Trigonometry & Inverse Trigonometry
Inverse Tangent Equations
Grade 12
Question:
<p>Considering only the principal values of inverse functions, the set \(A = \{x \geq 0 : \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}\}\)</p>
<p>(a) is an empty set</p>
<p>(b) is a singleton</p>
<p>(c) contains more than two elements</p>
<p>(d) contains two elements</p>
Step-by-Step Solution
Key Concept: Apply the addition formula for inverse tangent and solve the resulting algebraic equation, checking domain constraints.
<p>Using the addition formula for inverse tangent: $\tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a+b}{1-ab}\right)$ when $ab < 1$.</p><p>$\tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}$</p><p>$\tan^{-1}\left(\frac{2x + 3x}{1 - 6x^2}\right) = \frac{\pi}{4}$</p><p>$\frac{5x}{1 - 6x^2} = 1$</p><p>$5x = 1 - 6x^2$</p><p>$6x^2 + 5x - 1 = 0$</p><p>$(6x - 1)(x + 1) = 0$</p><p>$x = \frac{1}{6}$ or $x = -1$</p><p>Since $x \geq 0$, only $x = \frac{1}{6}$ is valid. Also, $6x^2 = \frac{1}{6} < 1$, so the formula applies.</p><p>∴ A is a singleton.</p>
Correct Answer: B