To find the product zw and the quotient z/w in polar and exponential form, we first compute zw and z/w using the given expressions for z and w.
Given:
z = 5 (cos(π/9) + i sin(π/9))
w = 5 (cos(π/10) + i sin(π/10))
Product zw:
zw = (5 * 5) (cos(π/9 + π/10) + i sin(π/9 + π/10))
zw = 25 (cos(19π/90) + i sin(19π/90))
Quotient z/w:
z/w = (5/5) (cos(π/9 - π/10) + i sin(π/9 - π/10))
z/w = (cos(π/90) + i sin(π/90))
Now, let's express the results in polar and exponential form.
Product zw:
zw = 25 (cos(19π/90) + i sin(19π/90))
In polar form: zw = 25 cis(19π/90)
In exponential form: zw = 25e^(i(19π/90))
Quotient z/w:
z/w = (cos(π/90) + i sin(π/90))
In polar form: z/w = cis(π/90)
In exponential form: z/w = e^(i(π/90))
So, the product zw in polar form is 25 cis(19π/90), and in exponential form is 25e^(i(19π/90)).
The quotient z/w in polar form is cis(π/90), and in exponential form is e^(i(π/90)).
