Question

In Exercises 27–32, select the representations that do not change the location of the given point. (2, − 3π/4) (2, − 7π/4)

Accepted Answer

Understand that the problem involves identifying different representations of points in polar coordinates that correspond to the same location. The point is given in the form $$(r, 	heta)$$, where $$r is $$the radius (distance from the origin) and $$	heta is $$the angle measured in radians. Recall that in polar coordinates, the same point can be represented by adding or subtracting full rotations of $$2\pi to $$the angle $$	heta. $$This means that $$	heta$$ and $$	heta + 2k\pi ($$where $$k is $$any integer) represent the same direction. Also remember that changing the sign of $$r$$ and adding $$\pi to $$the angle gives the same point: $$(r, 	heta) is $$equivalent to $$(-r, 	heta + \pi). $$This is because moving in the opposite direction by $$\pi$$ radians with a negative radius points to the same location. For the point $$(2, -\frac{3\pi}{4})$$, check if the alternative representations add or subtract multiples of $$2\pi to $$the angle or use the negative radius rule to see if the location remains unchanged. Similarly, for the point $$(2, -\frac{7\pi}{4})$$, apply the same checks: add or subtract $$2\pi$$ multiples to the angle or use the negative radius and angle shift to find equivalent points that do not change the location.

In Exercises 27–32, select the representations that do not change the location of the given point. (2, − 3π/4) (2, − 7π/4)

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Key Concepts

Polar Coordinates and Point Representation

Angle Coterminality in Polar Coordinates

Effect of Negative Radius and Angle Adjustments