For z1 = 4cis (7pi/6) and z2 = 3cis(pi/3), find z1 * z2 in rectangular form.
z1 z2 = (4*3)cis(7π/6+π/3) = 12cis(3π/2)
-12i
To find the product of two complex numbers in rectangular form, we can use the following formula:
z1 * z2 = (a1 + b1i) * (a2 + b2i) = (a1 * a2 - b1 * b2) + (a1 * b2 + a2 * b1)i
For z1 = 4cis (7π/6), we can rewrite it in rectangular form using the conversion formulas:
z1 = 4(cos(7π/6) + isin(7π/6))
The value of cos(7π/6) is (√3 / 2) and the value of sin(7π/6) is -1/2. Plugging these values into the rectangular form, we get:
z1 = 4(√3 / 2 - (1/2)i)
Similarly, for z2 = 3cis (π/3), we can rewrite it in rectangular form:
z2 = 3(cos(π/3) + isin(π/3))
The value of cos(π/3) is 1/2 and the value of sin(π/3) is (√3 / 2). Plugging these values into the rectangular form, we get:
z2 = 3(1/2 + (√3 / 2)i)
Now, we can multiply these two complex numbers using the formula mentioned earlier:
z1 * z2 = (4 * 3)(√3 / 2 - (1/2)i)(1/2 + (√3 / 2)i)
Simplifying the calculation:
z1 * z2 = 12(√3/2 * 1/2 - 1/2 * √3/2 + √3/2 * 1/2i + √3/2 * √3/2i)
= 12(√3/4 - √3/4 + √3/4i + 3/4i)
= 12(0 + √3/4i + 3/4i)
= 12(√3/4 + 3/4)i
= (9√3/2)i
Therefore, z1 * z2 in rectangular form is (9√3/2)i.