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Answer:
graph: A
(a) (5, -5π/4)
(b) (-5, 7π/4)
Explanation:
Graph
The angle -π/4 is a 4th-quadrant angle only shown correctly in graph A. The radius -5 will be 5 units from the origin in the direction opposite the terminal ray of angle -π/4. This, too is shown correctly in graph A.
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(a) Multiply by 1 in the form of (-1, -π) to get the desired sign and angle range:
(5, -5π/4)
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(b) Multiply by 1 in the form of (1, 2π) to get the angle in the desired range:
(-5, 7π/4)
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(c) Multiply the answer from (b) by 1 in the form of (-1, π) to get the desired sign and angle range:
(5, 11π/4)
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Additional comment
The product of two vectors in polar form is the product of their magnitudes and the sum of their angles.
(A; α) × (B; β) = (AB; α+β)
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Of course (1; 0) can be expressed as any of (-1; 2πn +π) or (1; 2πn) for any integer n. (The semicolon (;) is used in this comment to indicate the coordinates are (magnitude; angle), not (x, y).) In our answer text, we used the notation that was used in the problem statement.