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Details and assumptions
is the imaginary number satisfying .
KEY TECHNIQUES
De Moivre's Formula, Polar Coordinates, Complex Numbers, Trigonometric Functions
ANSWER
32
The complex number can be written as . What is the value of ?[2[cos( ) + i sin( )]]π
10
π
10
5
a + bi a + b
i = −1i2
From de Moivre's formula, we have
Thus, and . Hence, .
[2(cos( ) + i sin( ))]
π
10
π
10
5
= [cos( ) + i sin( )]25 5π
10
5π
10
= 32(cos( ) + i sin( ))
π
2
π
2
= 32(0 + i)
= 32i.
a = 0 b = 32 a + b = 0 + 32 = 32
Complex Representations
70 points
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Complex representations
This document contains a math problem involving complex numbers and De Moivre's formula. It asks for the value of a complex number written as [2(cos(π/10) + i sin(π/10))]5. The solution uses De Moivre's formula to rewrite the expression in exponential form, and then evaluates it to get the final answer of 32i.
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- 1. Get more freechallenging math and physics problems at briliant.org You're done with this section! Check back next week to view solutions. Go back home Details and assumptions is the imaginary number satisfying . KEY TECHNIQUES De Moivre's Formula, Polar Coordinates, Complex Numbers, Trigonometric Functions ANSWER 32 The complex number can be written as . What is the value of ?[2[cos( ) + i sin( )]]π 10 π 10 5 a + bi a + b i = −1i2 From de Moivre's formula, we have Thus, and . Hence, . [2(cos( ) + i sin( ))] π 10 π 10 5 = [cos( ) + i sin( )]25 5π 10 5π 10 = 32(cos( ) + i sin( )) π 2 π 2 = 32(0 + i) = 32i. a = 0 b = 32 a + b = 0 + 32 = 32 Complex Representations 70 points Brilliant Page 1 of 1Complex Representations | Brilliant 6/13/2013https://brilliant.org/assessment/s/algebra-and-number-theory/2301963/