The number of solutions of the equation cos2θcos2θ+cos25θ=2cos325θ in [−2π,2π] is:
Get the complete, step-by-step math solution for: "The number of solutions of the equation cos 2 (θ)/(2) + cos(5θ)/(2) = 2cos³(5θ)/(2) in [- {π}{2}, {π}{2}] is:". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Rearrange the equation
The given equation is cos2θcos2θ+cos25θ=2cos325θ. We can rearrange this equation by moving the term cos25θ to the right side.
Step 2: Factor the right side using the triple angle identity
We can factor out cos25θ from the right side. The expression 2cos2x−1 is the double angle identity for cos2x. Here, x=25θ, so 2cos225θ−1=cos(2⋅25θ)=cos5θ.
Step 3: Apply product-to-sum identities
We use the product-to-sum identity 2cosAcosB=cos(A+B)+cos(A−B) on both sides of the equation. This transforms the products of cosines into sums of cosines.
Step 4: Simplify the arguments and solve
After applying the product-to-sum identities and simplifying the arguments, we get cos25θ+cos23θ=cos215θ+cos25θ. The term cos25θ cancels out from both sides, leaving cos23θ=cos215θ. The general solution for cosA=cosB is A=2nπ±B, where n is an integer.
Step 5: Solve for θ in two cases
We consider two cases based on the general solution. In Case 1, we take the positive sign, leading to θ=3nπ. In Case 2, we take the negative sign, leading to θ=92nπ.
Step 6: Find solutions in the interval [−2π,2π]
We need to find the values of θ that lie in the interval [−2π,2π], which is approximately [−1.57,1.57].
For θ=3nπ: n=0,±1 give 0,±3π.
For θ=92nπ: n=0,±1,±2 give 0,±92π,±94π.
The unique solutions are 0,±3π,±92π,±94π.
Step 7: Count the number of unique solutions
Listing all the unique solutions found: 0,3π,−3π,92π,−92π,94π,−94π. There are 7 unique solutions.