Trigonometry & Inverse Trigonometry
tan 2α from cos(α+β) and sin(α−β) — r+s
nta_pyq_2026_jan
Grade 11
Question:
Let $\cos(\alpha+\beta)=-\dfrac{1}{10}$ and $\sin(\alpha-\beta)=\dfrac{3}{8}$, where $0<\alpha<\dfrac{\pi}{3}$ and $0<\beta<\dfrac{\pi}{4}$. If $\tan2\alpha=\dfrac{3(1-r\sqrt{5})}{\sqrt{11}(s+\sqrt{5})}$, $r,s\in\mathbb{N}$, then $r+s$ is equal to _____.
Step-by-Step Solution
Key Concept: $\tan2\alpha=\tan[(\alpha+\beta)+(\alpha-\beta)]=\dfrac{\tan(\alpha+\beta)+\tan(\alpha-\beta)}{1-\tan(\alpha+\beta)\tan(\alpha-\beta)}$. Compute $\tan(\alpha+\beta)$ and $\tan(\alpha-\beta)$ from given data.
$r=11$, $s=9$. $r+s=20$.
Correct Answer: 20