Express $z_{1}=-5 \sqrt{5}-5 \sqrt{15} i$ 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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Algebra 1

Find the inverse of a linear function

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Q. Express

z_{1}=-5 \sqrt{5}-5 \sqrt{15} 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}=

Magnitude Calculation: To express the complex number $z_1 = -5\sqrt{5} - 5\sqrt{15}i$ 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 $r(\cos(\theta) + i\sin(\theta))$ .
Angle Calculation: First, calculate the magnitude $r$ using the formula $r = \sqrt{a^2 + b^2}$ , where $a$ is the real part and $b$ is the imaginary part of the complex number. $\newline$ For $z_1$ , $a = -5\sqrt{5}$ and $b = -5\sqrt{15}$ . $\newline$ So, $r = \sqrt{(-5\sqrt{5})^2 + (-5\sqrt{15})^2}$ .
Polar Form Calculation: Perform the calculation for $r$ : $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $\newline$ $r = \sqrt{125 + 375}$ $\newline$ $r = \sqrt{500}$ $\newline$ $r = 5\sqrt{20}$ $\newline$ $r = 10\sqrt{5}$
Polar Form Calculation: Perform the calculation for $r$ : $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $\newline$ $r = \sqrt{125 + 375}$ $\newline$ $r = \sqrt{500}$ $\newline$ $r = 5\sqrt{20}$ $\newline$ $r = 10\sqrt{5}$ Next, we need to find the angle $\theta$ . The angle is determined by the arctangent of $b/a$ , but since both $a$ and $b$ are negative, $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $0$ lies in the third quadrant. Therefore, we add $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $1$ radians to the arctangent to get the angle in the correct quadrant. $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $2$ $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $3$
Polar Form Calculation: Perform the calculation for $r$ : $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $\newline$ $r = \sqrt{125 + 375}$ $\newline$ $r = \sqrt{500}$ $\newline$ $r = 5\sqrt{20}$ $\newline$ $r = 10\sqrt{5}$ Next, we need to find the angle $\theta$ . The angle is determined by the arctangent of $b/a$ , but since both $a$ and $b$ are negative, $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $0$ lies in the third quadrant. Therefore, we add $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $1$ radians to the arctangent to get the angle in the correct quadrant. $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $2$ $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $3$ Simplify the expression for $\theta$ : $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $5$ $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $6$ $\newline$ Since $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $7$ corresponds to $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $8$ radians, $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $9$ $\newline$ $r = \sqrt{125 + 375}$ $0$
Polar Form Calculation: Perform the calculation for $r$ : $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $\newline$ $r = \sqrt{125 + 375}$ $\newline$ $r = \sqrt{500}$ $\newline$ $r = 5\sqrt{20}$ $\newline$ $r = 10\sqrt{5}$ Next, we need to find the angle $\theta$ . The angle is determined by the arctangent of $b/a$ , but since both $a$ and $b$ are negative, $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $0$ lies in the third quadrant. Therefore, we add $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $1$ radians to the arctangent to get the angle in the correct quadrant. $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $2$ $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $3$ Simplify the expression for $\theta$ : $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $5$ $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $6$ $\newline$ Since $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $7$ corresponds to $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $8$ radians, $\newline$ $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $9$ $\newline$ $r = \sqrt{125 + 375}$ $0$ Now we can write the polar form of $r = \sqrt{25 \cdot 5 + 25 \cdot 15}$ $0$ using $r$ and $\theta$ : $\newline$ $r = \sqrt{125 + 375}$ $4$