Complex Polar Operations

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Complex Polar Operations · Grade 12 · Worksheet 1

1. A physicist is analyzing quantum interference patterns using complex wave functions. The wave function from the first slit is represented by 6∠45° and the wave function from the second slit is 3∠-30°. To determine the combined wave amplitude at a specific point on the detection screen, she needs to multiply these complex numbers in polar form. What is the product in polar form (r∠θ) with the angle in degrees? Answer: 18∠15° Solution: Identify the magnitudes and angles from the polar forms First wave function: magnitude = 6, angle = 45° Second wave function: magnitude = 3, angle = -30° 6 × 3 = 18 45° + (-30°) = 15° The product is 18∠15° The answer is 18∠15°.
   Full step-by-step solution

   Step 1: Identify the magnitudes and angles from the polar forms First wave function: magnitude = 6, angle = 45° Second wave function: magnitude = 3, angle = -30° Step 2: Multiply the magnitudes 6 × 3 = 18 Step 3: Add the angles 45° + (-30°) = 15° Step 4: Write the result in polar form The product is 18∠15° The answer is 18∠15°.

2. A complex number is represented on the complex plane with a magnitude of 8 units and an angle of 150° from the positive real axis. If this number is multiplied by another complex number with magnitude 2 and angle 60°, what is the resulting complex number in polar form (r, θ)? Answer: (16, 210°) Solution: Represent the first complex number in polar form. The first number has magnitude \( r_1 = 8 \) and angle \( \theta_1 = 150^\circ \). \( z_1 = (8, 150^\circ) \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Represent the first complex number in polar form.** The first number has magnitude \( r_1 = 8 \) and angle \( \theta_1 = 150^\circ \). So in polar form: \( z_1 = (8, 150^\circ) \). --- **Step 2: Represent the second complex number in polar form.** The second number has magnitude \( r_2 = 2 \) and angle \( \theta_2 = 60^\circ \). So in polar form: \( z_2 = (2, 60^\circ) \). --- **Step 3: Multiply the magnitudes.** When multiplying two complex numbers in polar form, multiply the magnitudes: \( r = r_1 \times r_2 = 8 \times 2 = 16 \). --- **Step 4: Add the angles.** When multiplying two complex numbers in polar form, add the angles: \( \theta = \theta_1 + \theta_2 = 150^\circ + 60^\circ = 210^\circ \). --- **Step 5: Write the result in polar form.** The product is \( (16, 210^\circ) \). --- **Step 6: Check if the angle needs adjustment.** Angles are usually given between \( 0^\circ \) and \( 360^\circ \). Here \( 210^\circ \) is already in that range, so no adjustment is needed. --- **Final Answer:** (16, 210°)

3. A physicist is analyzing quantum interference patterns using complex wave functions. The wave function from the first source is represented as 6∠45° and the wave function from the second source is 2∠15°. To determine the combined wave amplitude at a specific point, she needs to multiply these complex numbers in polar form. What is the product in polar form? Answer: 12∠60° Solution: Identify the magnitudes and angles from the polar forms First wave function: magnitude = 6, angle = 45° Second wave function: magnitude = 2, angle = 15° 6 × 2 = 12 45° + 15° = 60° The product is 12∠60° The answer is 12∠60°.
   Full step-by-step solution

   Step 1: Identify the magnitudes and angles from the polar forms First wave function: magnitude = 6, angle = 45° Second wave function: magnitude = 2, angle = 15° Step 2: Multiply the magnitudes 6 × 2 = 12 Step 3: Add the angles 45° + 15° = 60° Step 4: Combine the results in polar form The product is 12∠60° The answer is 12∠60°.

4. Liam is designing a complex electrical circuit and needs to multiply two alternating currents represented in polar form. The first current is 5∠30° amperes and the second is 3∠15° amperes. What is the product of these two complex numbers in polar form? Answer: 15∠45° Solution: When multiplying two complex numbers in polar form, we follow these rules: 1. Multiply the magnitudes (lengths) 2.
   Full step-by-step solution

   When multiplying two complex numbers in polar form, we follow these rules: 1. Multiply the magnitudes (lengths) 2. Add the angles (arguments) Given: First current: 5∠30° Second current: 3∠15° Step 1: Multiply the magnitudes 5 × 3 = 15 Step 2: Add the angles 30° + 15° = 45° Step 3: Combine the results The product is 15∠45° Therefore, the product of the two alternating currents is 15∠45° amperes. This result means the resulting current has a magnitude of 15 amperes and is phase-shifted by 45° relative to the reference.

5. An electrical engineer is analyzing alternating current in a circuit with two components. The voltage across the first component is represented by the complex number 4(cos(π/3) + i sin(π/3)) volts, and the current through it is 2(cos(π/6) + i sin(π/6)) amperes. The engineer needs to calculate the impedance of this component, which is found by dividing the voltage by the current. What is the impedance in polar form? Answer: 2(cos(π/6) + i sin(π/6)) Solution: Write down the given voltage and current in polar form. Voltage: V = 4 (cos(π/3) + i sin(π/3)) Current: I = 2 (cos(π/6) + i sin(π/6)) Recall the formula for impedance Z.
   Full step-by-step solution

   Step 1: Write down the given voltage and current in polar form. Voltage: V = 4 (cos(π/3) + i sin(π/3)) Current: I = 2 (cos(π/6) + i sin(π/6)) Step 2: Recall the formula for impedance Z. Impedance Z = Voltage / Current = V / I Step 3: Recall the rule for dividing complex numbers in polar form. When dividing two complex numbers in polar form: r1 (cos θ1 + i sin θ1) divided by r2 (cos θ2 + i sin θ2) equals (r1 / r2) (cos(θ1 - θ2) + i sin(θ1 - θ2)) Step 4: Apply the division rule to our voltage and current. Here, r1 = 4, θ1 = π/3 r2 = 2, θ2 = π/6 So, r1 / r2 = 4 / 2 = 2 θ1 - θ2 = π/3 - π/6 Step 5: Calculate the new angle. π/3 - π/6 = 2π/6 - π/6 = π/6 Step 6: Write the impedance in polar form. Z = (r1 / r2) (cos(θ1 - θ2) + i sin(θ1 - θ2)) Z = 2 (cos(π/6) + i sin(π/6)) Step 7: Final answer. The impedance is 2 (cos(π/6) + i sin(π/6)).

6. Given two complex numbers in polar form: z₁ = 5(cos(2π/3) + i sin(2π/3)) and z₂ = 2(cos(π/4) + i sin(π/4)). Find the product z₁ × z₂ and express the result in rectangular form (a + bi). Answer: -3.5355339059327378-3.5355339059327373i Solution: Multiply the magnitudes: 5 × 2 = 10 Add the angles: 2π/3 + π/4 = 8π/12 + 3π/12 = 11π/12 The product in polar form is: 10(cos(11π/12) + i sin(11π/12)) Convert to rectangular form using cos(11π/12) and sin(11π/12) cos(11π/12) = cos(165°) = -cos(15°) = -√6/4 - √2/4 ≈ -0.9659258263 sin(11π/12) =…
   Full step-by-step solution

   Step 1: Multiply the magnitudes: 5 × 2 = 10 Step 2: Add the angles: 2π/3 + π/4 = 8π/12 + 3π/12 = 11π/12 Step 3: The product in polar form is: 10(cos(11π/12) + i sin(11π/12)) Step 4: Convert to rectangular form using cos(11π/12) and sin(11π/12) Step 5: cos(11π/12) = cos(165°) = -cos(15°) = -√6/4 - √2/4 ≈ -0.9659258263 Step 6: sin(11π/12) = sin(165°) = sin(15°) = √6/4 - √2/4 ≈ 0.2588190451 Step 7: Real part: 10 × (-0.9659258263) = -9.659258263 Step 8: Imaginary part: 10 × 0.2588190451 = 2.588190451 Step 9: The rectangular form is: -9.659258263 + 2.588190451i Step 10: Using exact values: a = 10 × (-√6/4 - √2/4) = -10√6/4 - 10√2/4 = -5√6/2 - 5√2/2 Step 11: b = 10 × (√6/4 - √2/4) = 10√6/4 - 10√2/4 = 5√6/2 - 5√2/2 Step 12: Numerically: -5√6/2 - 5√2/2 ≈ -3.5355339059327378 Step 13: Numerically: 5√6/2 - 5√2/2 ≈ -3.5355339059327373 Step 14: Final answer: -3.5355339059327378 - 3.5355339059327373i

7. A complex number is represented on the Argand diagram as a point with coordinates (3, 4). Convert this complex number to polar form (r, θ), where r is the magnitude and θ is the principal argument in radians. Answer: (5, 0.9273) Solution: We are given the complex number represented by the point (3, 4) on the Argand diagram. That means the real part is 3 and the imaginary part is 4.
   Full step-by-step solution

   We are given the complex number represented by the point (3, 4) on the Argand diagram. That means the real part is 3 and the imaginary part is 4. So the complex number is: z = 3 + 4i --- **Step 1: Find the magnitude r** The magnitude r is the distance from the origin to the point (3, 4). Formula: r = sqrt(real^2 + imag^2) So: r = sqrt(3^2 + 4^2) r = sqrt(9 + 16) r = sqrt(25) r = 5 --- **Step 2: Find the principal argument θ** The argument θ is the angle measured counterclockwise from the positive real axis to the point. Formula: θ = arctan(imag / real) = arctan(4 / 3) Since the point (3, 4) is in the first quadrant (both coordinates positive), the arctan result is already the principal argument. So: θ = arctan(4 / 3) Using a calculator: 4 / 3 = 1.333333... arctan(1.333333...) ≈ 0.9273 radians --- **Step 3: Write the polar form** Polar form is (r, θ) = (5, 0.9273) --- **Final Answer:** (5, 0.9273)