Trigonometry & Inverse Trigonometry
Trigonometric Functions
Grade 11
Question:
<p>The range of values of \(k\) for which the equation \(2\cos 4x - \sin 4x + k = 0\) has at least one solution is \([l, m]\). Find the value of \(9m + l\).</p>
Step-by-Step Solution
Key Concept: Express a linear combination of sine and cosine in the form $R\sin(\theta + \phi)$ to find its range.
<p>Rearrange the equation as $k = \sin 4x - 2\cos 4x$. Find the range of the expression $\sin 4x - 2\cos 4x$ by writing it in the form $R\sin(4x + \phi)$ where $R = \sqrt{1 + 4} = \sqrt{5}$. Thus the range is $[-\sqrt{5}, \sqrt{5}]$, so $l = -\sqrt{5}$ and $m = \sqrt{5}$.</p>
Correct Answer: 7