Question

The number of solutions of the equation $ \cos 2\theta \cos \left( \frac{\theta}{2} \right) + \cos \left( \frac{5\theta}{2} \right) = 2 \cos^3 \left( \frac{5\theta}{2} \right) $ in the interval \(\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right ]\) is:

• A. 7
• B. 5
• C. 6
• D. 9

Hint:

When solving trigonometric equations, simplifying the equation and substituting auxiliary variables can make solving easier. Be sure to check all possible solutions for the given range.

Answer 1

The Correct Option is A

We are given the equation: \[ \cos 2\theta \cos \left( \frac{\theta}{2} \right) + \cos \left( \frac{5\theta}{2} \right) = 2 \cos^3 \left( \frac{5\theta}{2} \right) \] 
Step 1: Simplify the given equation.
First, rewrite the equation in a simpler form to recognize patterns. 
We can try substituting a simpler variable for trigonometric terms to make the equation easier to solve. 
Let \( x = \cos \left( \frac{\theta}{2} \right) \), \( y = \cos \left( \frac{5\theta}{2} \right) \). 
Thus, the equation becomes: \[ 2x^2 y + y = 2y^3 \] 
Simplify this to: \[ y(2x^2 + 1) = 2y^3 \] \[ y(2x^2 + 1 - 2y^2) = 0 \] 
Step 2: Solve for the possible solutions.
From the above factorization, we now solve for the possible values of \( y \). 
We can split the equation into two cases: 1. \( y = 0 \) 2. \( 2x^2 + 1 - 2y^2 = 0 \) In case 1, we check the values of \( y = 0 \) within the given interval and determine the corresponding values of \( \theta \). In case 2, we substitute the expression for \( y \) into the second equation and solve for \( \theta \). 
Step 3: Count the number of solutions.
We find that there are 7 distinct solutions in the given interval \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \).