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Express
z_(1)=24+0i in polar form.
Express your answer in exact terms, using radians, where your angle is between 0 and
2pi radians, inclusive.

z_(1)=

Express $z_{1}=24+0 i$ in polar form. $\newline$ Express your answer in exact terms, using radians, where your angle is between $0$ and $2 \pi$ radians, inclusive. $\newline$ $z_{1}=$

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Find the inverse of a linear function

Full solution

Q. Express

z_{1}=24+0 i

in polar form.

\newline

Express your answer in exact terms, using radians, where your angle is between

0

and

2 \pi

radians, inclusive.

\newline

z_{1}=

Calculate Magnitude: To express the complex number $z_{1}=24+0i$ in polar form, we need to find its magnitude ( $r$ ) and angle ( $\theta$ ) with respect to the positive x-axis. The polar form is given by $z = r(\cos(\theta) + i\sin(\theta))$ . $\newline$ First, we calculate the magnitude $r$ , which is the distance from the origin to the point $(24, 0)$ in the complex plane. $\newline$ $r = \sqrt{(\text{Re}(z_{1})^2 + \text{Im}(z_{1})^2)}$ $\newline$ $r = \sqrt{(24^2 + 0^2)}$ $\newline$ $r = \sqrt{(576)}$ $\newline$ $r = 24$
Calculate Angle: Next, we calculate the angle $\theta$ . Since the complex number is on the positive real axis and has no imaginary part, the angle $\theta$ is $0$ radians. $\newline$ $\theta = \text{arctan}(\frac{\text{Im}(z_{1})}{\text{Re}(z_{1})})$ $\newline$ $\theta = \text{arctan}(\frac{0}{24})$ $\newline$ $\theta = \text{arctan}(0)$ $\newline$ $\theta = 0$ radians
Write in Polar Form: Now we can write the complex number in polar form using the magnitude $r$ and angle $\theta$ .
$z_{1} = r(\cos(\theta) + i\sin(\theta))$
$z_{1} = 24(\cos(0) + i\sin(0))$
$z_{1} = 24(1 + i\cdot 0)$
$z_{1} = 24$

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