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ind The quantity of 1 + i all to the fifth power. and write the answer in standard form.

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Find The quantity of 1 + i all to the fifth power. and write the answer in standard form. 
A. image
B. image
C. image
D. image
When a complex number is in trigonometric form r times the quantity of cosine of theta + i times sine of theta., what does r represent? 
A. abscissa
B. argument
C. imaginary unit
D. modulus

 

asked Oct 20, 2014 in TRIGONOMETRY by tonymate Pupil

2 Answers

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Best answer

(b)

The complex number x+iy  corresponds to the point with coordinates (x, y) .

x is a real part of complex number . 

y is a imaginary part of complex number . 

The polar form of  x+iy is r(cos ø +i sin ø ) .

Where

r is  called the modulus - is the absolute value of the hypotenuse formed by sides "x" and "y" .

 r = (x2 + y2)1/2

 θ is called the argument .

 θ = arc tan (y/x) 

Hence r is called modulus .

So option (D) is correct .

answered Oct 21, 2014 by friend Mentor
selected Oct 21, 2014 by tonymate
0 votes

The complex number is  image .

Convert the complex number 1+i into polor form .

 r = (x2 + y2)1/2        θ = arc tan (y/x) 

 r = (12 + 12)1/2        θ = arc tan (1/1) 

 r = (2)1/2                   θ = arc tan (1/1) 

 r = √2                    θ = arc tan (1) 

 r = √2                    θ = π/4 

The polar form of  x+iy is r(cos ø +i sin ø ) .

1+ i = √2  (cos π/4 +i sin π/4 ) 

De Movire's Theorem states that: 

(cos x + i sin x)^n = cos(nx) + i sin(nx) 

So  (1 + i)^5 = (√2)^5[cos(5 * π/4) + i sin (5 * π/4)] 

 (1 + i)^5 = (4√2)[cos( 5π/4) + i sin (5π/4)] 

 (1 + i)^5 = (4√2)[-1/√2 -i/√2] 

 (1 + i)^5 = (4√2)(1/√2 )[-1-i] 

(1 + i)^5 = 4[-1-i] 

image =  -4 - 4i

Hence option (B) is correct .

answered Oct 21, 2014 by friend Mentor

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