<p>There is a unique angle \(\theta\) between \(0^\circ\) and \(90^\circ\) such that for non-negative integers \(n\), the value of \(\tan(2^n\theta)\) is positive when \(n\) is a multiple of 3, and negative otherwise. The degree measure of \(\theta\) is \(\dfrac{p}{q}\), where \(p\) and \(q\) are relatively prime integers. Find \(p + q\).</p>
Step-by-Step Solution
Key Concept: We need to track when tan(2^n θ) is positive or negative by analyzing which quadrants 2^n θ lies in. Since the sign pattern repeats with period 3 in n, the angle must satisfy a specific periodicity condition modulo 180°.
<p><strong>Step 1:</strong> Identify the sign pattern requirement. We need tan(2^n θ) to be positive when n ≡ 0 (mod 3) and negative otherwise. This means the pattern is: +, −, −, +, −, −, +, −, −, ... with period 3.</p><p><strong>Step 2:</strong> Determine quadrant constraints. For 0° < θ < 90°, we have tan(θ) > 0 (Quadrant I). For the pattern to work:
- n = 0: tan(θ) > 0 ✓ (automatically satisfied)
- n = 1: tan(2θ) < 0, so 90° < 2θ < 180° ⟹ 45° < θ < 90°
- n = 2: tan(4θ) < 0, so 90° < 4θ < 180° or 270° < 4θ < 360° ⟹ 22.5° < θ < 45° or 67.5° < θ < 90°
Combining: 67.5° < θ < 90°</p><p><strong>Step 3:</strong> Apply periodicity. Since the sign pattern repeats with period 3, we need tan(2^(n+3)θ) = tan(2^n θ) for all valid n. This means 2^(n+3)θ and 2^n θ differ by a multiple of 180°:
2^(n+3)θ − 2^n θ ≡ 0 (mod 180°)
8·2^n θ − 2^n θ ≡ 0 (mod 180°)
7·2^n θ ≡ 0 (mod 180°)</p><p><strong>Step 4:</strong> This must hold for all non-negative integers n. For n = 0: 7θ ≡ 0 (mod 180°), so 7θ = 180k for some positive integer k. Since 67.5° < θ < 90°, we have 472.5° < 7θ < 630°. The multiples of 180° in this range are 540°. Thus 7θ = 540°, giving θ = 540°/7.</p><p><strong>Step 5:</strong> Verify the solution θ = 540°/7 ≈ 77.14°:
- θ ≈ 77.14° ∈ (67.5°, 90°) ✓
- 2θ ≈ 154.29° ∈ (90°, 180°), so tan(2θ) < 0 ✓
- 4θ ≈ 308.57° ∈ (270°, 360°), so tan(4θ) < 0 ✓
- 8θ ≈ 617.14° ≡ 77.14° (mod 180°), so tan(8θ) > 0 ✓
The pattern continues correctly.</p><p><strong>Step 6:</strong> Express as p/q in lowest terms. θ = 540/7 degrees. Since gcd(540, 7) = 1 (7 is prime and doesn't divide 540), we have p = 540 and q = 7.</p><p><strong>∴ Answer:</strong> p + q = 540 + 7 = 547</p>
Correct Answer: 2