Trigonometry & Inverse Trigonometry
Multiple Angle Formulas
Grade 11
Question:
<p>The equation whose roots are \(\tan^2\left(\frac{\pi}{7}\right)\), \(\tan^2\left(\frac{3\pi}{7}\right)\), \(\tan^2\left(\frac{5\pi}{7}\right)\) is</p>
<p>(a) \(x^3 - 35x^2 + 7x - 21 = 0\)</p>
<p>(b) \(x^3 - 35x^2 + 21x - 7 = 0\)</p>
<p>(c) \(x^3 - 35x^2 + 35x - 7 = 0\)</p>
<p>(d) \(x^3 - 21x^2 + 7x - 35 = 0\)</p>
Step-by-Step Solution
Key Concept: From \(\tan(7\theta) = 0\), derive a polynomial whose roots are the squared tangent values using substitution and Vieta's formulas.
<p>Use the identity for \(\tan(7\theta) = 0\) to find relationships between \(\tan^2(\pi/7)\), \(\tan^2(3\pi/7)\), \(\tan^2(5\pi/7)\). Apply Vieta's formulas to construct the cubic equation.</p>
Correct Answer: C