Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Consider the complex number z=8(cos(240^(@))+i sin(240^(@)))". " Which of the following complex numbers best approximates z ? Choose 1 answer: (A) 6.9-4i (B) -6.9-4i (C) -4-6.9 i (D) 4-6.9 i

Consider the complex number z=8(cos⁡(240∘)+isin⁡(240∘)).  z=8\left(\cos \left(240^{\circ}\right)+i \sin \left(240^{\circ}\right)\right) \text {. } \newlineWhich of the following complex numbers best approximates z z ?\newlineChoose 11 answer:\newline(A) 6.9−4i 6.9-4 i \newline(B) −6.9−4i -6.9-4 i \newline(C) −4−6.9i -4-6.9 i \newline(D) 4−6.9i 4-6.9 i

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 1right chevron icon
Compare linear, exponential, and quadratic growthright chevron icon

Full solution

Q. Consider the complex number z=8(cos⁡(240∘)+isin⁡(240∘)).  z=8\left(\cos \left(240^{\circ}\right)+i \sin \left(240^{\circ}\right)\right) \text {. } \newlineWhich of the following complex numbers best approximates z z ?\newlineChoose 11 answer:\newline(A) 6.9−4i 6.9-4 i \newline(B) −6.9−4i -6.9-4 i \newline(C) −4−6.9i -4-6.9 i \newline(D) 4−6.9i 4-6.9 i
  1. Convert to trigonometric form: Convert the given complex number in trigonometric form to standard form.\newlineWe have z=8(cos⁡(240°)+isin⁡(240°))z = 8(\cos(240°) + i \sin(240°)). To convert this to standard form, we need to evaluate the cosine and sine of 240°240°.
  2. Calculate cosine and sine: Calculate the cosine and sine of 240°240°. The angle 240°240° is in the third quadrant, where cosine is negative and sine is also negative. We can use the reference angle of 240°−180°=60°240° - 180° = 60° to find the values. cos⁡(240°)=−cos⁡(60°)=−12\cos(240°) = -\cos(60°) = -\frac{1}{2} sin⁡(240°)=−sin⁡(60°)=−32\sin(240°) = -\sin(60°) = -\frac{\sqrt{3}}{2}
  3. Substitute values into complex number: Substitute the values of cosine and sine into the complex number. \newlinez=8(−12+i(−32))z = 8(-\frac{1}{2} + i(-\frac{\sqrt{3}}{2}))\newlinez=8×−12+8×i(−32)z = 8 \times -\frac{1}{2} + 8 \times i(-\frac{\sqrt{3}}{2})\newlinez=−4−4i3z = -4 - 4i\sqrt{3}
  4. Approximate imaginary part: Approximate the value of 3\sqrt{3} to find the imaginary part of zz.3\sqrt{3} is approximately 1.7321.732. So, the imaginary part of zz is:−4i3≈−4i×1.732≈−6.928i-4i\sqrt{3} \approx -4i \times 1.732 \approx -6.928i
  5. Write in standard form: Write the complex number in standard form with the approximated values. z≈−4−6.928iz \approx -4 - 6.928i
  6. Compare with options: Compare the approximated complex number with the given options.\newlineThe approximated complex number is −4−6.928i-4 - 6.928i. The option that best matches this number is (C) −4−6.9i-4 - 6.9i.

More problems from Compare linear, exponential, and quadratic 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 xx days to pick up the motorcycle.\newlineWrite an equation for the function. If it is linear, write it in the form g(x)=mx+bg(x) = mx + b. If it is exponential, write it in the form g(x)=a(b)xg(x) = a(b)^x.\newlineg(x)=‾g(x) = \underline{\hspace{3em}}
Get tutor helpright-arrow

Posted 8 months ago

Question
How does f(t)=−2t−7f(t)=-2t-7 change over the interval from t=−7t=-7 to t=−6t=-6?\newlineChoices:\newline[f(t) decreases by 2]\left[\text{f(t) decreases by } 2\right]\newline[f(t) increases by 2]\left[\text{f(t) increases by } 2\right]\newline[f(t) increases by 200%]\left[\text{f(t) increases by } 200\%\right]\newline[f(t) decreases by a factor of 2]\left[\text{f(t) decreases by a factor of } 2\right]
Get tutor helpright-arrow

Posted 8 months ago

Question
How does g(t)=9tg(t)=9^t change over the interval from t=1t=1 to t=3t=3?\newlineChoices: \newline[g(t) decreases by a factor of 81]\left[\text{g(t) decreases by a factor of } 81\right]\newline[g(t) increases by a factor of 81]\left[\text{g(t) increases by a factor of } 81\right]\newline[g(t) increases by 18%]\left[\text{g(t) increases by } 18\%\right]\newline[g(t) decreases by 9%]\left[\text{g(t) decreases by } 9\%\right]
Get tutor helpright-arrow

Posted 8 months ago

Question
Both of these functions grow as xx gets larger and larger. Which function eventually exceeds the other?\newline Choices:\newline(A)f(x)=8x+3.3(A)f(x) = 8x + 3.3\newline(B)g(x)=3.3x−5(B)g(x) = 3.3^x - 5
Get tutor helpright-arrow

Posted 9 months ago

Question
Both of these functions grow as x x gets larger and larger. Which function eventually exceeds the other?\newlineChoices:\newline(A) f(x)=2x+9(A) \ f(x) = 2x + 9 \newline(B) g(x)=2x(B)\ g(x) = 2^x
Get tutor helpright-arrow

Posted 9 months ago

Question
The functions f(x) f(x) and g(x) g(x) are differentiable. \newlineThe function h(x) h(x) is defined as: h(x)=g(x)f(x) h(x)=\frac{g(x)}{f(x)} \newlineIf f(8)=1 f(8)=1 , f′(8)=−6 f'(8)=-6 , g(8)=8 g(8)=8 , and g′(8)=−10 g'(8)=-10 , what is h′(8) h'(8) ? \newlineSimplify any fractions. \newlineh′(8)= h'(8)= ______
Get tutor helpright-arrow

Posted 8 months ago

Question
The functions p(x) p(x) and q(x) q(x) are differentiable. \newlineThe function r(x) r(x) is defined as: r(x)=p(x)q(x) r(x)= \frac{p(x)}{q(x)} \newlineIf p(3)=2 p(3)= 2 , p′(3)=4 p'(3)= 4 , q(3)=6 q(3)= 6 , and q′(3)=9 q'(3)= 9 , what is r′(3) r'(3) ? \newlineSimplify any fractions. \newliner′(3)= r'(3)= _____
Get tutor helpright-arrow

Posted 10 months ago

Question
The functions u(x) u(x) and v(x) v(x) are differentiable. \newlineThe function w(x) w(x) is defined as: w(x)=u(x)v(x) w(x)= \frac{u(x)}{v(x)} \newlineIf u(5)=3 u(5)= 3 , u′(5)=−2 u'(5)= -2 , v(5)=7 v(5)= 7 , and v′(5)=1 v'(5)= 1 , what is w′(5) w'(5) ? \newlineSimplify any fractions. \newlinew′(5)= w'(5)= _____
Get tutor helpright-arrow

Posted 10 months ago

Question
The functions a(x) a(x) and b(x) b(x) are differentiable. \newlineThe function c(x) c(x) is defined as: c(x)=a(x)b(x) c(x)= \frac{a(x)}{b(x)} \newlineIf a(2)=4 a(2)= 4 , a′(2)=5 a'(2)= 5 , b(2)=1 b(2)= 1 , and b′(2)=−3 b'(2)= -3 , what is c′(2) c'(2) ? \newlineSimplify any fractions. \newlinec′(2)= c'(2)= _____
Get tutor helpright-arrow

Posted 10 months ago

Question
The functions m(x) m(x) and n(x) n(x) are differentiable. \newlineThe function o(x) o(x) is defined as: o(x)=m(x)n(x) o(x)= \frac{m(x)}{n(x)} \newlineIf m(7)=2 m(7)= 2 , m′(7)=−1 m'(7)= -1 , n(7)=4 n(7)= 4 , and n′(7)=8 n'(7)= 8 , what is o′(7) o'(7) ? \newlineSimplify any fractions. \newlineo′(7)= o'(7)= _____
Get tutor helpright-arrow

Posted 10 months ago

View all (100)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant