Question

For \(\theta>\pi/3\) the value of \(f(\theta) = \sec^2\theta + \cos^2\theta\) always lies in the interval

• A. \((0, 2)\)
• B. \([0, 1]\)
• C. \((1, 2)\)
• D. \([2, \infty)\)

Hint:

For positive numbers, AM \(\ge\) GM, equality when numbers equal.

Answer 1

The Correct Option is D

To determine the range of \( f(\theta) = \sec^2\theta + \cos^2\theta \) for \(\theta > \pi/3\), let's analyze and simplify the expression step-by-step.

1. Recall that \( \sec\theta = \frac{1}{\cos\theta} \). Therefore, \(\sec^2\theta = \frac{1}{\cos^2\theta}\).
2. Substituting this back into the function gives: \(f(\theta) = \frac{1}{\cos^2\theta} + \cos^2\theta\).
3. To find the range of this expression for \(\theta > \pi/3\), notice that:
   • At \(\theta = \pi/3\), \(\cos(\pi/3) = \frac{1}{2}\), so \(\cos^2(\pi/3) = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\).
   • Thus, at \(\theta = \pi/3\): \(f(\theta) = \frac{1}{(\frac{1}{2})^2} + (\frac{1}{2})^2 = 4 + \frac{1}{4} = \frac{17}{4} \approx 4.25\).
   • The value of cos \(\theta\) becomes smaller as \(\theta\) increases beyond \(\pi/3\), causing \(\sec^2\theta\) to increase. Thus, \(f(\theta)\) also increases as \(\theta\) increases beyond \(\pi/3\).
4. Since the smallest value of \( f(\theta) \) when \(\theta = \pi/3\) is \(\frac{17}{4} \approx 4.25\), and it increases indefinitely as \(\theta\) continues to increase, the range of \( f(\theta) \) is \([2, \infty)\).

Therefore, the correct interval for the value of \( f(\theta) \) is \([2, \infty)\).