Final answer:
The correct answer is option d) α = 81° and β = -36°.
Step-by-step explanation:
The correct answer is option d) α = 81° and β = -36°.
To find the values of α and β, we can use the identities cos(a-b) = cos(a)cos(b) + sin(a)sin(b) and sin(a+b) = sin(a)cos(b) + cos(a)sin(b).
Using the given values of cos(81°) = 0.123 and sin(81°) = 0.993, and cos(36°) = 0.809 and sin(36°) = 0.588, we can substitute these values into the identities to find that:
cos(81°)cos(36°) + sin(81°)sin(36°) = 0.123 * 0.809 + 0.993 * 0.588 ≈ 0.099 + 0.583 ≈ 0.682
Therefore, the values of α and β that make the equation true are α = 81° and β = -36°.
The expression given can be simplified using the cosine addition formula: cos(α - β) = cos(α)cos(β) + sin(α)sin(β). Applying this formula, we get cos(81° - 36°) = cos(81°)cos(36°) + sin(81°)sin(36°), which simplifies to cos(45°). Therefore, the values of α and β that make the equality true correspond to option (a).