A point has polar coordinates (2, π/3). What are its rectangular coordinates?
A(1, √3)
B(√3, 1)
C(2, 2)
D(√3/2, 1/2)
x = r·cos(θ) = 2·cos(π/3) = 2·(1/2) = 1. y = r·sin(θ) = 2·sin(π/3) = 2·(√3/2) = √3. So the point is (1, √3). A common error is swapping the cosine and sine assignments, yielding (√3, 1).
Question 2 True / False
The polar points (3, π/4) and (3, π/4 + 2π) represent different locations in the plane.
TTrue
FFalse
Answer: False
Polar representations are not unique. Adding 2π to the angle completes one full rotation and returns to the same point. Both expressions describe the same location. Similarly, (-3, π/4 + π) also represents the same point.
Question 3 Short Answer
What is the polar equation of the circle x² + y² = 9?
Think about your answer, then reveal below.
Model answer: r = 3
Substituting the identity r² = x² + y² gives r² = 9, so r = 3. This illustrates why polar coordinates are natural for circles centered at the origin — a complex rectangular equation collapses to a single constant.