To find the product of the complex numbers z1 and z2, we first write them in their respective forms:
z1 = 8cos(40°) + i sin(40°) and z2 = 4cos(135°) + i sin(135°).
Next, we calculate the values of z1 and z2:
- For z1:
- cos(40°) ≈ 0.7660, so 8cos(40°) ≈ 8 * 0.7660 ≈ 6.128.
- sin(40°) ≈ 0.6428, so isin(40°) ≈ 8 * 0.6428i ≈ 5.1424i.
- Thus, z1 ≈ 6.128 + 5.1424i.
- For z2:
- cos(135°) = -√2/2 ≈ -0.7071, so 4cos(135°) ≈ 4 * (-0.7071) ≈ -2.8284.
- sin(135°) = √2/2 ≈ 0.7071, so isin(135°) ≈ 4 * 0.7071i ≈ 2.8284i.
- Thus, z2 ≈ -2.8284 + 2.8284i.
To find the product of z1 and z2, we use the formula for multiplying complex numbers:
z1 * z2 = (a + bi)(c + di) = ac + adi + bci + bidi
So we compute:
- Real part: 6.128 * (-2.8284) + 5.1424 * 2.8284 = -17.3037 + 14.5500 = -2.7537.
- Imaginary part: 6.128 * 2.8284 + 5.1424 * (-2.8284) = 17.3037 – 14.5500 = 2.7537.
Thus, the product of z1 and z2 is:
z1 * z2 ≈ -2.7537 + 2.7537i.
This means that the final answer can be expressed as a complex number:
z1 * z2 = -2.7537 + 2.7537i.