Trigonometry & Inverse Trigonometry
Trigonometric Functions in Triangles
Grade 11
Question:
<p>In triangle ABC, if \(\tan\frac{A}{4} + \tan\frac{B}{4} + \tan\frac{C}{4} = 1\), then triangle ABC is</p>
<p>(a) equilateral</p>
<p>(b) isosceles</p>
<p>(c) scalene</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Using properties of tangent and the angle sum in a triangle to establish equality of angles
<p><strong>Step 1:</strong> Let \(\tan A = \tan\alpha\), \(\tan B = \tan\beta\), \(\tan C = \tan\gamma\)</p><p><strong>Step 2:</strong> From the given condition, we have \(\tan\alpha \tan\beta = 1\)</p><p><strong>Step 3:</strong> This implies \(\tan\alpha = \tan\beta = \tan\gamma\)</p><p><strong>Step 4:</strong> Therefore, \(A = B = C\)</p><p>∴ Triangle ABC is equilateral.</p>
Correct Answer: a