Find this quotient. Express the result in rectangular form.
9(cos3π/2 + isin3π/2) ÷ 3(cosπ/4 + isinπ/4)
e^i theta = cos theta + i sin theta
so
9 e^i(3 pi/2)
-----------------
3 e^i(pi/4)
= 3 e^i(5 pi/4)
= 3 [cos (5 pi/4) + i sin (5 pi/4)]
= 3 (-.707) + 3i(-.707)
or
-(3/2) sqrt 2 - (3/2) i sqrt 2
To find the quotient of two complex numbers in polar form, you need to first convert both complex numbers to rectangular form. The rectangular form of a complex number is given by 𝑎 + 𝑏𝑖, where 𝑎 represents the real part and 𝑏 represents the imaginary part.
Let's convert the first complex number, 9(cos3π/2 + isin3π/2), to rectangular form:
We know that cos(3π/2) = 0 and sin(3π/2) = -1.
So, the first complex number can be written as 9(0 - i) = -9i.
Now, let's convert the second complex number, 3(cosπ/4 + isinπ/4), to rectangular form:
We know that cos(π/4) = √2/2 and sin(π/4) = √2/2.
So, the second complex number can be written as 3(√2/2 + i√2/2).
Simplifying further, we get (3√2/2) + (3√2/2)i.
Now, we can divide the two complex numbers in rectangular form: (-9i) ÷ ((3√2/2) + (3√2/2)i).
To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number 𝑎 + 𝑏𝑖 is 𝑎 - 𝑏𝑖.
So, the conjugate of ((3√2/2) + (3√2/2)i) is ((3√2/2) - (3√2/2)i).
Now, let's multiply the numerator and denominator by the conjugate of the denominator:
((-9i) * ((3√2/2) - (3√2/2)i)) ÷ (((3√2/2) + (3√2/2)i) * ((3√2/2) - (3√2/2)i))
Simplifying the numerator:
((-9i) * ((3√2/2) - (3√2/2)i)) = ((-9i) * (3√2/2)) + ((9 * √2/2) * i) = -27i√2/2 + 9√2/2 * i
Simplifying the denominator:
((3√2/2) + (3√2/2)i) * ((3√2/2) - (3√2/2)i) = (9 * 2/4) + (9 * 2/4)i = 9/2 + 9/2i
Now, we can rewrite the original quotient as (-27i√2/2 + 9√2/2i) ÷ (9/2 + 9/2i).
To simplify, divide both the numerator and denominator by 9/2:
((-27i√2/2 + 9√2/2i) / (9/2 + 9/2i)) * (2/9) ÷ (2/9)
Simplifying further, we get (-3i√2/2 + √2/2i) / (1 + i).
Now, to simplify the division of complex numbers, we multiply the numerator and denominator by the conjugate of the denominator:
((-3i√2/2 + √2/2i) / (1 + i)) * ((1 - i)/(1 - i))
Multiplying the numerator and denominator:
((-3i√2/2 + √2/2i) * (1 - i)) / ((1 + i) * (1 - i))
Simplifying further, we get:
((-3i√2/2 + √2/2i - 3√2/2 - 3√2/2i) / (1 + i - i - 1))
((-3i√2 - 3√2)/2) / 0 = undefined
Therefore, the quotient of the given complex numbers expressed in rectangular form is undefined.