Answer :
To determine which pair represents the same complex number, we need to convert the polar form of the complex numbers into their Cartesian form and then compare with the given Cartesian forms.
### Pair A.
Cartesian Form:
[tex]\[ -3 - 3i \][/tex]
Polar Form:
[tex]\[ 3\sqrt{2}\left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right) \][/tex]
### Pair B.
Cartesian Form:
[tex]\[ 1 + i \][/tex]
Polar Form:
[tex]\[ \sqrt{2}\left(\cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right)\right) \][/tex]
### Pair C.
Cartesian Form:
[tex]\[ 3 + 4i \][/tex]
Polar Form:
[tex]\[ 5\left(\cos\left(\frac{4}{3}\right) + i\sin\left(\frac{4}{3}\right)\right) \][/tex]
### Pair D.
Cartesian Form:
[tex]\[ 1 - i \][/tex]
Polar Form:
[tex]\[ -\sqrt{2}\left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right) \][/tex]
Now, let's identify the pair that matches.
### Detailed Verification:
1. Pair A:
- Cartesian form: [tex]\(-3 - 3i\)[/tex]
- Polar form: [tex]\(3 \sqrt{2}\left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right)\)[/tex]
Convert the polar form:
[tex]\[ 3\sqrt{2}\left(\frac{\sqrt{2}}{2} + i \cdot \frac{-\sqrt{2}}{2}\right) = 3(\frac{\sqrt{2} \cdot \sqrt{2}}{2} + i\cdot \frac{-\sqrt{2} \cdot \sqrt{2}}{2}) = 3(1 - i) = 3 - 3i \][/tex]
This does not match [tex]\(-3 -3i\)[/tex].
2. Pair B:
- Cartesian form: [tex]\(1 + i\)[/tex]
- Polar form: [tex]\(\sqrt{2} \left(\cos\left(\frac{\pi}{4}\right) + i \sin\left(\frac{\pi}{4}\right)\right)\)[/tex]
Convert the polar form:
[tex]\[ \sqrt{2} \left(\frac{\sqrt{2}}{2} + i \cdot \frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2} \cdot \sqrt{2}}{2} + i \cdot \frac{\sqrt{2} \cdot \sqrt{2}}{2} = 1 + i \][/tex]
This matches [tex]\(1 + i\)[/tex].
3. Pair C:
- Cartesian form: [tex]\(3 + 4i\)[/tex]
- Polar form: [tex]\(5 \left(\cos\left(\frac{4}{3}\right) + i \sin\left(\frac{4}{3}\right)\right)\)[/tex]
This pair's Cartesian form does not match because [tex]\(5 \left(\cos\left(\frac{4}{3}\right) + i \sin\left(\frac{4}{3}\right)\right)\)[/tex] is not straightforward without actual value, but 3 + 4i = 5…?
4. Pair D:
- Cartesian form: [tex]\(1 - i\)[/tex]
- Polar form: [tex]\(-\sqrt{2} \left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right)\)[/tex]
Convert the polar form:
[tex]\[ -\sqrt{2}\left(\frac{\sqrt{2}}{2} + i\cdot \frac{-\sqrt{2}}{2}\right) = -1 -i \][/tex]
This does not match [tex]\(1 - i\)[/tex].
Thus, the pair that represents the same complex number is:
[tex]\[ \boxed{2} \][/tex]
### Pair A.
Cartesian Form:
[tex]\[ -3 - 3i \][/tex]
Polar Form:
[tex]\[ 3\sqrt{2}\left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right) \][/tex]
### Pair B.
Cartesian Form:
[tex]\[ 1 + i \][/tex]
Polar Form:
[tex]\[ \sqrt{2}\left(\cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right)\right) \][/tex]
### Pair C.
Cartesian Form:
[tex]\[ 3 + 4i \][/tex]
Polar Form:
[tex]\[ 5\left(\cos\left(\frac{4}{3}\right) + i\sin\left(\frac{4}{3}\right)\right) \][/tex]
### Pair D.
Cartesian Form:
[tex]\[ 1 - i \][/tex]
Polar Form:
[tex]\[ -\sqrt{2}\left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right) \][/tex]
Now, let's identify the pair that matches.
### Detailed Verification:
1. Pair A:
- Cartesian form: [tex]\(-3 - 3i\)[/tex]
- Polar form: [tex]\(3 \sqrt{2}\left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right)\)[/tex]
Convert the polar form:
[tex]\[ 3\sqrt{2}\left(\frac{\sqrt{2}}{2} + i \cdot \frac{-\sqrt{2}}{2}\right) = 3(\frac{\sqrt{2} \cdot \sqrt{2}}{2} + i\cdot \frac{-\sqrt{2} \cdot \sqrt{2}}{2}) = 3(1 - i) = 3 - 3i \][/tex]
This does not match [tex]\(-3 -3i\)[/tex].
2. Pair B:
- Cartesian form: [tex]\(1 + i\)[/tex]
- Polar form: [tex]\(\sqrt{2} \left(\cos\left(\frac{\pi}{4}\right) + i \sin\left(\frac{\pi}{4}\right)\right)\)[/tex]
Convert the polar form:
[tex]\[ \sqrt{2} \left(\frac{\sqrt{2}}{2} + i \cdot \frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2} \cdot \sqrt{2}}{2} + i \cdot \frac{\sqrt{2} \cdot \sqrt{2}}{2} = 1 + i \][/tex]
This matches [tex]\(1 + i\)[/tex].
3. Pair C:
- Cartesian form: [tex]\(3 + 4i\)[/tex]
- Polar form: [tex]\(5 \left(\cos\left(\frac{4}{3}\right) + i \sin\left(\frac{4}{3}\right)\right)\)[/tex]
This pair's Cartesian form does not match because [tex]\(5 \left(\cos\left(\frac{4}{3}\right) + i \sin\left(\frac{4}{3}\right)\right)\)[/tex] is not straightforward without actual value, but 3 + 4i = 5…?
4. Pair D:
- Cartesian form: [tex]\(1 - i\)[/tex]
- Polar form: [tex]\(-\sqrt{2} \left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right)\)[/tex]
Convert the polar form:
[tex]\[ -\sqrt{2}\left(\frac{\sqrt{2}}{2} + i\cdot \frac{-\sqrt{2}}{2}\right) = -1 -i \][/tex]
This does not match [tex]\(1 - i\)[/tex].
Thus, the pair that represents the same complex number is:
[tex]\[ \boxed{2} \][/tex]
