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.
Using the identity
cos(x)+i sin(x) = eix
the expression simplifies considerably:
3(cos (5pi/12) + i sin (5pi/12) / 6(cos (pi/12) + i sin (pi/12))
=3(e5iπ/12)/6(eiπ/12)
=(1/2)e(5iπ-iπ)/12
=(1/2)eiπ/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 eix
=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
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.