Find the quotient 3(cos (5pi/12) + i sin (5pi/12) / 6(cos (pi/12) i sin (pi/12)). Express the quotient in rectangular form.

Question

Find the quotient 3(cos (5pi/12) + i sin (5pi/12) / 6(cos (pi/12) i sin (pi/12)). Express the quotient in rectangular form.

I have no idea what I did but I got (1/4) - 433i/1000.

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2 answers
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Answer 1

MathMate answered
15 years ago

Using the identity

cos(x)+i sin(x) = e^ix
the expression simplifies considerably:
3(cos (5pi/12) + i sin (5pi/12) / 6(cos (pi/12) + i sin (pi/12))
=3(e^5iπ/12)/6(e^iπ/12)
=(1/2)e^{(5iπ-iπ)/12}
=(1/2)e^iπ/3
=(1/2)(cos(π/3)+i sin(π/3))
=(1/2)(1/2+(√3/3)i)
=1/4+(√3/6)i

Note: the identity can be derived by the expansion of e^ix
=1 + ix + (ix)²/2! + (ix)³/3! + ...
=1 +ix -x²/2! - x³/3! + ...
=1 -x²/2! + x⁴/4! - x⁶/6! + ...
+ i( x - x³/3! + x⁵/5! - ...)
= cos(x) + i sin(x)

Alternatively, multiply both numerator and denominator by the conjugate of the denominator, namely (cos (pi/12)- i sin (pi/12))
to reduce the denominator to:
6(cos (pi/12)+i sin (pi/12))(cos (pi/12)-i sin (pi/12))
=6(cos²(π/12)+sin²(π/12))
=6
The numerator becomes
3(cos (5pi/12) + i sin (5pi/12)(cos (pi/12)-i sin (pi/12))
=3(cos(5&pi/12)cos(&pi/12)+sin(5&pi/12)sin(&pi/12))
+ 3i(sin(5&pi/12)cos(&pi/12)-cos(5π/12)sin(&pi/12))
=3(cos(&pi/3)+isin(&pi/3))
So the result is also
3(cos(&pi/3)+isin(&pi/3))/6
=(1/2)(1/2+(√3/3)i)
=1/4+(√3/6)i

Answer 2

Explain Bot answered
1 year ago

To find the quotient in rectangular form, we can simplify the expression by multiplying the numerator and denominator by the complex conjugate of the denominator. Let's break down the steps:

Step 1: Convert the given complex numbers in the form of complex exponentials.
We have:
Numerator: 3(cos(5π/12) + i sin(5π/12))
Denominator: 6(cos(π/12) + i sin(π/12))

Step 2: Multiply the numerator and denominator by the complex conjugate of the denominator.
The complex conjugate of a complex number is obtained by changing the sign of the imaginary part.
So the complex conjugate of the denominator - 6(cos(π/12) + i sin(π/12)) is 6(cos(π/12) - i sin(π/12)).
Multiplying both the numerator and denominator by the complex conjugate gives us:
(3(cos(5π/12) + i sin(5π/12))) * (6(cos(π/12) - i sin(π/12))) / (6(cos(π/12) + i sin(π/12))) * (6(cos(π/12) - i sin(π/12)))

Step 3: Simplify the expression by multiplying and combining terms.
Applying the distributive property to the numerator and denominator, we have:
(18(cos(5π/12) cos(π/12) + sin(5π/12) sin(π/12)) + 18i(cos(5π/12) sin(π/12) - sin(5π/12) cos(π/12))) /
(6(cos(π/12) cos(π/12) + sin(π/12) sin(π/12)) + 6i(cos(π/12) sin(π/12) - sin(π/12) cos(π/12)))

Simplifying further, we have:
(18(cos((5π/12)+(π/12)) + 18i(sin((5π/12)-(π/12)))) /
(6(cos((π/12)+(π/12)) + 6i(sin((π/12)-(π/12))))

Which simplifies to:
(18(cos(2π/3)) + 18i(sin(4π/12))) / (6(cos(π/6)) + 6i(sin(0)))

The angles in the trigonometric functions can be simplified:
cos(2π/3) = -1/2
sin(4π/12) = sin(π/3) = √3/2
cos(π/6) = √3/2
sin(0) = 0

Substituting these values, we get:
(18(-1/2) + 18i(√3/2)) / (6(√3/2) + 6i(0))

Simplifying further, we have:
(-9 + 9i√3) / (3√3)

Now, to express the quotient in rectangular form, we need to rationalize the denominator.
Multiplying the numerator and denominator by (√3 / √3), we get:
((-9 + 9i√3) * (√3 / √3)) / ((3√3) * (√3 / √3))

Simplifying:
(-9√3 + 27i) / 9

Dividing each term by 9, we have:
-√3 + 3i

So the quotient in rectangular form is -√3 + 3i.