Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
Grade 12
Question:
<p>If in a \(\triangle ABC\), \(\angle A = \tan^{-1}2\) and \(\angle B = \tan^{-1}3\), then \(\angle C\) is equal to</p>
<p>(a) \(\frac{\pi}{2}\)</p>
<p>(b) \(\frac{\pi}{3}\)</p>
<p>(c) \(\frac{\pi}{4}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use the constraint that angles in a triangle sum to π and tangent addition formula
<p>Since $A + B + C = \pi$, we have $C = \pi - A - B$. Using $\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A\tan B} = \frac{2+3}{1-6} = -1$, so $A + B = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$, thus $C = \frac{\pi}{4}$.</p>
Correct Answer: A