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Consider the complex number z=3(cos(355^(@))+i sin(355^(@)))". " Which of the following complex numbers best approximates z ? Choose 1 answer: (A) 3-0.3 i (B) 0.3+3i (C) 3+0.3 i (D) 0.3-3i

Consider the complex number\newlinez=3(cos⁡(355∘)+isin⁡(355∘)).  z=3\left(\cos \left(355^{\circ}\right)+i \sin \left(355^{\circ}\right)\right) \text {. } \newlineWhich of the following complex numbers best approximates z z ?\newlineChoose 11 answer:\newline(A) 3−0.3i 3-0.3 i \newline(B) 0.3+3i 0.3+3 i \newline(C) 3+0.3i 3+0.3 i \newline(D) 0.3−3i 0.3-3 i

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Full solution

Q. Consider the complex number\newlinez=3(cos⁡(355∘)+isin⁡(355∘)).  z=3\left(\cos \left(355^{\circ}\right)+i \sin \left(355^{\circ}\right)\right) \text {. } \newlineWhich of the following complex numbers best approximates z z ?\newlineChoose 11 answer:\newline(A) 3−0.3i 3-0.3 i \newline(B) 0.3+3i 0.3+3 i \newline(C) 3+0.3i 3+0.3 i \newline(D) 0.3−3i 0.3-3 i
  1. Convert to radians: Convert the given angle from degrees to radians.\newlineSince we are dealing with trigonometric functions, we need to convert the angle 355355 degrees to radians. The conversion factor is π\pi radians = 180180 degrees. Therefore, 355355 degrees is (355/180)π(355/180)\pi radians.
  2. Evaluate cosine and sine: Evaluate the cosine and sine of the angle in radians.\newlineUsing a calculator or trigonometric tables, we find the values of cosine and sine for the angle close to 355355 degrees (which is close to 360360 degrees or 2π2\pi radians). We know that cos⁡(360°)=1\cos(360°) = 1 and sin⁡(360°)=0\sin(360°) = 0. Since 355°355° is close to 360°360°, cos⁡(355°)\cos(355°) will be slightly less than 11 and sin⁡(355°)\sin(355°) will be slightly greater than 36036000 but still very small.
  3. Approximate cosine and sine: Approximate the values of cosine and sine for 355355 degrees.\newlineGiven that 355355 degrees is very close to 360360 degrees, we can approximate:\newlinecos⁡(355∘)≈1\cos(355^\circ) \approx 1 (slightly less)\newlinesin⁡(355∘)≈0\sin(355^\circ) \approx 0 (slightly more)
  4. Multiply by modulus: Multiply the approximated cosine and sine values by the modulus of the complex number.\newlineThe modulus of the complex number zz is 33. Therefore, we multiply the approximated values of cosine and sine by 33:\newlineReal part: 3×cos⁡(355°)≈3×1≈33 \times \cos(355°) \approx 3 \times 1 \approx 3\newlineImaginary part: 3×sin⁡(355°)≈3×0≈03 \times \sin(355°) \approx 3 \times 0 \approx 0 (but slightly positive)
  5. Choose best match: Choose the answer that best matches the approximated real and imaginary parts.\newlineLooking at the options, we need a real part close to 33 and an imaginary part that is positive but very small. The best match is:\newline(C) 3+0.3i3 + 0.3 i

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