A complex number z_(1) has a magnitude |z_(1)|=13 and an angle theta_(1)=315^(@). Express z_(1) in rectangular form, as z_(1)=a+bi. Express a+bi in exact terms. z_(1)=◻

Equations for Conversion: 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.Convert Angle to Radians: First, we need to convert the angle from degrees to radians. The angle given is 315 degrees. To convert degrees to radians, we multiply by π/180. So, theta_(1) in radians is 315 * (π/180) = (7π/4) radians.Find Real Part a: Now we can find the real part a of the complex number z_(1). We have |z_(1)| = 13 and theta_(1) = (7π/4) radians. So, a = 13 * cos(7π/4).Find Imaginary Part b: The cosine of 7π/4 is √2/2. Therefore, a = 13 * (√2/2).Simplify Expressions: Now we can find the imaginary part b of the complex number z_(1). We have |z_(1)| = 13 and theta_(1) = (7π/4) radians. So, b = 13 * sin(7π/4).Final Result: The sine of 7π/4 is -√2/2. Therefore, b = 13 * (-√2/2).Simplifying the expressions for a and b, we get a = 13√2/2 and b = -13√2/2.Therefore, the complex number z_(1) in rectangular form is z_(1) = 13√2/2 - 13√2/2 * i.

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A complex number
z_(1) has a magnitude
|z_(1)|=13 and an angle
theta_(1)=315^(@).
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 $\left|z_{1}\right|=13$ and an angle $\theta_{1}=315^{\circ}$ . $\newline$ 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|=13

and an angle

\theta_{1}=315^{\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

Equations for Conversion: 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.
Convert Angle to Radians: First, we need to convert the angle from degrees to radians. The angle given is $315$ degrees. To convert degrees to radians, we multiply by $\pi/180$ . So, $\theta_{1}$ in radians is $315 \times (\pi/180) = (7\pi/4)$ radians.
Find Real Part $a$ : Now we can find the real part $a$ of the complex number $z_{1}$ . We have $|z_{1}| = 13$ and $\theta_{1} = \frac{7\pi}{4}$ radians. So, $a = 13 \cdot \cos\left(\frac{7\pi}{4}\right)$ .
Find Imaginary Part $b$ : The cosine of $\frac{7\pi}{4}$ is $\frac{\sqrt{2}}{2}$ . Therefore, $a = 13 \times \left(\frac{\sqrt{2}}{2}\right)$ .
Simplify Expressions: Now we can find the imaginary part $b$ of the complex number $z_{1}$ . We have $|z_{1}| = 13$ and $\theta_{1} = \frac{7\pi}{4}$ radians. So, $b = 13 \times \sin(\frac{7\pi}{4})$ .
Final Result: The sine of $7\pi/4$ is $-\sqrt{2}/2$ . Therefore, $b = 13 \times (-\sqrt{2}/2)$ .
Final Result: The sine of $\frac{7\pi}{4}$ is $-\frac{\sqrt{2}}{2}$ . Therefore, $b = 13 \times \left(-\frac{\sqrt{2}}{2}\right)$ . Simplifying the expressions for $a$ and $b$ , we get $a = \frac{13\sqrt{2}}{2}$ and $b = -\frac{13\sqrt{2}}{2}$ .
Final Result: The sine of $7\pi/4$ is $-\sqrt{2}/2$ . Therefore, $b = 13 \times (-\sqrt{2}/2)$ . Simplifying the expressions for $a$ and $b$ , we get $a = 13\sqrt{2}/2$ and $b = -13\sqrt{2}/2$ . Therefore, the complex number $z_{1}$ in rectangular form is $z_{1} = 13\sqrt{2}/2 - 13\sqrt{2}/2 \times i$ .