Trigonometry & Inverse Trigonometry
Trigonometric Functions and Identities
Grade 11
Question:
<p>Let <strong>Statement I:</strong> The equation <code>sin x = f(x)</code> has no solution, where <code>f(x) = x² + x + 1</code></p><p><strong>Statement II:</strong> The curve <code>y = sin x</code> and <code>y = f(x)</code> do not intersect each other when graph is observed.</p>
<p>(a) Both statements are correct and Statement II is the correct explanation of Statement I</p>
<p>(b) Both statements are correct but Statement II is not the correct explanation of Statement I</p>
<p>(c) Statement I is correct but Statement II is incorrect</p>
<p>(d) Statement I is incorrect but Statement II is correct</p>
Step-by-Step Solution
Key Concept: The curve y = sin x is bounded between -1 and 1, while f(x) = x² + x + 1 has minimum value 3/4, so they cannot intersect.
<p>Since \(-1 < \sin x < 1\) for all \(x\), and \(y = x^2 + x + 1 = \left(x + \frac{1}{2}\right)^2 + \frac{3}{4} \geq \frac{3}{4}\) for all \(x\).</p><p>It is clear from the graph that the minimum value of \(f(x)\) is \(\frac{3}{4}\), which is greater than the maximum value of \(\sin x\), which is \(1\). Therefore, the two curves do not intersect.</p><p>∴ Both statements are correct and Statement II correctly explains Statement I.</p>
Correct Answer: a