How do you find the product of the complex numbers z1 = 8cos(40°) + i*sin(40°) and z2 = 4cos(135°) + i*sin(135°)?

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To find the product of the complex numbers z1 = 8(cos(40°) + i sin(40°) and z2 = 4(cos(135°) + i sin(135°), we can follow a systematic approach using properties of complex numbers.

Step 1: Understand the forms

Both z1 and z2 are expressed in their polar forms, where:

  • Magnitude (or modulus) of z1 = 8, phase (or argument) of z1 = 40°
  • Magnitude of z2 = 4, phase of z2 = 135°

Step 2: Use the formula for multiplication

When multiplying two complex numbers in polar form, the moduli multiply and the arguments add:

  • |z1| × |z2| = 8 × 4 = 32
  • arg(z1) + arg(z2) = 40° + 135° = 175°

Step 3: Write the result in polar form

Thus, the product z1z2 can be expressed as:

z1z2 = 32(cos(175°) + i sin(175°))

Step 4: Convert to rectangular form if necessary

To express this in rectangular coordinates (a + bi), we use the trigonometric values:

  • cos(175°) ≈ -0.9962
  • sin(175°) ≈ 0.0872

So, we multiply:

  • Real part: 32 × -0.9962 ≈ -31.9994
  • Imaginary part: 32 × 0.0872 ≈ 2.7904

Putting it all together, we get:

z1z2 ≈ -32 + 2.79i

Final Answer

The product of z1 and z2 is approximately:

z1z2 = 32(cos(175°) + i sin(175°)), or z1z2 ≈ -32 + 2.79i

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