Straight Lines
Region defined by inequalities
Grade 11
Question:
<p>Possible values of \(\theta\) for which the point \((\cos\theta, \sin\theta)\) lies inside the triangle formed by lines \(x + y = 2\); \(x - y = 1\) and \(6x + 2y = 10\) are:</p>
<p>(a) \(\frac{\pi}{8}\)</p>
<p>(b) \(\frac{\pi}{4}\)</p>
<p>(c) \(\frac{3\pi}{8}\)</p>
<p>(d) \(\frac{\pi}{2}\)</p>
Step-by-Step Solution
Key Concept: The point \((\cos\theta, \sin\theta)\) lies on the unit circle; determine which angular ranges correspond to the interior of the given triangle.
<p>The point \((\cos\theta, \sin\theta)\) lies on the unit circle. We need to find which values of \(\theta\) place this point inside the triangle formed by the three lines. The triangle has vertices at the intersections of these lines. Testing the given angles, we find that \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{3\pi}{8}\) satisfy the interior conditions.</p>
Correct Answer: b, c