Evaluate and write your answer in a + bi form.

[4(cos 113° + i sin 113°)]^3 =

Round to two decimal places.

To evaluate [ 4 ( cos 113° + i s in 113° ) ] 3 , we used De Moivre's theorem and calculated the result to be approximately 28.16 − 57.60 i . This involves computing the angle and applying trigonometric functions. The final answer is given in the form a + bi .
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Answered by Anonymous | 2025-07-04

To evaluate [ 4 ( cos 11 3 ∘ + i sin 11 3 ∘ ) ] 3 and express it in the form a + bi , we use De Moivre's Theorem.
Step-by-step Solution:

Understand De Moivre's Theorem:
De Moivre's Theorem states that for any complex number in polar form r ( cos θ + i sin θ ) and any integer n ,


it can be expressed as:
\[[r(\cos \theta + i \sin \theta)]^n = r^n (\cos(n\theta) + i \sin(n\theta)).\]


Apply De Moivre's Theorem to the given problem:
Here, r = 4 , θ = 11 3 ∘ , and n = 3 .
So, compute:



r n = 4 3 = 64

n θ = 3 × 11 3 ∘ = 33 9 ∘



Find the trigonometric values of cos ( 33 9 ∘ ) and sin ( 33 9 ∘ ) :
Both values can be calculated using standard trigonometric tables or a calculator:



cos ( 33 9 ∘ ) ≈ 0.93 (rounded to two decimal places)

sin ( 33 9 ∘ ) ≈ − 0.35 (rounded to two decimal places)



Substitute these values back into the equation:
64 ( cos ( 33 9 ∘ ) + i sin ( 33 9 ∘ )) = 64 ( 0.93 − 0.35 i ) .

Multiply through by 64:



Real part: 64 × 0.93 = 59.52

Imaginary part: 64 × ( − 0.35 ) = − 22.40



Combine the results to form a complex number in a + bi form:
Therefore, the result is:
59.52 − 22.40 i .


So, [ 4 ( cos 11 3 ∘ + i sin 11 3 ∘ ) ] 3 simplifies to approximately 59.52 − 22.40 i .

Answered by JessicaJessy | 2025-07-06