Trigonometry & Inverse Trigonometry
Trigonometric Extrema
Grade 11
Question:
<p>The least value of \(\csc^2 x + 25\sec^2 x\) is</p>
<p>(a) \(0\)</p>
<p>(b) \(26\)</p>
<p>(c) \(28\)</p>
<p>(d) \(36\)</p>
Step-by-Step Solution
Key Concept: Apply optimization techniques (AM-GM or calculus) to find the minimum of a sum of reciprocal trigonometric functions.
<p>Use the AM-GM inequality or calculus to find the minimum value of $\csc^2 x + 25\sec^2 x = \frac{1}{\sin^2 x} + \frac{25}{\cos^2 x}$. The minimum occurs when the partial derivatives equal zero.</p>
Correct Answer: D