This question tests your ability to convert complex numbers between rectangular form a + bi and polar form r(cos θ + i sin θ), which represent the same number using Cartesian coordinates versus magnitude and direction. Rectangular form a + bi uses horizontal (real) and vertical (imaginary) components, while polar form r(cos θ + i sin θ) uses distance from origin (modulus r) and angle from positive real axis (argument θ, measured counterclockwise). To convert 5(cos 30° + i sin 30°) to rectangular: a = 5 cos 30° = 5(√3/2) = (5√3)/2, b = 5 sin 30° = 5(1/2) = 5/2, giving (5√3)/2 + (5/2)i. Choice A correctly applies the formulas a = r cos θ and b = r sin θ, using the special angle values from the 30-60-90 triangle. Choice B swaps sin and cos, a common mix-up, but remember cos is for the real part (adjacent over hypotenuse) and sin for imaginary (opposite over hypotenuse)—think of the unit circle! Polar to rectangular recipe: (1) Identify r and θ; (2) a = r cos θ; (3) b = r sin θ; (4) Write a + bi, and verify by converting back if needed. Keep practicing these conversions—they'll become second nature, and you're already on the right track!

This question tests your ability to convert complex numbers between rectangular form a + bi and polar form r(cos θ + i sin θ), which represent the same number using Cartesian coordinates versus magnitude and direction. Rectangular form a + bi uses horizontal (real) and vertical (imaginary) components, while polar form r(cos θ + i sin θ) uses distance from origin (modulus r) and angle from positive real axis (argument θ, measured counterclockwise). For z = -3 + 3i: r = √((-3)² + 3²) = √(9 + 9) = √18 = 3√2; reference = arctan(3/3) = 45°, Quadrant 2 (a < 0, b > 0), θ = 180° - 45° = 135°, so 3√2(cos 135° + i sin 135°). Choice B correctly finds the modulus and adjusts the argument for the AC current context. Choice A ignores the quadrant, using 45° which fits Quadrant 1—remember, negative real means Q2 or Q3! Rectangular to polar recipe: (1) r = √(a² + b²); (2) Reference = arctan(|b|/|a|); (3) Adjust per quadrant; (4) Apply to real-world like currents. You're brilliant—applying to contexts like AC builds deeper understanding!

This question tests your ability to convert complex numbers between rectangular form a + bi and polar form r(cos θ + i sin θ), which represent the same number using Cartesian coordinates versus magnitude and direction. Rectangular form a + bi uses horizontal (real) and vertical (imaginary) components, while polar form r(cos θ + i sin θ) uses distance from origin (modulus r) and angle from positive real axis (argument θ, measured counterclockwise). For z = 1 - i: r = √(1² + (-1)²) = √2; reference angle = arctan(1/1) = 45°, in Quadrant 4 (a > 0, b < 0), θ = 360° - 45° = 315°, so √2(cos 315° + i sin 315°). Choice B correctly calculates r and chooses the appropriate θ with quadrant adjustment. Choice A uses 45° without adjustment, which is Quadrant 1—always adjust for negative imaginary part in Q4! Rectangular to polar recipe: (1) r = √(a² + b²); (2) Reference = arctan(|b|/|a|); (3) For Q4: θ = 360° - reference; (4) Write polar form. You're handling quadrants well—practice more to make it intuitive!

This question tests your ability to convert complex numbers between rectangular form a + bi and polar form $r(\cos \theta + i \sin \theta)$, which represent the same number using Cartesian coordinates versus magnitude and direction. Rectangular form a + bi uses horizontal (real) and vertical (imaginary) components, while polar form $r(\cos \theta + i \sin \theta)$ uses distance from origin (modulus r) and angle from positive real axis (argument $\theta$, measured counterclockwise). For z = -2 + 2i: $r = \sqrt{(-2)^2 + 2^2} = \sqrt{8} = 2\sqrt{2}$; reference angle = $\arctan(2/2) = 45^\circ$, but in Quadrant 2 (a < 0, b > 0), $\theta = 180^\circ - 45^\circ = 135^\circ$, so $2\sqrt{2}(\cos 135^\circ + i \sin 135^\circ)$. Choice B correctly computes the modulus and adjusts the argument for Quadrant 2, ensuring the angle is measured properly from the positive real axis. Choice A forgets the quadrant adjustment, using 45° which would place it in Quadrant 1—always check signs of a and b to determine the quadrant! Rectangular to polar recipe: (1) $r = \sqrt{a^2 + b^2}$; (2) Reference = $\arctan(|b|/|a|)$; (3) For Q2: $\theta = 180^\circ - $ reference; (4) Write $r(\cos \theta + i \sin \theta)$. Great job tackling quadrants—keep going, you've got this!

This question tests your ability to convert complex numbers between rectangular form a + bi and polar form r(cos θ + i sin θ), which represent the same number using Cartesian coordinates versus magnitude and direction. Rectangular form a + bi uses horizontal (real) and vertical (imaginary) components, while polar form r(cos θ + i sin θ) uses distance from origin (modulus r) and angle from positive real axis (argument θ, measured counterclockwise). To convert √2(cos 45° + i sin 45°) to rectangular: a = √2 cos 45° = √2 (√2/2) = 1, b = √2 sin 45° = √2 (√2/2) = 1, giving 1 + i. Choice A properly uses a = r cos θ and b = r sin θ, with cos 45° = sin 45° = √2/2 for exact simplification. Choice B doubles the values, perhaps by forgetting to multiply correctly or confusing with 2(cos 45° + i sin 45°)—check your calculations carefully! Polar to rectangular recipe: (1) Identify r and θ; (2) a = r cos θ; (3) b = r sin θ; (4) Simplify and write a + bi. Nice effort—special angles like 45° are perfect for practice!

This question tests your ability to convert complex numbers between rectangular form a + bi and polar form r(cos θ + i sin θ), which represent the same number using Cartesian coordinates versus magnitude and direction. Rectangular form a + bi uses horizontal (real) and vertical (imaginary) components, while polar form r(cos θ + i sin θ) uses distance from origin (modulus r) and angle from positive real axis (argument θ, measured counterclockwise). To convert 2(cos 120° + i sin 120°) to rectangular: a = 2 cos 120° = 2(-1/2) = -1, b = 2 sin 120° = 2(√3/2) = √3, giving -1 + √3 i. Choice A accurately applies a = r cos θ and b = r sin θ, recalling cos 120° = -1/2 and sin 120° = √3/2 from the unit circle. Choice D confuses the values, perhaps using sin for real and cos for imaginary or mixing with another angle—memorize key angles like 120° in Quadrant 2! Polar to rectangular recipe: (1) Identify r and θ; (2) Compute a and b using trig functions; (3) Write a + bi. Fantastic work—keep verifying with back-conversion to build accuracy!

This question tests your ability to convert complex numbers between rectangular form a + bi and polar form r(cos θ + i sin θ), which represent the same number using Cartesian coordinates versus magnitude and direction. Rectangular form a + bi uses horizontal (real) and vertical (imaginary) components, while polar form r(cos θ + i sin θ) uses distance from origin (modulus r) and angle from positive real axis (argument θ, measured counterclockwise). To convert 3 + 4i to polar form: (1) Find modulus r = √(3² + 4²) = √(9 + 16) = √25 = 5; (2) Find argument θ = arctan(4/3) ≈ 53.13°, and since a > 0 and b > 0 (Quadrant 1), no adjustment needed; (3) Write 5(cos 53.13° + i sin 53.13°). Choice B correctly calculates the modulus using the Pythagorean theorem and determines the argument with proper quadrant consideration, matching the conversion formulas. Choice A might tempt if you swap the angle calculation, using arctan(3/4) ≈ 36.87° instead of arctan(4/3), but remember θ = arctan(b/a), not a/b—keep the order straight! Rectangular to polar recipe: (1) r = √(a² + b²); (2) Reference angle = arctan(|b|/|a|); (3) Adjust for quadrant based on signs of a and b. You're doing great—practice with different quadrants to build confidence!

This question tests your ability to convert complex numbers between rectangular form a + bi and polar form r(cos θ + i sin θ), which represent the same number using Cartesian coordinates versus magnitude and direction. Rectangular form a + bi uses horizontal (real) and vertical (imaginary) components, while polar form r(cos θ + i sin θ) uses distance from origin (modulus r) and angle from positive real axis (argument θ, measured counterclockwise). For -√3 - i: r = √((-√3)² + (-1)²) = √(3 + 1) = √4 = 2; reference angle = arctan(1/√3) = 30°, in Quadrant 3 (a < 0, b < 0), θ = 180° + 30° = 210°, so 2(cos 210° + i sin 210°). Choice A correctly finds the modulus and adjusts θ for Quadrant 3 using the proper formulas. Choice B might result from incorrect quadrant adjustment, like using 150° for Quadrant 2 instead—double-check signs: both negative means Quadrant 3! Rectangular to polar recipe: (1) r = √(a² + b²); (2) Reference = arctan(|b|/|a|); (3) For Q3: θ = 180° + reference; (4) Write the polar form. You're making excellent progress—remember special angles like 30° for exact values!

This question tests your ability to convert complex numbers between rectangular form a + bi and polar form r(cos θ + i sin θ), which represent the same number using Cartesian coordinates versus magnitude and direction. Rectangular form a + bi uses horizontal (real) and vertical (imaginary) components, while polar form r(cos θ + i sin θ) uses distance from origin (modulus r) and angle from positive real axis (argument θ, measured counterclockwise). To convert 3(cos 270° + i sin 270°) to rectangular: a = 3 cos 270° = 3(0) = 0, b = 3 sin 270° = 3(-1) = -3, giving -3i (or 0 - 3i). Choice B accurately computes using the unit circle values: cos 270° = 0, sin 270° = -1. Choice A uses sin 270° as +1, forgetting the negative direction on the imaginary axis—visualize the angle at the bottom of the unit circle! Polar to rectangular recipe: (1) Identify r and θ; (2) a = r cos θ; (3) b = r sin θ; (4) Write a + bi, omitting zero terms if needed. Excellent—angles like 270° highlight pure imaginary numbers perfectly!

This question tests your ability to convert complex numbers between rectangular form a + bi and polar form r(cos θ + i sin θ), which represent the same number using Cartesian coordinates versus magnitude and direction. Rectangular form a + bi uses horizontal (real) and vertical (imaginary) components, while polar form r(cos θ + i sin θ) uses distance from origin (modulus r) and angle from positive real axis (argument θ, measured counterclockwise). To convert from rectangular to polar: find r = sqrt(a² + b²) using Pythagorean theorem, then find θ = arctan(b/a) but adjust for quadrant (arctan only gives reference angle!). For -2 + 2i, r = sqrt(4 + 4) = sqrt(8) = 2√2, reference angle = arctan(2/2) = 45°, and since a < 0 and b > 0 (quadrant 2), θ = 180° - 45° = 135°, so polar form is 2√2(cos 135° + i sin 135°). Choice D correctly calculates the modulus using the Pythagorean theorem, determines the argument with proper quadrant adjustment for quadrant 2, and writes the polar form accurately. Choice A fails by using θ = 45° without quadrant adjustment, ignoring that the point is in quadrant 2, not 1—remember, arctan(b/a) needs correction based on signs of a and b! Rectangular to polar recipe: (1) Calculate r = sqrt(a² + b²) (always positive), (2) Reference angle = arctan(|b|/|a|), (3) Adjust for quadrant: Q2 θ = 180° - reference, then write r(cos θ + i sin θ)—keep practicing these adjustments, you've got this!