Trigonometry
Trigonometry
Allen Star Batch
Grade 11
Question:
If $x + \sin y = 2014$ and $x + 2014\cos y = 2013, 0 \leq y \leq \frac{\pi}{2}$, then find the value of $[x + y] - 2005$ (where $[.]$ denotes greatest integer function)
Step-by-Step Solution
Key Concept: Linear combinations of sine and cosine have a fixed maximum value determined by the Cauchy-Schwarz inequality.
The equation $\sin y - 2014 \cos y = 1$ can be solved by recognizing that the maximum of $\sin y - 2014\cos y$ is $\sqrt{1 + 2014^2} \approx 2014$. For the equation to equal 1, we require $\sin y = 1$ and $\cos y = 0$, giving $y = \frac{\pi}{2}$.
Correct Answer: 9