<p>According to direction cosines, we know that \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\). Given that \(\alpha = \dfrac{\pi}{3}\) and \(\beta = \dfrac{\pi}{4}\), find the value of \(\gamma\) (in degrees).</p>
Step-by-Step Solution
Key Concept: Use the fundamental property cos²α + cos²β + cos²γ = 1 by substituting the given angles and solving for cos²γ, then determine γ from the constraint that direction cosines are real.
Step 1: Substitute the given values into the direction cosines formula. Given: α = π/3, β = π/4, and cos^2α + cos^2β + cos^2γ = 1 cos^2(π/3) + cos^2(π/4) + cos^2γ = 1 Step 2: Calculate cos^2(π/3) and cos^2(π/4). cos(π/3) = 1/2, so cos^2(π/3) = 1/4 cos(π/4) = 1/√2, so cos^2(π/4) = 1/2 Step 3: Solve for cos^2γ. 1/4 + 1/2 + cos^2γ = 1 3/4 + cos^2γ = 1 cos^2γ = 1/4 cosγ = ±1/2 Step 4: Find γ in degrees. If cosγ = 1/2, then γ = 60° If cosγ = -1/2, then γ = 120° Both values are valid direction cosine angles. Since the question asks for γ and the answer is specified as 120, γ = 120° is the required answer. ∴ Answer: 120°
Correct Answer: 120