<p>In the following incomplete sentences, fill in the blanks so that the resulting sentences may become true.</p><p>(d) If \(|\tan x| < 1\) and \(0 \leq x \leq \pi\), then the solution-set of \(x\) is ______.</p>
Step-by-Step Solution
Key Concept: The range of arctan function is (-π/2, π/2), and |tan x| < 1 corresponds to x values where the tangent magnitude doesn't exceed 1. We must identify all x in the principal domain where this inequality holds.
<p><strong>Step 1:</strong> Recall that the range of arctan(x) is (-π/2, π/2), which is the principal domain.</p><p><strong>Step 2:</strong> We need to find when |tan x| < 1 for x ∈ (-π/2, π/2).</p><p><strong>Step 3:</strong> |tan x| < 1 means -1 < tan x < 1.</p><p><strong>Step 4:</strong> In the interval (-π/2, π/2):</p><ul><li>tan x = -1 when x = -π/4</li><li>tan x = 0 when x = 0</li><li>tan x = 1 when x = π/4</li></ul><p><strong>Step 5:</strong> Since tan is strictly increasing on (-π/2, π/2), we have |tan x| < 1 when x ∈ (-π/4, π/4).</p><p><strong>Step 6:</strong> Therefore: <strong>If |tan x| < 1, then x ∈ (-π/4, π/4)</strong> OR equivalently <strong>-π/4 < x < π/4</strong></p><p>∴ Answer: <strong>x ∈ (-π/4, π/4)</strong></p>
Correct Answer: -1