Final answer:
To solve the expression i₁i₂ / i₁+i₂, multiply i₁ and i₂, then divide the product by the sum of i₁ and i₂. The final answer is 4 cos(377t+30).
Step-by-step explanation:
To solve for the expression i₁i₂ / (i₁+i₂) with given currents i₁ = 16 sin(377t-30°) and i₂ = 4 cos(377t+30°), we must first express i₂ in terms of sine using the identity cos(θ) = sin(90° + θ). Thus, i₂ = 4 sin(377t+30° + 90°) or i₂ = 4 sin(377t+120°). To solve the given expression, first multiply i₁ and i₂:
i₁ * i₂ = (16 sin(377t-30)) * (4 cos(377t+30))
Using the identity sin(A + B) = sinAcosB + cosAsinB, we can expand the expression:
i₁ * i₂ = 64 sin(377t-30)cos(377t+30)
Next, divide i₁ * i₂ by i₁ + i₂:
i₁ * i₂ / (i₁ + i₂) = 64 sin(377t-30)cos(377t+30) / (16 sin(377t-30) + 4 cos(377t+30))
Simplifying the expression, we get:
i₁ * i₂ / (i₁ + i₂) = 4 cos(377t+30)