Trigonometry & Inverse Trigonometry
Half-Angle and Sum Formulas
Grade 11
Question:
<p>If \(\tan\frac{a}{2}\) and \(\tan\frac{b}{2}\) are the roots of the equation \(8x^2 - 26x + 15 = 0\), then \(\cos(a + b)\) is equal to</p>
<p>(a) \(-\frac{627}{72}\)</p>
<p>(b) \(\frac{627}{72}\)</p>
<p>(c) Some other value</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas to find sums and products, then apply half-angle formulas to find the cosine of the sum.
<p>Using Vieta's formulas: $\tan\frac{a}{2} + \tan\frac{b}{2} = \frac{26}{8}$ and $\tan\frac{a}{2}\tan\frac{b}{2} = \frac{15}{8}$. Apply the formula $\cos(a+b) = \frac{1-\tan^2\frac{a+b}{2}}{1+\tan^2\frac{a+b}{2}}$ with $\tan\frac{a+b}{2} = \frac{\tan\frac{a}{2} + \tan\frac{b}{2}}{1 - \tan\frac{a}{2}\tan\frac{b}{2}}$.</p>
Correct Answer: A