Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
Grade 12
Question:
<p>If sides AB, BC and CA of a triangle ABC are represented by <math>x + 2 = 0</math>, <math>3x + y = 0</math> and <math>x + 3y + 2 = 0</math> respectively, then identify the correct statement.</p><p>(a) <math>\sum \tan A = -\frac{4}{3}</math></p><p>(b) <math>\pi - (\tan^{-1}2 + \tan^{-1}3)</math></p><p>(c) <math>\frac{\pi}{4}</math></p><p>(d) <math>\pi + (\tan^{-1}2 + \tan^{-1}3)</math></p>
<p>(a) <math>\sum \tan A = -\frac{4}{3}</math></p>
<p>(b) <math>\pi - (\tan^{-1}2 + \tan^{-1}3)</math></p>
<p>(c) <math>\frac{\pi}{4}</math></p>
<p>(d) <math>\pi + (\tan^{-1}2 + \tan^{-1}3)</math></p>
Step-by-Step Solution
Key Concept: Use the telescoping property of inverse trigonometric functions and the properties of triangle angles to find the sum of tangent values.
<p><strong>Solution:</strong></p><p>Using the telescoping series method:</p><p><math>T_r = \tan^{-1}(r+3) - \tan^{-1}(r+1)</math></p><p>Sum = <math>\pi - (\tan^{-1}(2) + \tan^{-1}(3)) = \frac{\pi}{4}</math></p>
Correct Answer: B