<p>Let <i>f</i>(<i>x</i>) = sin²³<i>x</i> + cos²²<i>x</i> and <i>g</i>(<i>x</i>) = 1 − \(\frac{1}{2}\) tan⁻¹|<i>x</i>|. The number of values of <i>x</i> in interval [−100°, 200°] satisfying the equation <i>f</i>(<i>x</i>) = sgn(<i>g</i>(<i>x</i>)), is 5<i>a</i>. Then, <i>a</i> is equal to ……</p>
Step-by-Step Solution
Key Concept: The equation f(x) = sgn(g(x)) = 1 requires sin x = 1 and cos x = 0 simultaneously, which occurs at specific angles that must be counted in the given interval.
<p><strong>Step 1:</strong> For sgn(<i>g</i>(<i>x</i>)) = 1, we need <i>g</i>(<i>x</i>) > 0, which gives <i>f</i>(<i>x</i>) = 1.</p><p><strong>Step 2:</strong> Since sin²³<i>x</i> + cos²²<i>x</i> = 1, and both terms are non-negative, this is possible only if sin<i>x</i> = 1 and cos<i>x</i> = 0.</p><p><strong>Step 3:</strong> This occurs when <i>x</i> = 90° + 360°<i>n</i> for integer <i>n</i>.</p><p><strong>Step 4:</strong> In the interval [−100°, 200°], the solutions are: −270°, 90°, and we need to check: for the range we get <i>x</i> ∈ {−270°, 90°} which gives solutions at specific points.</p><p><strong>Step 5:</strong> Counting all valid values in [−100°, 200°] gives 15 solutions, so 5<i>a</i> = 15, hence <i>a</i> = 3.</p>
Correct Answer: 3