Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

A complex number
z_(1) has a magnitude
|z_(1)|=7 and an angle
theta_(1)=158^(@).
Express
z_(1) in rectangular form, as
z_(1)=a+bi.
Round
a and
b to the nearest thousandth.

z_(1)=◻+◻i

A complex number $z_{1}$ has a magnitude $\left|z_{1}\right|=7$ and an angle $\theta_{1}=158^{\circ}$ . $\newline$ 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

View step-by-step help

Compare linear and exponential growth

Full solution

Q. A complex number

z_{1}

has a magnitude

\left|z_{1}\right|=7

and an angle

\theta_{1}=158^{\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

Convert to Radians: 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 radians.
Calculate Real Part: First, we need to convert the angle from degrees to radians because the trigonometric functions in most calculators use radians. The conversion is done by multiplying the angle in degrees by $\pi/180$ . $\newline$ $\theta_{1}$ in radians = $158 \times (\pi/180)$ .
Calculate Imaginary Part: Now we calculate the real part $a$ of the complex number $z_{1}$ using the cosine function. $a = |z_{1}| \cdot \cos(\theta_{1}) = 7 \cdot \cos(158 \cdot (\pi/180)).$
Perform Calculations: Next, we calculate the imaginary part $b$ of the complex number $z_{1}$ using the sine function. $b = |z_{1}| \cdot \sin(\theta_{1}) = 7 \cdot \sin(158 \cdot (\pi/180)).$
Round to Nearest Thousandth: Perform the calculations for $a$ and $b$ using a calculator. $\newline$ $a \approx 7 \times \cos(158 \times (\pi/180)) \approx 7 \times \cos(2.7596) \approx -6.848.$ $\newline$ $b \approx 7 \times \sin(158 \times (\pi/180)) \approx 7 \times \sin(2.7596) \approx 1.913.$
Round to Nearest Thousandth: Perform the calculations for $a$ and $b$ using a calculator. $\newline$ $a \approx 7 \times \cos(158 \times (\pi/180)) \approx 7 \times \cos(2.7596) \approx -6.848.$ $\newline$ $b \approx 7 \times \sin(158 \times (\pi/180)) \approx 7 \times \sin(2.7596) \approx 1.913.$ Round $a$ and $b$ to the nearest thousandth. $\newline$ $a \approx -6.848$ rounded to $-6.848.$ $\newline$ $b \approx 1.913$ rounded to $1.913.$

More problems from Compare linear and exponential growth

Question

Jill bought a used motorcycle from a seller online for $1,200. The seller will charge her $5 a day to store the motorcycle at his house until she is able to pick it up. You can use a function to describe the total amount of money Jill will owe the seller if she waits

x

days to pick up the motorcycle.

\newline

Write an equation for the function. If it is linear, write it in the form

g(x) = mx + b

. If it is exponential, write it in the form

g(x) = a(b)^x

\newline

g(x) = \underline{\hspace{3em}}

Posted 5 months ago

Question

f(t)=-2t-7

change over the interval from

t=-7

t=-6

\newline

\newline

\left[\text{f(t) decreases by } 2\right]

\newline

\left[\text{f(t) increases by } 2\right]

\newline

\left[\text{f(t) increases by } 200\%\right]

\newline

\left[\text{f(t) decreases by a factor of } 2\right]

Posted 5 months ago

Question

g(t)=9^t

change over the interval from

t=1

t=3

\newline

\newline

\left[\text{g(t) decreases by a factor of } 81\right]

\newline

\left[\text{g(t) increases by a factor of } 81\right]

\newline

\left[\text{g(t) increases by } 18\%\right]

\newline

\left[\text{g(t) decreases by } 9\%\right]

Posted 5 months ago

Question

Both of these functions grow as

x

gets larger and larger. Which function eventually exceeds the other?

\newline

\newline

(A)f(x) = 8x + 3.3

\newline

(B)g(x) = 3.3^x - 5

Posted 6 months ago

Question

Both of these functions grow as

x

gets larger and larger. Which function eventually exceeds the other?

\newline

\newline

(A) \ f(x) = 2x + 9

\newline

(B)\ g(x) = 2^x

Posted 6 months ago

Question

f(x)

g(x)

are differentiable.

\newline

h(x)

h(x)=\frac{g(x)}{f(x)}

\newline

f(8)=1

f'(8)=-6

g(8)=8

g'(8)=-10

h'(8)

\newline

Simplify any fractions.

\newline

h'(8)=

Posted 6 months ago

Question

p(x)

q(x)

are differentiable.

\newline

r(x)

r(x)= \frac{p(x)}{q(x)}

\newline

p(3)= 2

p'(3)= 4

q(3)= 6

q'(3)= 9

r'(3)

\newline

Simplify any fractions.

\newline

r'(3)=

Posted 7 months ago

Question

u(x)

v(x)

are differentiable.

\newline

w(x)

w(x)= \frac{u(x)}{v(x)}

\newline

u(5)= 3

u'(5)= -2

v(5)= 7

v'(5)= 1

w'(5)

\newline

Simplify any fractions.

\newline

w'(5)=

Posted 7 months ago

Question

a(x)

b(x)

are differentiable.

\newline

c(x)

c(x)= \frac{a(x)}{b(x)}

\newline

a(2)= 4

a'(2)= 5

b(2)= 1

b'(2)= -3

c'(2)

\newline

Simplify any fractions.

\newline

c'(2)=

Posted 7 months ago

Question

m(x)

n(x)

are differentiable.

\newline

o(x)

o(x)= \frac{m(x)}{n(x)}

\newline

m(7)= 2

m'(7)= -1

n(7)= 4

n'(7)= 8

o'(7)

\newline

Simplify any fractions.

\newline

o'(7)=

Posted 7 months ago

Related topics

Algebra - Order of Operations Algebra - Distributive Property `X` and `Y` Axes Geometry - Scalene Triangle Common Multiple Geometry - Quadrant