Find each product or quotient. Express the result in rectangular form.
2(cosπ/6+isinπ/6) X 4(cos2π/3 +i2π/3)
2 e^i(pi/6) * 4 e^i(4pi/6)
= 8 e^i(5 pi/6)
= 8 [cos (5 pi/6) + i sin (5 pi/6)]
150 degrees , quadrant 2 , 30 above -x axis
8 [ -sqrt3/2) + i (1/2) ]
-4 sqrt 3 + 4 i
To find the product of these complex numbers, we can multiply the magnitudes and add the angles.
Let's break down the steps:
Step 1: Simplify the complex numbers.
2(cosπ/6 + i sinπ/6) = √3/2 + i/2
4(cos2π/3 + i sin2π/3) = -2 + 2i√3
Step 2: Multiply the magnitudes.
The magnitudes are 2 and 4, so the product of the magnitudes is 8.
Step 3: Add the angles.
The angles are π/6 and 2π/3, so the sum of the angles is π/6 + 2π/3 = π/6 + 4π/6 = 5π/6.
Step 4: Express the result in rectangular form.
The rectangular form of a complex number can be obtained using the formula: a + bi.
So, the product will be:
8(cos(5π/6) + i sin(5π/6))
Using the trigonometric identity cos(θ) + i sin(θ) = e^(iθ), we can rewrite the expression as:
8e^(i(5π/6))
Simplifying further, we get:
8(cos(5π/6) + i sin(5π/6))
Thus, the final result in rectangular form is:
8(cos(5π/6) + i sin(5π/6))
To find the product, we can multiply the magnitudes and add the angles of the complex numbers in polar form. The magnitudes can be calculated using the formula:
|z| = √(a^2 + b^2)
where "a" and "b" represent the real and imaginary parts, respectively.
First, let's convert the complex numbers to rectangular form:
2(cos(π/6) + isin(π/6)) = 2(cos(π/6)) + 2isin(π/6))
= 2(√3/2 + i(1/2))
= √3 + i
4(cos(2π/3) + i*sin(2π/3)) = 4(cos(2π/3)) + 4i*sin(2π/3))
= 4(-1/2 + i(√3/2))
= -2 + 2√3i
Now, multiply the two complex numbers:
(√3 + i)(-2 + 2√3i) = √3 * -2 + √3 * 2√3i + i * -2 + i * 2√3i
= -2√3 - 2 + 2√3i - 2i
Finally, combine like terms to express the result in rectangular form:
-2√3 - 2 + 2√3i - 2i = (-2 - 2) + (-2√3 + 2√3)i
= -4 + 0i
= -4
Therefore, the product is -4 in rectangular form.