A complex number z_(1) has a magnitude |z_(1)|=11 and an angle theta_(1)=180^(@). Express z_(1) in rectangular form, as z_(1)=a+bi. Express a+bi in exact terms. z_(1)=◻

Conversion formulas: A complex number in polar form can be expressed in rectangular form (a + bi) using the conversion formulas 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 180 degrees, which is equivalent to π radians since 180 degrees * (π radians / 180 degrees) = π radians.Calculate rectangular form: Now we can use the magnitude |z_(1)| = 11 and the angle theta_(1) = π to find the rectangular form. We calculate a = 11 * cos(π) and b = 11 * sin(π).Calculate a: Calculating a gives us a = 11 * cos(π) = 11 * (-1) = -11, since cos(π) = -1.Calculate b: Calculating b gives us b = 11 * sin(π) = 11 * 0 = 0, since sin(π) = 0.Final rectangular form: Therefore, the complex number z_(1) in rectangular form is z_(1) = -11 + 0i, which simplifies to z_(1) = -11.

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A complex number
z_(1) has a magnitude
|z_(1)|=11 and an angle
theta_(1)=180^(@).
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 $\left|z_{1}\right|=11$ and an angle $\theta_{1}=180^{\circ}$ . $\newline$ 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|=11

and an angle

\theta_{1}=180^{\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

Conversion formulas: A complex number in polar form can be expressed in rectangular form $(a + bi)$ using the conversion formulas $a = r \cdot \cos(\theta)$ 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 $180$ degrees, which is equivalent to $\pi$ radians since $180$ degrees $\times (\pi$ radians $/ 180$ degrees) $= \pi$ radians.
Calculate rectangular form: Now we can use the magnitude $|z_{1}| = 11$ and the angle $\theta_{1} = \pi$ to find the rectangular form. We calculate $a = 11 \times \cos(\pi)$ and $b = 11 \times \sin(\pi)$ .
Calculate $a$ : Calculating $a$ gives us $a = 11 \times \cos(\pi) = 11 \times (-1) = -11$ , since $\cos(\pi) = -1$ .
Calculate $b$ : Calculating $b$ gives us $b = 11 \times \sin(\pi) = 11 \times 0 = 0$ , since $\sin(\pi) = 0$ .
Final rectangular form: Therefore, the complex number $z_{1}$ in rectangular form is $z_{1} = -11 + 0i$ , which simplifies to $z_{1} = -11$ .