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 -5i (0 - 5i), r = sqrt(0 + 25) = 5, and since it's on the negative imaginary axis (a=0, b<0), θ = 270° (or -90°), so polar form is 5(cos 270° + i sin 270°)—verify: cos 270°=0, sin 270°=-1, giving 0 -5i. Choice C correctly identifies r positive and θ=270° for the negative imaginary axis. Choice D uses negative r, but modulus is always positive—r represents distance, so it's non-negative! Rectangular to polar recipe: For axis cases, like negative imaginary: θ=270°, r=|b|—you're handling special cases well, stay encouraged!

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 60° + i sin 60°) to rectangular form: (1) Identify r = 3 and θ = 60°. (2) Calculate real part: a = r cos θ = 3 cos 60° = 3(1/2) = 3/2. (3) Calculate imaginary part: b = r sin θ = 3 sin 60° = 3(√3/2) = 3√3/2. (4) Write rectangular form: 3/2 + (3√3/2)i. Choice A correctly applies the conversion formulas a = r cos θ and b = r sin θ, using the special angle values cos 60° = 1/2 and sin 60° = √3/2. Choice B swaps the values, incorrectly using a = r sin θ and b = r cos θ—remember cosine goes with the real part (horizontal) and sine with the imaginary part (vertical), just like x = r cos θ and y = r sin θ in parametric equations! Polar to rectangular recipe: From r(cos θ + i sin θ), calculate a = r cos θ for the real part and b = r sin θ for the imaginary part, then write a + bi. For special angles like 30°, 45°, 60°, memorize the exact values from the unit circle—they appear frequently and give cleaner answers than decimal approximations!

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. The notation "cis" is shorthand: r cis θ means r(cos θ + i sin θ), so 4 cis 150° = 4(cos 150° + i sin 150°). To convert to rectangular form: (1) Identify r = 4 and θ = 150°. (2) Calculate real part: a = r cos θ = 4 cos 150° = 4(-√3/2) = -2√3. (3) Calculate imaginary part: b = r sin θ = 4 sin 150° = 4(1/2) = 2. (4) Write rectangular form: -2√3 + 2i. Choice A correctly applies the conversion formulas using cos 150° = -√3/2 (negative in Quadrant 2) and sin 150° = 1/2 (positive in Quadrant 2). Choice B incorrectly makes the imaginary part negative; Choice D swaps the coefficients, confusing which trig value goes where—remember cos 150° = -cos 30° = -√3/2 and sin 150° = sin 30° = 1/2. Polar to rectangular recipe: From r cis θ or r(cos θ + i sin θ), calculate a = r cos θ and b = r sin θ, then write a + bi. For angles like 150° = 180° - 30°, use reference angle relationships: cos 150° = -cos 30° and sin 150° = sin 30°, making calculations easier with special angle 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 FROM rectangular TO polar: find r = √(a² + b²) using Pythagorean theorem, then find θ = arctan(b/a) BUT adjust for quadrant (arctan only gives reference angle!). For 3 - 3i: (1) Find modulus: r = √(3² + (-3)²) = √(9 + 9) = √18 = 3√2. (2) Find argument: reference angle = arctan(|-3|/|3|) = arctan(1) = 45°. Since a = 3 > 0 and b = -3 < 0, we're in Quadrant 4, so θ = 360° - 45° = 315°. (3) Write polar form: 3√2(cos 315° + i sin 315°). Choice B correctly calculates modulus using Pythagorean theorem and determines argument with proper quadrant adjustment for Quadrant 4. Choice A would be correct if the complex number were 3 + 3i (Quadrant 1), but fails to recognize that negative imaginary part places us in Quadrant 4, requiring the 360° - reference angle adjustment. Rectangular to polar recipe: (1) Calculate r = √(a² + b²) (always positive). (2) Calculate reference angle = arctan(|b|/|a|). (3) Determine quadrant from signs of a and b: Q1 (++), Q2 (-+), Q3 (--), Q4 (+-). (4) Adjust angle: Q1: θ = reference; Q2: θ = 180° - reference; Q3: θ = 180° + reference; Q4: θ = 360° - reference. For 3 - 3i, we get r = 3√2 and θ = 315°, giving 3√2(cos 315° + i sin 315°).

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 + 2i to polar form: (1) Find modulus: r = √((-2)² + 2²) = √(4 + 4) = √8 = 2√2. (2) Find argument: reference angle = arctan(|2|/|-2|) = arctan(1) = 45°. Since a = -2 < 0 and b = 2 > 0, we're in Quadrant 2, so θ = 180° - 45° = 135°. (3) Write polar form: 2√2(cos 135° + i sin 135°). Choice B correctly calculates the modulus as 2√2 and determines the argument as 135° with proper quadrant adjustment for Quadrant 2. Choice A incorrectly places the complex number in Quadrant 1 with θ = 45°, forgetting that negative real part means we're in Quadrant 2 or 3; Choice D uses -45° which would be for 2 - 2i in Quadrant 4. Rectangular to polar recipe: Calculate r = √(a² + b²), find reference angle, determine quadrant from signs (Q2 for -+), adjust angle (Q2: θ = 180° - reference), then write r(cos θ + i sin θ). Always check your quadrant—the signs of a and b tell you where your complex number lives on the complex plane!

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: reference angle = arctan(4/3) ≈ 53.13°. Since a = 3 > 0 and b = 4 > 0, we're in Quadrant 1, so θ = 53.13° (no adjustment needed for Q1). (3) Write polar form: 5(cos 53.13° + i sin 53.13°). Choice A correctly calculates modulus using Pythagorean theorem and determines argument with proper quadrant consideration. Choice B incorrectly calculates r = 3 + 4 = 7 instead of using the Pythagorean theorem—remember, modulus is the distance from origin, not the sum of components! Rectangular to polar recipe: (1) Calculate r = √(a² + b²) (always positive). (2) Calculate reference angle = arctan(|b|/|a|). (3) Determine quadrant from signs of a and b, then adjust angle accordingly. For 3 + 4i in Q1, no adjustment needed, giving 5(cos 53.13° + i sin 53.13°).

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). To convert from polar to rectangular: use $a = r \cos \theta$ and $b = r \sin \theta$—both forms describe the same point, just different perspectives! For $4(\cos 120^\circ + i \sin 120^\circ)$, $a = 4 \cos 120^\circ = 4(-1/2) = -2$, $b = 4 \sin 120^\circ = 4(\sqrt{3}/2) = 2\sqrt{3}$, giving $-2 + 2\sqrt{3} i$—note $120^\circ$ is in quadrant 2, where cos is negative and sin is positive. Choice B correctly applies $a = r \cos \theta$ and $b = r \sin \theta$ with trig values for $120^\circ$ ($\cos 120^\circ = -1/2$, $\sin 120^\circ = \sqrt{3}/2$). Choice A omits the negative sign from $\cos 120^\circ$, forgetting quadrant effects—always evaluate trig functions based on the angle's position! Polar to rectangular recipe: (1) Identify $r$ and $\theta$, (2) $a = r \cos \theta$, (3) $b = r \sin \theta$, (4) Write $a + bi$—for angles like $120^\circ$, use reference $60^\circ$ and adjust signs—you're getting stronger with each one!

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 1 - √3i to polar form: (1) Find modulus: r = √(1² + (-√3)²) = √(1 + 3) = √4 = 2. (2) Find argument: reference angle = arctan(|-√3|/|1|) = arctan(√3) = 60°. Since a = 1 > 0 and b = -√3 < 0, we're in Quadrant 4, so θ = 360° - 60° = 300° (or equivalently -60°). (3) Write polar form: 2(cos 300° + i sin 300°). Choice A correctly calculates the modulus as 2 and determines the argument as 300° with proper quadrant adjustment for Quadrant 4. Choice D also correctly represents the same angle as -60°, which is coterminal with 300°; both are valid! Choice B incorrectly uses 60°, which would be for 1 + √3i in Quadrant 1. Rectangular to polar recipe: Calculate r = √(a² + b²), find reference angle, determine quadrant from signs (Q4 for +-), adjust angle (Q4: θ = 360° - reference or use negative angle), then write r(cos θ + i sin θ). In Quadrant 4, you can use either positive angles (300° to 360°) or negative angles (-60° to 0°)—they represent the same direction!

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 - 3i to polar form: (1) Find modulus: r = √((-3)² + (-3)²) = √(9 + 9) = √18 = 3√2. (2) Find argument: reference angle = arctan(|-3|/|-3|) = arctan(1) = 45°. Since a = -3 < 0 and b = -3 < 0, we're in Quadrant 3, so θ = 180° + 45° = 225°. (3) Write polar form: 3√2(cos 225° + i sin 225°). Choice C correctly calculates the modulus as 3√2 and determines the argument as 225° with proper quadrant adjustment for Quadrant 3. Choice A incorrectly uses 45° for Quadrant 1; Choice B has the wrong modulus of 6 instead of 3√2; Choice D uses 135° which would be for -3 + 3i in Quadrant 2. Rectangular to polar recipe: Calculate r = √(a² + b²), find reference angle, determine quadrant from signs (Q3 for --), adjust angle (Q3: θ = 180° + reference), then write r(cos θ + i sin θ). When both components have the same absolute value, the reference angle is always 45°—the quadrant adjustment determines the final angle!