A complex number $z_{1}$ has a magnitude $\left|z_{1}\right|=9$ and an angle $\theta_{1}=50^{\circ}$ . $\newline$ Express $z_{1}$ in rectangular form, as $z_{1}=a+b i$ . $\newline$ Round $a$ and $b$ to the nearest thousandth. $\newline$ $z_{1}=\square+\square i$

View step-by-step help

Home

Math Problems

Algebra 1

Compare linear and exponential growth

Full solution

Q. A complex number

z_{1}

has a magnitude

\left|z_{1}\right|=9

and an angle

\theta_{1}=50^{\circ}

\newline

Express

z_{1}

in rectangular form, as

z_{1}=a+b i

\newline

Round

a

and

b

to the nearest thousandth.

\newline

z_{1}=\square+\square i

Formulas Used: To convert a complex number from polar to rectangular form, we use the formulas $a = r \cdot \cos(\theta)$ and $b = r \cdot \sin(\theta)$ , where $r$ is the magnitude and $\theta$ is the angle in degrees.
Calculate Real Part: First, we calculate the real part $a$ of the complex number $z_{1}$ . We have $r = 9$ and $\theta = 50$ degrees. So, $a = 9 \times \cos(50 \text{ degrees})$ .
Calculate Imaginary Part: Using a calculator, we find that $\cos(50^\circ) \approx 0.64279$ . Therefore, $a = 9 \times 0.64279$ .
Calculate Imaginary Part: Using a calculator, we find that $\cos(50^\circ) \approx 0.64279$ . Therefore, $a = 9 \times 0.64279$ .Calculating the value of $a$ gives us $a \approx 9 \times 0.64279 \approx 5.78511$ . We round this to the nearest thousandth, which gives us $a \approx 5.785$ .
Calculate Imaginary Part: Using a calculator, we find that $\cos(50^\circ) \approx 0.64279$ . Therefore, $a = 9 \times 0.64279$ . Calculating the value of $a$ gives us $a \approx 9 \times 0.64279 \approx 5.78511$ . We round this to the nearest thousandth, which gives us $a \approx 5.785$ . Next, we calculate the imaginary part $b$ of the complex number $z_{1}$ . We have $r = 9$ and $\theta = 50^\circ$ . So, $b = 9 \times \sin(50^\circ)$ .
Calculate Imaginary Part: Using a calculator, we find that $\cos(50^\circ) \approx 0.64279$ . Therefore, $a = 9 \times 0.64279$ . Calculating the value of $a$ gives us $a \approx 9 \times 0.64279 \approx 5.78511$ . We round this to the nearest thousandth, which gives us $a \approx 5.785$ . Next, we calculate the imaginary part $b$ of the complex number $z_{1}$ . We have $r = 9$ and $\theta = 50^\circ$ . So, $b = 9 \times \sin(50^\circ)$ . Using a calculator, we find that $a = 9 \times 0.64279$ $0$ . Therefore, $a = 9 \times 0.64279$ $1$ .
Calculate Imaginary Part: Using a calculator, we find that $\cos(50^\circ) \approx 0.64279$ . Therefore, $a = 9 \times 0.64279$ . Calculating the value of $a$ gives us $a \approx 9 \times 0.64279 \approx 5.78511$ . We round this to the nearest thousandth, which gives us $a \approx 5.785$ . Next, we calculate the imaginary part $b$ of the complex number $z_{1}$ . We have $r = 9$ and $\theta = 50^\circ$ . So, $b = 9 \times \sin(50^\circ)$ . Using a calculator, we find that $a = 9 \times 0.64279$ $0$ . Therefore, $a = 9 \times 0.64279$ $1$ . Calculating the value of $b$ gives us $a = 9 \times 0.64279$ $3$ . We round this to the nearest thousandth, which gives us $a = 9 \times 0.64279$ $4$ .