The point $(3, \frac{7\pi}{4})$ is given in polar coordinates, where $3$ is the radius and $\frac{7\pi}{4}$ is the angle in radians.

In polar coordinates, a point can have multiple representations by:

1. Adding or subtracting multiples of $2\pi$ to the angle (since $2\pi$ represents one full revolution).
2. Reversing the sign of the radius and adding $\pi$ to the angle (this reflects the point across the origin).

Step-by-step analysis:

1. Equivalent angles for $\frac{7\pi}{4}$: Adding $2\pi$: $\frac{7\pi}{4} + 2\pi = \frac{7\pi}{4} + \frac{8\pi}{4} = \frac{15\pi}{4}$ Subtracting $2\pi$: $\frac{7\pi}{4} - 2\pi = \frac{7\pi}{4} - \frac{8\pi}{4} = -\frac{\pi}{4}$

2. Reversing the radius $3$: For $-3$, the angle becomes: $\frac{7\pi}{4} + \pi = \frac{7\pi}{4} + \frac{4\pi}{4} = \frac{11\pi}{4}$

Check the options:

• A: $(-3, \frac{11\pi}{4})$ matches the reversed radius and the adjusted angle. ✅
• B: $(3, \frac{9\pi}{4})$: $\frac{9\pi}{4} \neq \frac{7\pi}{4}$. ❌
• C: $(3, \frac{11\pi}{4})$: $\frac{11\pi}{4} \neq \frac{7\pi}{4}$. ❌
• D: $(-3, \frac{9\pi}{4})$: $\frac{9\pi}{4} \neq \frac{7\pi}{4}$. ❌
• E: $(-3, \frac{15\pi}{4})$: $\frac{15\pi}{4} \neq \frac{11\pi}{4}$. ❌
• F: $(3, \frac{15\pi}{4})$ matches the equivalent angle of $\frac{15\pi}{4}$ by adding $2\pi$. ✅

Final Answer:

The two points that refer to the same point as $(3, \frac{7\pi}{4})$ are:

• A: $(-3, \frac{11\pi}{4})$
• F: $(3, \frac{15\pi}{4})$

Would you like further clarification or more examples?

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Related Questions:

1. How are polar coordinates converted to Cartesian coordinates?
2. What is the significance of adding $2\pi$ to an angle in polar coordinates?
3. How do you reflect points across the origin in polar coordinates?
4. Why does reversing the radius require adding $\pi$ to the angle?
5. Can you convert between degrees and radians for polar coordinates?

Tip: Remember, angles in polar coordinates are periodic with a period of $2\pi$, so adding/subtracting $2\pi$ does not change the point's position.