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A complex number
z_(1) has a magnitude
|z_(1)|=5 and an angle
theta_(1)=150^(@).
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 $\left|z_{1}\right|=5$ and an angle $\theta_{1}=150^{\circ}$ . $\newline$ 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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Write equations of sine and cosine functions using properties

Full solution

Q. A complex number

z_{1}

has a magnitude

\left|z_{1}\right|=5

and an angle

\theta_{1}=150^{\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

Rectangular Form Expression: To express a complex number in rectangular form, we use the polar form of the complex number, which is $z = r(\cos(\theta) + i\sin(\theta))$ , where $r$ is the magnitude and $\theta$ is the angle.
Substitute Values: Given $\lvert z_{1} \rvert=5$ and $\theta_{1}=150$ degrees, we can substitute these values into the polar form equation. $z_{1} = 5(\cos(150 \text{ degrees}) + i\sin(150 \text{ degrees}))$
Calculate Cosine and Sine: We need to calculate the cosine and sine of $150$ degrees. Since $150$ degrees is in the second quadrant, we know that cosine will be negative and sine will be positive. $\newline$ $\cos(150 \text{ degrees}) = \cos(180 \text{ degrees} - 30 \text{ degrees}) = -\cos(30 \text{ degrees})$ $\newline$ $\sin(150 \text{ degrees}) = \sin(180 \text{ degrees} - 30 \text{ degrees}) = \sin(30 \text{ degrees})$
Exact Trigonometric Values: We know the exact values for $\cos(30^\circ)$ and $\sin(30^\circ)$ : $\cos(30^\circ) = \frac{\sqrt{3}}{2}$ $\sin(30^\circ) = \frac{1}{2}$
Substitute Exact Values: Substitute the exact values into the equation: $\newline$ $z_{1} = 5(-\sqrt{3}/2 + i*(1/2))$
Distribute Magnitude: Now, distribute the magnitude ( $5$ ) to both the real and imaginary parts: $\newline$ $z_{1} = 5 \times -\sqrt{3}/2 + 5 \times i\times(1/2)$
Simplify Expression: Simplify the expression: $z_{1} = -\frac{5\sqrt{3}}{2} + \left(\frac{5}{2}\right)i$

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