Limits, Continuity & Differentiability
Limits involving trigonometric functions
Grade 12
Question:
<p>If $a, b \in \left(-\frac{\pi}{2}, 0\right)$ such that $(\sin a + \sin b) + \frac{\sin a}{\sin b} = 0$ and $(\sin a + \sin b) \frac{\sin a}{\sin b} = -1$ and $\lambda = \lim_{n \to \infty} \frac{1 + (2\sin a)^{2n}}{(2\sin b)^{2n}}$ then:</p>
<p>(a) $a = -\frac{\pi}{6}$</p>
<p>(b) $\lambda = 2$</p>
<p>(c) $a = -\frac{\pi}{3}$</p>
<p>(d) $\lambda = 1$</p>
Step-by-Step Solution
Key Concept: Solve the system of equations for $\sin a$ and $\sin b$, then evaluate the limit based on the magnitudes of these values.
<p>The correct answers are (a) and (b).</p>
Correct Answer: A, B