Rectangular form conversion: Express the polar-form complex number 40∠55° as its equivalent rectangular form (a + jb).
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A22.94 + j32.76
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B40 + j40
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C45.88 + j65.52
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D55 + j55
Answer
Correct Answer: 22.94 + j32.76
Explanation
Introduction / Context:Converting between polar (magnitude ∠ angle) and rectangular (a + jb) forms is essential for phasor arithmetic in AC circuits. Rectangular form is convenient for addition and subtraction; polar form is handy for multiplication and division.
Given Data / Assumptions:
- Magnitude M = 40.
- Angle θ = 55° (degrees).
- Use a = M * cosθ and b = M * sinθ for rectangular components.
Concept / Approach:
Apply trigonometric projections to map the polar vector onto the real and imaginary axes. Ensure the calculator is in degrees when using degree angles.
Step-by-Step Solution:
Compute real part: a = 40 * cos(55°) ≈ 40 * 0.5736 ≈ 22.94.Compute imaginary part: b = 40 * sin(55°) ≈ 40 * 0.8192 ≈ 32.77 ≈ 32.76 (rounded).Therefore, rectangular form ≈ 22.94 + j32.76.Verification / Alternative check:
Back-convert: magnitude √(22.94^2 + 32.76^2) ≈ 40; angle arctan(32.76/22.94) ≈ 55°, confirming correctness.
Why Other Options Are Wrong:
40 + j40 and 55 + j55 imply θ = 45° with different magnitudes. 45.88 + j65.52 corresponds to a magnitude much larger than 40.
Common Pitfalls:
Using radians inadvertently; swapping sin and cos; rounding too early leading to noticeable errors.
Final Answer:
22.94 + j32.76