A complex number $z_{1}$ has a magnitude $\left|z_{1}\right|=4$ and an angle $\theta_{1}=330^{\circ}$ . $\newline$ 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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Find the inverse of a linear function

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Q. A complex number

z_{1}

has a magnitude

\left|z_{1}\right|=4

and an angle

\theta_{1}=330^{\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

Convert angle to radians: To express a complex number in rectangular form $(a+bi)$ given its magnitude and angle, we use the polar to rectangular conversion formula: $z = r(\cos(\theta) + i\sin(\theta))$ , where $r$ is the magnitude and $\theta$ is the angle in radians. Here, $r=4$ and $\theta=330^\circ$ .
Calculate radians: First, convert the angle from degrees to radians because the trigonometric functions in the formula require the angle in radians. The conversion formula is $\text{radians} = \text{degrees} \times (\pi/180)$ . So, $\theta = 330 \times (\pi/180)$ .
Substitute values into formula: Calculating the radians: $\theta = 330 \times (\pi/180) = 11\pi/6$ radians.
Calculate trigonometric functions: Now, substitute $r=4$ and $\theta=\frac{11\pi}{6}$ into the polar to rectangular conversion formula: $z = 4(\cos(\frac{11\pi}{6}) + i\sin(\frac{11\pi}{6}))$ .
Simplify expression: Calculate $\cos(\frac{11\pi}{6})$ and $\sin(\frac{11\pi}{6})$ . From the unit circle, we know that $\cos(\frac{11\pi}{6}) = \frac{\sqrt{3}}{2}$ and $\sin(\frac{11\pi}{6}) = -\frac{1}{2}$ .
Simplify expression: Calculate $\cos(\frac{11\pi}{6})$ and $\sin(\frac{11\pi}{6})$ . From the unit circle, we know that $\cos(\frac{11\pi}{6}) = \frac{\sqrt{3}}{2}$ and $\sin(\frac{11\pi}{6}) = -\frac{1}{2}$ .Substitute the values of $\cos(\frac{11\pi}{6})$ and $\sin(\frac{11\pi}{6})$ into the formula: $z = 4(\frac{\sqrt{3}}{2} + i*(-\frac{1}{2}))$ .
Simplify expression: Calculate $\cos(\frac{11\pi}{6})$ and $\sin(\frac{11\pi}{6})$ . From the unit circle, we know that $\cos(\frac{11\pi}{6}) = \frac{\sqrt{3}}{2}$ and $\sin(\frac{11\pi}{6}) = -\frac{1}{2}$ .Substitute the values of $\cos(\frac{11\pi}{6})$ and $\sin(\frac{11\pi}{6})$ into the formula: $z = 4(\frac{\sqrt{3}}{2} + i*(-\frac{1}{2}))$ .Simplify the expression: $z = 4*\frac{\sqrt{3}}{2} + 4*(-\frac{1}{2})i = 2\sqrt{3} - 2i$ .