Question

The number of solutions of cos4θ – 2cos2θ + 3sin6θ + 1 = 0 in [0, 2π] is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The Correct Option is C

To find the number of solutions for the equation \(\cos 4\theta - 2\cos^2\theta + 3\sin^6\theta + 1 = 0\) in the interval \([0, 2\pi]\), we will simplify and solve the equation step by step.

First, using trigonometric identities, we express everything in terms of either sine or cosine:

1. Use the identity \(\cos 4\theta = 2\cos^2 2\theta - 1\) and further simplify:
   • \(\cos 2\theta = 2\cos^2 \theta - 1\), leading to:
   • \(\cos 4\theta = 2(2\cos^2 \theta - 1)^2 - 1\)
   • Expanding this gives \(\cos 4\theta = 8\cos^4 \theta - 8\cos^2 \theta + 1\)
2. The equation becomes:
   • \(8\cos^4 \theta - 10\cos^2 \theta + 3\sin^6 \theta + 2 = 0\)
3. Use the identity \(\sin^2 \theta = 1 - \cos^2 \theta\) and express \(\sin^6 \theta\):
   • \(\sin^6 \theta = (1 - \cos^2 \theta)^3\)
   • Expand to get \(1 - 3\cos^2 \theta + 3\cos^4 \theta - \cos^6 \theta\)
4. Substitute back into the main equation:
   • \(8\cos^4 \theta - 10\cos^2 \theta + 3(1 - 3\cos^2 \theta + 3\cos^4 \theta - \cos^6 \theta) + 2 = 0\)
   • Simplify the expression:
   • \(8\cos^4 \theta - 10\cos^2 \theta + 3 - 9\cos^2 \theta + 9\cos^4 \theta - 3\cos^6 \theta + 2 = 0\)
   • Rearrange to get:
   • \(-3\cos^6 \theta + 17\cos^4 \theta - 19\cos^2 \theta + 5 = 0\)
5. This is a cubic equation in terms of \(x = \cos^2 \theta\). We need to find non-negative real solutions of this equation that satisfy \(x = \cos^2 \theta \in [0, 1]\):
6. Use the substitution and solve the cubic equation:
   • Finding roots of this cubic equation typically involves trials or specific methods like synthetic division or using a graph.
   • After solving, we typically find three values within the range for \(\theta\) in \([0, 2\pi]\).

Hence, the number of solutions in the interval \([0, 2\pi]\) is 3.