A complex number z_(1) has a magnitude |z_(1)|=12 and an angle theta_(1)=45^(@). Express z_(1) in rectangular form, as z_(1)=a+bi. Express a+bi in exact terms. z_(1)=

Convert angle to radians: To express a complex number in rectangular form, we use the polar to rectangular conversion formula: z = r(cos(θ) + i sin(θ)), where r is the magnitude and θ is the angle in radians.Calculate trigonometric values: First, we need to convert the angle from degrees to radians. The angle given is 45 degrees. To convert degrees to radians, we multiply by π/180. Thus, 45 degrees is 45 * (π/180) = π/4 radians.Substitute values into formula: Now we can use the magnitude r = 12 and the angle θ = π/4 radians to find the rectangular form. We calculate the cosine and sine of π/4, which are both √2/2.Simplify expression: Substitute r and the trigonometric values into the formula: z = 12(cos(π/4) + i sin(π/4)) = 12(√2/2 + i √2/2).Simplify the expression by multiplying 12 by √2/2 to get 12√2/2 = 6√2. So, z = 6√2 + 6√2i.

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A complex number
z_(1) has a magnitude
|z_(1)|=12 and an angle
theta_(1)=45^(@).
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 $\left|z_{1}\right|=12$ and an angle $\theta_{1}=45^{\circ}$ . $\newline$ Express $z_{1}$ in rectangular form, as $z_{1}=a+b i$ . $\newline$ Express $a+b i$ in exact terms. $\newline$ $z_{1}=$

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Algebra 2

Write equations of cosine functions using properties

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Q. A complex number

z_{1}

has a magnitude

\left|z_{1}\right|=12

and an angle

\theta_{1}=45^{\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}=

Convert angle to radians: To express a complex number in rectangular form, 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.
Calculate trigonometric values: First, we need to convert the angle from degrees to radians. The angle given is $45$ degrees. To convert degrees to radians, we multiply by $\pi/180$ . Thus, $45$ degrees is $45 \times (\pi/180) = \pi/4$ radians.
Substitute values into formula: Now we can use the magnitude $r = 12$ and the angle $\theta = \frac{\pi}{4}$ radians to find the rectangular form. We calculate the cosine and sine of $\frac{\pi}{4}$ , which are both $\sqrt{2}/2$ .
Simplify expression: Substitute $r$ and the trigonometric values into the formula: $z = 12(\cos(\pi/4) + i \sin(\pi/4)) = 12(\sqrt{2}/2 + i \sqrt{2}/2)$ .
Simplify expression: Substitute $r$ and the trigonometric values into the formula: $z = 12(\cos(\pi/4) + i \sin(\pi/4)) = 12(\sqrt{2}/2 + i \sqrt{2}/2)$ . Simplify the expression by multiplying $12$ by $\sqrt{2}/2$ to get $12\sqrt{2}/2 = 6\sqrt{2}$ . So, $z = 6\sqrt{2} + 6\sqrt{2}i$ .