<p>Let <em>x + y + z = θ</em> and <em>k = 2</em>. If <br/> \(\cos x + \cos y + \cos z = k\cos\theta\) and \(\sin x + \sin y + \sin z = k\sin\theta\), <br/> find the value of \(\cos(x+y) + \cos(y+z) + \cos(z+x)\).</p>
Step-by-Step Solution
Key Concept: Use the constraint equations to establish relationships between the angles, then leverage x + y + z = θ to express products like cos(x+y) in terms of known quantities. The key is recognizing that cos(x+y) = cos(θ - z), allowing you to rewrite the sum using the given conditions.
<p><strong>Step 1:</strong> Given: x + y + z = θ, cos x + cos y + cos z = 2cos θ, sin x + sin y + sin z = 2sin θ</p><p><strong>Step 2:</strong> Since x + y + z = θ, we have x + y = θ - z, y + z = θ - x, and z + x = θ - y</p><p><strong>Step 3:</strong> Therefore:<br>cos(x+y) + cos(y+z) + cos(z+x) = cos(θ-z) + cos(θ-x) + cos(θ-y)</p><p><strong>Step 4:</strong> Using cos(θ - A) = cos θ cos A + sin θ sin A:<br>= cos θ(cos x + cos y + cos z) + sin θ(sin x + sin y + sin z)</p><p><strong>Step 5:</strong> Substitute the given conditions:<br>= cos θ · (2cos θ) + sin θ · (2sin θ)<br>= 2cos²θ + 2sin²θ<br>= 2(cos²θ + sin²θ)<br>= 2(1)<br>= <strong>2</strong></p><p>∴ Answer: C</p>
Correct Answer: C