Question:What is the value of \(\mathrm{sec}\left(-\frac{23\pi}{4}\right)\)?Answer Choices:-{sqrt(2)}-{(sqrt(2))/2}(sqrt(2))/2sqrt(2)

1. INFER the key strategy

• When dealing with large angles like \(-23\pi/4\), the key insight is to find a coterminal angle
• Coterminal angles have the same trigonometric function values
• We need to add or subtract multiples of \(2\pi\) (full revolutions)

2. TRANSLATE the problem setup

• Given: \(\mathrm{sec}(-23\pi/4)\)
• Need to find: a coterminal angle between \(0\) and \(2\pi\)
• One full revolution = \(2\pi = 8\pi/4\)

3. SIMPLIFY to find the coterminal angle

• Start with: \(-23\pi/4 + k(8\pi/4)\) where \(k\) is an integer
• We need: \(-23 + 8k \geq 0\)
• Solving: \(k \geq 23/8 = 2.875\)
• So \(k = 3\) (smallest integer that works)
• Calculate:
  \(-23\pi/4 + 3(8\pi/4)\)
  \(= -23\pi/4 + 24\pi/4\)
  \(= \pi/4\)

4. INFER that secant equals reciprocal of cosine

• \(\mathrm{sec}(-23\pi/4) = \mathrm{sec}(\pi/4) = 1/\mathrm{cos}(\pi/4)\)
• From the unit circle: \(\mathrm{cos}(\pi/4) = \sqrt{2}/2\)

5. SIMPLIFY the reciprocal calculation

• \(\mathrm{sec}(\pi/4) = 1/(\sqrt{2}/2) = 2/\sqrt{2}\)
• Rationalize:
  \((2/\sqrt{2}) \times (\sqrt{2}/\sqrt{2})\)
  \(= 2\sqrt{2}/2\)
  \(= \sqrt{2}\)

Answer: D. \(\sqrt{2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to find a coterminal angle and instead try to work directly with \(-23\pi/4\), leading to confusion about which quadrant the angle is in or what the reference angle should be.

Without the coterminal angle strategy, students might guess that since the angle is negative and large, the answer should be negative, leading them to select Choice A (\(-\sqrt{2}\)) or get completely lost and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need a coterminal angle but make arithmetic errors when calculating \(-23\pi/4 + 24\pi/4\), or they forget to rationalize the denominator \(2/\sqrt{2}\).

This can lead them to stop at \(2/\sqrt{2}\) and try to match it to an answer choice, potentially selecting Choice C (\(\sqrt{2}/2\)) thinking it looks similar.

The Bottom Line:

The key challenge is recognizing that large angles require the coterminal angle strategy, then executing the multi-step simplification accurately. Students who miss the strategic insight typically struggle with systematic approaches to complex trigonometric evaluations.