Trigonometry & Inverse Trigonometry
General
Grade 12
Question:
<p>If <span class="math-inline">\(\cos^{-1}x+\cos^{-1}y+\cos^{-1}z=3\pi\)</span>, then <span class="math-inline">\(xy+yz+zx=\)</span></p>
1
x^2 + y^2 + z^2
xyz
x + y + z
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>Each <span class="math-inline">$\cos^{-1}$</span> term has max value <span class="math-inline">$\pi$</span>. Sum = <span class="math-inline">$3\pi$</span> forces each = <span class="math-inline">$\pi$</span>, so <span class="math-inline">$x=y=z=-1$</span>. Then <span class="math-inline">$xy+yz+zx=3$</span>.</p><div class="key-concept"><strong>Key Concept:</strong> Bounded sum at maximum forces every term to its maximum</div></div>
Correct Answer: 3