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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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

Conversion formula: To express a complex number in rectangular form, we use the polar to rectangular conversion formula, which is $z = r(\cos(\theta) + i\sin(\theta))$ , 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 because the trigonometric functions in the formula require the angle to be in radians. The conversion is done by multiplying the degree measure by $\pi/180$ . $\newline$ $\theta_{1}$ in radians = $330 \times (\pi/180) = (11/6)\pi$
Calculate cosine and sine: Now we can calculate the cosine and sine of $\theta_{1}$ in radians. $\newline$ $\cos(\theta_{1}) = \cos\left(\frac{11}{6}\pi\right)$ and $\sin(\theta_{1}) = \sin\left(\frac{11}{6}\pi\right)$
Substitute values into formula: Using the unit circle or trigonometric identities, we know that $\cos\left(\frac{11}{6}\pi\right) = \cos\left(\pi - \frac{1}{6}\pi\right) = \cos\left(\frac{5}{6}\pi\right) = -\cos\left(\frac{1}{6}\pi\right)$ and $\sin\left(\frac{11}{6}\pi\right) = -\sin\left(\frac{1}{6}\pi\right)$ , since $\frac{11}{6}\pi$ is in the fourth quadrant where cosine is positive and sine is negative.
Simplify expression: The exact values for $\cos\left(\frac{1}{6}\pi\right)$ and $\sin\left(\frac{1}{6}\pi\right)$ are $\sqrt{3}/2$ and $1/2$ , respectively. Therefore, $\cos\left(\frac{11}{6}\pi\right)$ = $-\sqrt{3}/2$ and $\sin\left(\frac{11}{6}\pi\right)$ = $-1/2$ .
Simplify expression: The exact values for $\cos\left(\frac{1}{6}\pi\right)$ and $\sin\left(\frac{1}{6}\pi\right)$ are $\sqrt{3}/2$ and $1/2$ , respectively. Therefore, $\cos\left(\frac{11}{6}\pi\right) = -\sqrt{3}/2$ and $\sin\left(\frac{11}{6}\pi\right) = -1/2$ .Now we can substitute the values of $r$ , $\cos(\theta_{1})$ , and $\sin(\theta_{1})$ into the formula to get the rectangular form of $z_{1}$ . $\sin\left(\frac{1}{6}\pi\right)$ $0$
Simplify expression: The exact values for $\cos\left(\frac{1}{6}\pi\right)$ and $\sin\left(\frac{1}{6}\pi\right)$ are $\sqrt{3}/2$ and $1/2$ , respectively. Therefore, $\cos\left(\frac{11}{6}\pi\right) = -\sqrt{3}/2$ and $\sin\left(\frac{11}{6}\pi\right) = -1/2$ .Now we can substitute the values of $r$ , $\cos(\theta_{1})$ , and $\sin(\theta_{1})$ into the formula to get the rectangular form of $z_{1}$ . $\newline$ $\sin\left(\frac{1}{6}\pi\right)$ $0$ Simplify the expression to get the final rectangular form of $z_{1}$ . $\newline$ $\sin\left(\frac{1}{6}\pi\right)$ $2$