$ degrees is in the third quadrant where both sine and cosine are negative.
Use exact values: Use the exact values for $\cos(30^\circ)$ and $\sin(30^\circ)$ , which are $\frac{\sqrt{3}}{2}$ and $\frac{1}{2}$ , respectively. $\newline$ $z_{1} = 10\left(-\frac{\sqrt{3}}{2} + i\left(-\frac{1}{2}\right)\right)$
Multiply magnitude: Multiply the magnitude $10$ by the cosine and sine values to get the rectangular form. $\newline$ $z_{1} = 10 \times -\sqrt{3}/2 + 10 \times i \times -1/2$
Simplify expression: Simplify the expression to get the final rectangular form. $z_{1} = -5\sqrt{3} + i*(-5)$