Question

Recall from class that e^(iθ) = cos(θ) + isin(θ). Also, recall that complex exponentials have the same algebraic properties as real exponentials. For example, e^(i(θ₁ + θ₂)) = e^(iθ₁) * e^(iθ₂).

Take advantage of that fact to discover expressions for cos(θ₁ + θ₂) and sin(θ₁ + θ₂) as combinations of cosines and sines of θ₁ and θ₂.

Answer 1

Using Euler's formula, we can express cos(θ1 + θ2) as cos(θ1)cos(θ2) - sin(θ1)sin(θ2) and sin(θ1 + θ2) as sin(θ1)cos(θ2) + cos(θ1)sin(θ2), by equating the real and imaginary parts of the complex exponential expressions.

Step-by-step explanation:

To find expressions for cos(θ_1 + θ_2) and sin(θ_1 + θ_2) as combinations of cosines and sines of θ_1 and θ_2, we can use Euler's formula: e^(iθ) = cos(θ) + i sin(θ).

First, we write the complex exponential for the sum of two angles:

e^(i(θ_1 + θ_2))
= e^(iθ_1) * e^(iθ_2)
= (cos(θ_1) + i sin(θ_1)) * (cos(θ_2) + i sin(θ_2))

Expanding the right-hand side, we get:

(cos(θ1)cos(θ2) - sin(θ_1)sin(θ_2)) + i(sin(θ_1)cos(θ_2) + cos(θ_1)sin(θ_2))

Since the left-hand side, by Euler's formula, is also equal to cos(θ_1 + θ_2) + i sin(θ_1 + θ_2), we can identify the real part and the imaginary part of the left and right sides:

cos(θ1 + θ2) = cos(θ1)cos(θ2) - sin(θ1)sin(θ2)

sin(θ1 + θ2) = sin(θ1)cos(θ2) + cos(θ1)sin(θ2)

This gives us the desired expressions for the cosine and sine of the sum of two angles in terms of the cosines and sines of the individual angles.