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)=◻

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 |z_(1)|=5 and theta_(1)=150 degrees, we can substitute these values into the polar form equation. z_(1) = 5(cos(150 degrees) + i*sin(150 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. cos(150 degrees) = cos(180 degrees - 30 degrees) = -cos(30 degrees) sin(150 degrees) = sin(180 degrees - 30 degrees) = sin(30 degrees)Exact Trigonometric Values: We know the exact values for cos(30 degrees) and sin(30 degrees): cos(30 degrees) = sqrt(3)/2 sin(30 degrees) = 1/2Substitute Exact Values: Substitute the exact values into the equation: z_(1) = 5(-sqrt(3)/2 + i*(1/2))Distribute Magnitude: Now, distribute the magnitude (5) to both the real and imaginary parts: z_(1) = 5 * -sqrt(3)/2 + 5 * i*(1/2)Simplify Expression: Simplify the expression: z_(1) = -5sqrt(3)/2 + (5/2)i

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

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. $\newline$ $z_{1}=\square$

View step-by-step help

Home

Math Problems

Algebra 2

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: $z_{1} = -\frac{5\sqrt{3}}{2} + \left(\frac{5}{2}\right)i$