Question 1185527: Find polar forms for zw, z/w, and 1/z by first putting z and w into polar form.
z = 27(3 + i), w = −9 − 9i
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! You haven't provided specific values for z and w. I'll give you the general process and formulas, and you can plug in your values.
**1. Polar Form of z:**
Let z = a + bi, where a and b are real numbers. The polar form of z is z = r(cos θ + i sin θ), where:
* r = |z| = √(a² + b²) (the modulus or magnitude of z)
* θ = arctan(b/a) (the argument of z). Be careful with the quadrant of θ; if 'a' is negative, you may need to add or subtract π (180 degrees) from the result of arctan(b/a) to get the correct angle.
**2. Polar Form of w:**
Let w = c + di, where c and d are real numbers. The polar form of w is w = s(cos φ + i sin φ), where:
* s = |w| = √(c² + d²)
* φ = arctan(d/c) (again, be mindful of the quadrant).
**3. Polar Form of zw:**
If z = r(cos θ + i sin θ) and w = s(cos φ + i sin φ), then:
zw = rs[cos(θ + φ) + i sin(θ + φ)]
So, the modulus of zw is rs, and the argument of zw is θ + φ.
**4. Polar Form of z/w:**
If z = r(cos θ + i sin θ) and w = s(cos φ + i sin φ), then:
z/w = (r/s)[cos(θ - φ) + i sin(θ - φ)]
So, the modulus of z/w is r/s, and the argument of z/w is θ - φ.
**5. Polar Form of 1/z:**
If z = r(cos θ + i sin θ), then:
1/z = (1/r)[cos(-θ) + i sin(-θ)] or equivalently,
1/z = (1/r)[cos(θ) - i sin(θ)]
So, the modulus of 1/z is 1/r, and the argument of 1/z is -θ.
**In summary:**
* **zw:** Multiply moduli, add arguments.
* **z/w:** Divide moduli, subtract arguments.
* **1/z:** Take the reciprocal of the modulus, negate the argument.
**Example (Let z = 1 + i and w = 2 - 2i):**
1. **z:** r = √2, θ = π/4. So, z = √2(cos(π/4) + i sin(π/4)).
2. **w:** s = √8 = 2√2, φ = -π/4. So, w = 2√2(cos(-π/4) + i sin(-π/4)).
Now you can use the formulas above to find zw, z/w, and 1/z.
Answer by ikleyn(52778)  (Show Source):
You can put this solution on YOUR website! .
In the post by @CPhill, the first phrase is
" You haven't provided specific values for z and w. "
This starting statement is factually incorrect, since the specific values for z and w are provided in the second line of the original post.
Simply instead of solving the problem, @CPhill preferred to place the material, widely known from any standard textbook on complex numbers.
|
|
|
