find the quotient z/w
z=8(cospi/3+isinpi/3)
w=4(cospi/18+isinpi/18)
Joe, we are not going to do your work for you. Sorry.
are you aware that cos x + i sin x = e^(i x) ?
To find the quotient z/w, we need to divide the complex number z by w.
First, let's simplify the expressions for z and w:
z = 8(cos(pi/3) + i sin(pi/3))
w = 4(cos(pi/18) + i sin(pi/18))
Now, let's divide z by w:
z/w = (8(cos(pi/3) + i sin(pi/3))) / (4(cos(pi/18) + i sin(pi/18)))
When dividing complex numbers, we multiply the numerator and denominator by the conjugate of the denominator.
Conjugate of w = 4(cos(pi/18) - i sin(pi/18))
Multiplying the numerator and denominator by the conjugate gives us:
z/w = (8(cos(pi/3) + i sin(pi/3))) / (4(cos(pi/18) + i sin(pi/18))) * (4(cos(pi/18) - i sin(pi/18))) / (4(cos(pi/18) - i sin(pi/18)))
Now, let's simplify the expression:
z/w = (8 * 4)(cos(pi/3) * cos(pi/18) + i sin(pi/3) * cos(pi/18) + i sin(pi/3) * -i sin(pi/18)) / (4 * 4)(cos(pi/18) * cos(pi/18) - i sin(pi/18) * cos(pi/18) - i sin(pi/18) * -i sin(pi/18)))
z/w = (32)(cos(pi/3 + pi/18) + i sin(pi/3 + pi/18)) / (16)(cos^2(pi/18) + sin^2(pi/18))
Simplifying further:
z/w = (32)(cos(7pi/18) + i sin(7pi/18)) / (16)(1)
Finally:
z/w = 2(cos(7pi/18) + i sin(7pi/18))
Therefore, the quotient z/w is 2(cos(7pi/18) + i sin(7pi/18)).
To find the quotient z/w, we need to divide z by w.
Let's first simplify z and w:
z = 8(cos(π/3) + i*sin(π/3))
w = 4(cos(π/18) + i*sin(π/18))
Now, let's divide z by w:
z/w = (8(cos(π/3) + i*sin(π/3))) / (4(cos(π/18) + i*sin(π/18)))
To divide complex numbers, we use the concept of multiplying by the conjugate. The conjugate of a complex number a + bi is a - bi.
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator. In this case, the conjugate of w is 4(cos(π/18) - i*sin(π/18)).
Applying the conjugate, we have:
z/w = (8(cos(π/3) + i*sin(π/3))) / (4(cos(π/18) + i*sin(π/18))) * (4*cos(π/18) - i*sin(π/18)) / (4*cos(π/18) - i*sin(π/18))
Now, let's simplify the numerator and denominator separately:
Numerator:
8(cos(π/3) + i*sin(π/3)) * (4*cos(π/18) - i*sin(π/18))
= 8 * 4 * (cos(π/3)*cos(π/18) + sin(π/3)*sin(π/18)) + 8 * (cos(π/3)*(-i*sin(π/18)) + sin(π/3)*(4*cos(π/18)))
= 32 * cos((π/3) - (π/18)) + 32i*sin((π/3) - (π/18)) - 8i*sin((π/3) + (π/18)) + 32cos((π/3) + (π/18))
= 32 * cos(5π/18) + 32i * sin(5π/18) - 8i * sin(7π/18) + 32 * cos(7π/18)
Denominator:
4(cos(π/18) + i*sin(π/18)) * (4*cos(π/18) - i*sin(π/18))
= 4 * 4 * (cos(π/18)*cos(π/18) + sin(π/18)*sin(π/18))
= 16 * cos^2(π/18) + 16 * sin^2(π/18)
= 16 * (cos^2(π/18) + sin^2(π/18))
= 16 * 1
= 16
Now, putting it all together:
z/w = (32 * cos(5π/18) + 32i * sin(5π/18) - 8i * sin(7π/18) + 32 * cos(7π/18)) / 16
Simplifying the expression further, we get:
z/w = (2 * cos(5π/18) + 2i * sin(5π/18) - 0.5i * sin(7π/18) + 2 * cos(7π/18))
/ 1
Therefore, the quotient z/w is:
2 * cos(5π/18) + 2i * sin(5π/18) - 0.5i * sin(7π/18) + 2 * cos(7π/18)