Express z = 2[cos(120°) + i sin(120°)] in rectangular form.

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Multiple Choice

Express z = 2[cos(120°) + i sin(120°)] in rectangular form.

Converting from polar (trigonometric) form to rectangular form uses the relationships x = r cos θ and y = r sin θ, where z = r[cos θ + i sin θ] = x + iy. Here r = 2 and θ = 120°. Knowing cos 120° = -1/2 and sin 120° = √3/2, the real part is 2(-1/2) = -1 and the imaginary part is 2(√3/2) = √3. So z = -1 + i√3. This places z in the second quadrant, where cosine is negative and sine is positive, matching the given angle.

Converting from polar (trigonometric) form to rectangular form uses the relationships x = r cos θ and y = r sin θ, where z = r[cos θ + i sin θ] = x + iy. Here r = 2 and θ = 120°. Knowing cos 120° = -1/2 and sin 120° = √3/2, the real part is 2(-1/2) = -1 and the imaginary part is 2(√3/2) = √3. So z = -1 + i√3. This places z in the second quadrant, where cosine is negative and sine is positive, matching the given angle.

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