Complex Numbers in Rectangular, Polar Form - Algebra 2

Convert the complex number $1-\sqrt{3}i$ from rectangular form $a+bi$ to polar form $r(\cos\theta+i\sin\theta)$ with $\theta$ in degrees. Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\left(\frac{b}{a}\right)$ with quadrant adjustment. (Argument is measured from the positive real axis.)
Answer: $2\left(\cos 300^\circ+i\sin 300^\circ\right)$ 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 -60°). (3) Write polar form: 2(cos 300° + i sin 300°). Choice B correctly calculates modulus using Pythagorean theorem and determines argument with proper quadrant adjustment for Q4. Choice A incorrectly places the complex number in Q1 with θ = 60°, but 1 - √3i has negative imaginary part, placing it in Q4 where θ = 300°! Rectangular to polar recipe: (1) Calculate r = √(a² + b²). (2) Find reference angle. (3) Determine quadrant from signs. (4) For Q4 (+-), use θ = 360° - reference. For 1 - √3i, this gives 2(cos 300° + i sin 300°).
If $z_1 = \sqrt{3} + i$ and $z_2 = 1 + \sqrt{3}i$ , and both numbers are converted to polar form $r(\cos\theta + i\sin\theta)$ with $\theta \in [0, 2\pi)$ , what is the relationship between their arguments $\theta_1$ and $\theta_2$ ?
Answer: $\theta_2 = \theta_1 + \frac{\pi}{6}$ When working with complex numbers in polar form, you need to convert from rectangular form $a + bi$ to polar form $r(\cos\theta + i\sin\theta)$ by finding the modulus $r = \sqrt{a^2 + b^2}$ and argument $\theta = \arctan\left(\frac{b}{a}\right)$ , adjusted for the correct quadrant. For $z_1 = \sqrt{3} + i$ : The modulus is $r_1 = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{4} = 2$ . Since both components are positive (first quadrant), $\theta_1 = \arctan\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}$ . For $z_2 = 1 + \sqrt{3}i$ : The modulus is $r_2 = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2$ . Again in the first quadrant, $\theta_2 = \arctan\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}$ . The relationship is $\theta_2 = \frac{\pi}{3} = \frac{\pi}{6} + \frac{\pi}{6} = \theta_1 + \frac{\pi}{6}$ , confirming answer D. Looking at the wrong answers: A) $\theta_2 = 2\theta_1$ would give $\frac{\pi}{3} = 2 \cdot \frac{\pi}{6} = \frac{\pi}{3}$ , which seems right but misses that this is coincidental—the relationship $\theta_1 + \frac{\pi}{6}$ is the fundamental pattern. B) $\theta_1 + \frac{\pi}{3} = \frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{2} \neq \frac{\pi}{3}$ . C) $\theta_1 + \frac{\pi}{2} = \frac{\pi}{6} + \frac{\pi}{2} = \frac{2\pi}{3} \neq \frac{\pi}{3}$ . Remember: always verify your arctangent calculations by checking which quadrant your complex number lies in, and watch for coincidental relationships that might mask the true pattern.
Convert $3\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)$ to rectangular form $a+bi$ . Use $a=r\cos\theta$ and $b=r\sin\theta$ .
Answer: $\frac{3}{2}+\frac{3\sqrt{3}}{2}i$ 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. Polar form r(cos θ + i sin θ) uses distance from origin (modulus r) and angle from positive real axis (argument θ), while rectangular form a + bi uses horizontal (real) and vertical (imaginary) components. To convert 3(cos(π/3) + i sin(π/3)) to rectangular: a = r cos θ = 3 cos(π/3) = 3(1/2) = 3/2, and b = r sin θ = 3 sin(π/3) = 3(√3/2) = 3√3/2, giving 3/2 + (3√3/2)i. Choice B correctly applies the conversion formulas a = r cos θ and b = r sin θ with the special angle values cos(π/3) = 1/2 and sin(π/3) = √3/2. Choice A swaps the trig values, using cos(π/3) = √3/2 and sin(π/3) = 1/2, but π/3 = 60°, and from the 30-60-90 triangle, cos 60° = 1/2 and sin 60° = √3/2! Polar to rectangular recipe: (1) Identify r and θ. (2) Calculate a = r cos θ. (3) Calculate b = r sin θ. (4) Write a + bi. For π/3 radians (60°), remember cos(π/3) = 1/2 and sin(π/3) = √3/2, giving 3(cos(π/3) + i sin(π/3)) = 3/2 + (3√3/2)i.
Convert $3\left(\cos 120^\circ+i\sin 120^\circ\right)$ to rectangular form $a+bi$ using $a=r\cos\theta$ and $b=r\sin\theta$ .
Answer: $-\frac{3}{2}+\frac{3\sqrt{3}}{2}i$ 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 polar TO rectangular: use a = r cos θ and b = r sin θ. For 3(cos 120° + i sin 120°): (1) Identify r = 3 and θ = 120°. (2) Calculate real part: a = 3 cos 120° = 3(-1/2) = -3/2. (3) Calculate imaginary part: b = 3 sin 120° = 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 with the special angle values cos 120° = -1/2 and sin 120° = √3/2. Choice C swaps the trig values: using cos 120° = -√3/2 and sin 120° = 1/2, but these are actually the values for 150°! Remember: 120° = 180° - 60°, so cos 120° = -cos 60° = -1/2 and sin 120° = sin 60° = √3/2. Use reference angles! Polar to rectangular recipe: (1) From r(cos θ + i sin θ), identify r and θ. (2) Calculate a = r cos θ. (3) Calculate b = r sin θ. (4) Write a + bi. For 120°, think reference angle: 120° is in Quadrant 2, reference angle is 60°. In Q2, cosine is negative, sine is positive: cos 120° = -cos 60° = -1/2, sin 120° = sin 60° = √3/2.
Convert the complex number $-\sqrt{3}-i$ from rectangular form $a+bi$ to polar form $r(\cos\theta+i\sin\theta)$ , with $\theta$ in degrees. Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan(b/a)$ with quadrant adjustment (argument measured from the positive real axis).
Answer: $2\big(\cos 210^\circ+i\sin 210^\circ\big)$ 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!
A complex number in polar form is given as $z = 6\left(\cos\frac{11\pi}{6} + i\sin\frac{11\pi}{6}\right)$ . When $z$ is written in rectangular form $a + bi$ and then converted back to polar form using the principal argument in $(-\pi, \pi]$ , what is the new polar representation?
Answer: $6\left(\cos\left(-\frac{\pi}{6}\right) + i\sin\left(-\frac{\pi}{6}\right)\right)$ First, convert to rectangular form: $z = 6\cos\frac{11\pi}{6} + 6i\sin\frac{11\pi}{6} = 6 \cdot \frac{\sqrt{3}}{2} + 6i \cdot \left(-\frac{1}{2}\right) = 3\sqrt{3} - 3i$ . When converting back to polar form with principal argument in $(-\pi, \pi]$ , we need $\theta$ such that $\tan\theta = \frac{-3}{3\sqrt{3}} = -\frac{1}{\sqrt{3}}$ . Since the number is in the fourth quadrant, $\theta = -\frac{\pi}{6}$ . Note that $\frac{11\pi}{6} = 2\pi - \frac{\pi}{6}$ , so $\frac{11\pi}{6}$ and $-\frac{\pi}{6}$ are coterminal angles. Choice A keeps the original angle outside the principal range. Choice C uses the reference angle with wrong sign. Choice D places the number in the wrong quadrant.
Convert $\sqrt{2}\left(\cos 315^\circ+i\sin 315^\circ\right)$ to rectangular form $a+bi$ using $a=r\cos\theta$ and $b=r\sin\theta$ .
Answer: $1-i$ 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 polar TO rectangular: use a = r cos θ and b = r sin θ. For √2(cos 315° + i sin 315°): (1) Identify r = √2 and θ = 315°. (2) Calculate real part: a = √2 cos 315° = √2(√2/2) = 2/2 = 1. (3) Calculate imaginary part: b = √2 sin 315° = √2(-√2/2) = -2/2 = -1. (4) Write rectangular form: 1 + (-1)i = 1 - i. Choice B correctly applies the conversion formulas with the special angle values cos 315° = √2/2 and sin 315° = -√2/2. Choice A would be correct for √2(cos 45° + i sin 45°) = 1 + i, confusing 315° (Quadrant 4) with 45° (Quadrant 1). Remember: 315° = 360° - 45°, so it's in Q4 where cosine is positive but sine is negative! Polar to rectangular recipe: (1) Recognize 315° = 360° - 45°, which is in Quadrant 4. (2) In Q4, cos 315° = cos 45° = √2/2 (positive), sin 315° = -sin 45° = -√2/2 (negative). (3) Multiply by r = √2: a = √2(√2/2) = 1, b = √2(-√2/2) = -1. The angle 315° points down and right at 45° below the positive x-axis, giving positive real and negative imaginary parts!
Convert the rectangular form $-4-4i$ to polar form $r(\cos\theta+i\sin\theta)$ with $\theta$ in degrees. Use $r=\sqrt{a^2+b^2}$ and determine the argument as the angle from the positive real axis (counterclockwise), adjusting the quadrant.
Answer: $4\sqrt{2}\left(\cos 225^\circ+i\sin 225^\circ\right)$ 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 -4 - 4i: (1) Find modulus: r = √((-4)² + (-4)²) = √(16 + 16) = √32 = 4√2. (2) Find argument: reference angle = arctan(|-4|/|-4|) = arctan(1) = 45°. Since a = -4 < 0 and b = -4 < 0, we're in Quadrant 3, so θ = 180° + 45° = 225°. (3) Write polar form: 4√2(cos 225° + i sin 225°). Choice C correctly calculates modulus using Pythagorean theorem and determines argument with proper quadrant adjustment for Quadrant 3. Choice D incorrectly calculates the modulus as 8 instead of 4√2, perhaps adding 4 + 4 instead of using √(16 + 16) = √32 = 4√2. Remember: modulus is always the distance formula, not simple addition! 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 -4 - 4i in Quadrant 3, we get r = 4√2 and θ = 225°!
Convert $5\left(\cos 30^\circ+i\sin 30^\circ\right)$ to rectangular form $a+bi$ using $a=r\cos\theta$ and $b=r\sin\theta$ .
Answer: $\frac{5\sqrt{3}}{2}+\frac{5}{2}i$ 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 polar TO rectangular: use a = r cos θ and b = r sin θ. For 5(cos 30° + i sin 30°): (1) Identify r = 5 and θ = 30°. (2) Calculate real part: a = 5 cos 30° = 5(√3/2) = 5√3/2. (3) Calculate imaginary part: b = 5 sin 30° = 5(1/2) = 5/2. (4) Write rectangular form: (5√3/2) + (5/2)i. Choice A correctly applies the conversion formulas a = r cos θ and b = r sin θ with the special angle values cos 30° = √3/2 and sin 30° = 1/2. Choice B swaps the conversion formulas: using a = r sin θ and b = r cos θ instead of a = r cos θ and b = r sin θ. Remember: cos goes with real (horizontal), sin goes with imaginary (vertical). The x-coordinate uses cosine, y-coordinate uses sine, just like parametric equations! Polar to rectangular recipe: (1) From r(cos θ + i sin θ), identify r and θ. (2) Calculate a = r cos θ (use calculator or special angle knowledge). (3) Calculate b = r sin θ. (4) Write a + bi. For special angles like 30°, memorize: cos 30° = √3/2, sin 30° = 1/2 from the 30-60-90 triangle. This gives exact values without calculator!
Convert $4\,\text{cis}\,150^\circ$ to rectangular form $a+bi$ . (Recall $r\,\text{cis}\,\theta=r(\cos\theta+i\sin\theta)$ and use $a=r\cos\theta$ , $b=r\sin\theta$ .)
Answer: $-2\sqrt{3}+2i$ 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!
Convert the complex number $-4+0i$ to polar form $r(\cos\theta+i\sin\theta)$ with $\theta$ in degrees. Use $r=\sqrt{a^2+b^2}$ and choose an argument measured counterclockwise from the positive real axis.
Answer: $4\left(\cos 180^\circ+i\sin 180^\circ\right)$ 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 -4 + 0i to polar form: (1) Find modulus: r = √((-4)² + 0²) = √16 = 4. (2) Find argument: the complex number -4 + 0i lies on the negative real axis, so θ = 180°. (3) Write polar form: 4(cos 180° + i sin 180°). Choice C correctly identifies that -4 + 0i has modulus 4 and lies on the negative real axis at angle 180° from the positive real axis. Choice A incorrectly places it at 0° (positive real axis), but -4 is negative, so it's at 180°! Choice D places it at 270° (negative imaginary axis), but there's no imaginary component. Rectangular to polar recipe for real numbers: positive real numbers have θ = 0°, negative real numbers have θ = 180°, positive imaginary numbers have θ = 90°, negative imaginary numbers have θ = 270°. For -4 + 0i, a negative real number, θ = 180°, giving 4(cos 180° + i sin 180°).
Convert the rectangular complex number $1-\sqrt{3}i$ to polar form $r(\cos\theta+i\sin\theta)$ . Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\!\left(\frac{b}{a}\right)$ with quadrant adjustment. (Angle in degrees.)
Answer: $2\left(\cos 300^\circ+i\sin 300^\circ\right)$ 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!
Convert the complex number $3+4i$ from rectangular form $a+bi$ to polar form $r(\cos\theta+i\sin\theta)$ . Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan(b/a)$ (with quadrant adjustment). The argument $\theta$ is measured counterclockwise from the positive real axis, in degrees.
Answer: $5\big(\cos 53.13^\circ+i\sin 53.13^\circ\big)$ 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!
Convert to rectangular form $a+bi$ : $4\left(\cos 120^\circ + i\sin 120^\circ\right).$ Use $a=r\cos\theta$ and $b=r\sin\theta$ .
Answer: $-2 + 2\sqrt{3}\,i$ 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!
Convert $5\big(\cos 30^\circ+i\sin 30^\circ\big)$ from polar form to rectangular form $a+bi$ . Use $a=r\cos\theta$ and $b=r\sin\theta$ . (The argument is the angle from the positive real axis.)
Answer: $\frac{5\sqrt{3}}{2}+\frac{5}{2}i$ 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!
Find the modulus $r$ and a correct argument $\theta$ (in degrees) for $z=-\sqrt{3}-i$ , then write $z$ in polar form $r(\cos\theta+i\sin\theta)$ . Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\!\left(\frac{b}{a}\right)$ with quadrant adjustment.
Answer: $2\left(\cos 210^\circ+i\sin 210^\circ\right)$ 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 - i to polar form: (1) Find modulus: r = √((-√3)² + (-1)²) = √(3 + 1) = √4 = 2. (2) Find argument: reference angle = arctan(|-1|/|-√3|) = arctan(1/√3) = 30°. Since a = -√3 < 0 and b = -1 < 0, we're in Quadrant 3, so θ = 180° + 30° = 210°. (3) Write polar form: 2(cos 210° + i sin 210°). Choice A correctly calculates the modulus as 2 and determines the argument as 210° with proper quadrant adjustment for Quadrant 3. Choice B incorrectly uses 30°, which would be the angle for √3 + i in Quadrant 1; Choice D uses 150° which would be for -√3 + i in Quadrant 2. Rectangular to polar recipe: Calculate r = √(a² + b²), find reference angle using arctan(|b|/|a|), determine quadrant from signs (Q3 for --), adjust angle (Q3: θ = 180° + reference), then write r(cos θ + i sin θ). The reference angle arctan(1/√3) = 30° is a special angle—recognizing these patterns helps you work with exact values instead of decimal approximations!
The complex number $w = 2\left(\cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6}\right)$ can be written in rectangular form as $a + bi$ . What is the value of $a - b$ ?
Answer: $-\sqrt{3} - 1$ Converting to rectangular form: $w = 2\cos\frac{5\pi}{6} + 2i\sin\frac{5\pi}{6} = 2\left(-\frac{\sqrt{3}}{2}\right) + 2i\left(\frac{1}{2}\right) = -\sqrt{3} + i$ . So $a = -\sqrt{3}$ and $b = 1$ , giving $a - b = -\sqrt{3} - 1$ . Choice B incorrectly adds instead of subtracting. Choice C uses wrong signs for the trigonometric values. Choice D forgets to include the imaginary part in the calculation.
Find the modulus $r$ and argument $\theta$ (in degrees) of the complex number $3 - 3i$ . Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\left(\frac{b}{a}\right)$ with quadrant adjustment (argument is the angle from the positive real axis). Which polar form is correct?
Answer: $3\sqrt{2}\left(\cos 315^\circ + i\sin 315^\circ\right)$ 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 3 - 3i, r = sqrt(9 + 9) = sqrt(18) = 3√2, reference angle = arctan(3/3) = 45°, and since a > 0 and b < 0 (quadrant 4), θ = 360° - 45° = 315°, so polar form is 3√2(cos 315° + i sin 315°). Choice A correctly calculates the modulus and adjusts the argument for quadrant 4 using the formulas accurately. Choice B uses θ = 45° without adjustment, placing it in quadrant 1 instead of 4—always check signs of a and b to determine the quadrant and adjust θ accordingly! Rectangular to polar recipe: (1) r = sqrt(a² + b²), (2) Reference = arctan(|b|/|a|), (3) For Q4: θ = 360° - reference, then write the form—practice with quadrants, you're doing awesome!
Convert $2\,\text{cis}\,45^\circ$ to rectangular form $a+bi$ . (Recall $r\,\text{cis}\,\theta=r(\cos\theta+i\sin\theta)$ and $a=r\cos\theta$ , $b=r\sin\theta$ .)
Answer: $\sqrt{2}+\sqrt{2}i$ 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 = r(cos θ + i sin θ), combining cosine and i sine into one compact form. To convert 2 cis 45° to rectangular: a = r cos θ = 2 cos 45° = 2(√2/2) = √2, and b = r sin θ = 2 sin 45° = 2(√2/2) = √2, giving √2 + √2i. Choice A correctly applies the conversion formulas a = r cos θ and b = r sin θ with the special angle values cos 45° = sin 45° = √2/2 from the 45-45-90 triangle. Choice B makes the common error of thinking cos 45° = sin 45° = 1, but these values equal √2/2 ≈ 0.707, not 1—the 45-45-90 triangle has sides 1, 1, √2, so cos 45° = adjacent/hypotenuse = 1/√2 = √2/2! Polar to rectangular recipe: (1) From r cis θ or r(cos θ + i sin θ), identify r and θ. (2) Calculate a = r cos θ. (3) Calculate b = r sin θ. (4) Write a + bi. For 45°, remember cos 45° = sin 45° = √2/2, giving 2 cis 45° = √2 + √2i.
Convert $\sqrt{2}\big(\cos 45^\circ+i\sin 45^\circ\big)$ from polar form to rectangular form $a+bi$ . Use $a=r\cos\theta$ and $b=r\sin\theta$ .
Answer: $1+i$ 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!

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