What are all the solutions to the equation cos(4x) * cos(2x) = 0 within the interval [0, 2π]?

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Finding Solutions for the Equation cos(4x) * cos(2x) = 0

To solve the equation cos(4x) * cos(2x) = 0, we need to identify when either of the cosine terms equals zero, as the product of two terms equals zero if at least one of them is zero.

Step 1: Solve cos(4x) = 0

The cosine function is equal to zero at odd multiples of
frac{ ext{π}}{2}. Thus, we can express the solutions for cos(4x) as follows:

4x = (2n + 1) rac{ ext{π}}{2}, where n is an integer.

Dividing both sides by 4 gives:

x = rac{(2n + 1) rac{ ext{π}}{2}}{4} = rac{(2n + 1) ext{π}}{8}

Now we need to find suitable values of n such that 0 ≤ x ≤ 2π.

Calculating some integer values for n:

  • For n = 0: x = rac{1 ext{π}}{8} = rac{ ext{π}}{8}
  • For n = 1: x = rac{3}{8} ext{π}
  • For n = 2: x = rac{5}{8} ext{π}
  • For n = 3: x = rac{7}{8} ext{π}
  • For n = 4: x = rac{9}{8} ext{π}
  • For n = 5: x = rac{11}{8} ext{π}
  • For n = 6: x = rac{13}{8} ext{π}
  • For n = 7: x = rac{15}{8} ext{π}

Thus, from cos(4x) = 0, our solutions are:

  • x = rac{ ext{π}}{8}
  • x = rac{3}{8} ext{π}
  • x = rac{5}{8} ext{π}
  • x = rac{7}{8} ext{π}
  • x = rac{9}{8} ext{π}
  • x = rac{11}{8} ext{π}
  • x = rac{13}{8} ext{π}
  • x = rac{15}{8} ext{π}

Step 2: Solve cos(2x) = 0

Similarly, for cos(2x) = 0, we have:

2x = (2m + 1) rac{ ext{π}}{2}, where m is an integer.

Dividing both sides by 2 provides:

x = rac{(2m + 1) rac{ ext{π}}{2}}{2} = rac{(2m + 1) ext{π}}{4}

Now we need to find appropriate values of m such that 0 ≤ x ≤ 2π.

Calculating suitable integer values for m:

  • For m = 0: x = rac{ ext{π}}{4}
  • For m = 1: x = rac{3}{4} ext{π}
  • For m = 2: x = rac{5}{4} ext{π}
  • For m = 3: x = rac{7}{4} ext{π}

Thus, from cos(2x) = 0, our solutions are:

  • x = rac{ ext{π}}{4}
  • x = rac{3}{4} ext{π}
  • x = rac{5}{4} ext{π}
  • x = rac{7}{4} ext{π}

Step 3: Combine the Solutions

Now, we compile the solutions obtained from both cos(4x) and cos(2x). The complete set of solutions within the interval [0, 2π] is:

  • x = rac{ ext{π}}{8}
  • x = rac{ ext{π}}{4}
  • x = rac{3}{8} ext{π}
  • x = rac{5}{8} ext{π}
  • x = rac{7}{8} ext{π}
  • x = rac{3}{4} ext{π}
  • x = rac{9}{8} ext{π}
  • x = rac{11}{8} ext{π}
  • x = rac{5}{4} ext{π}
  • x = rac{13}{8} ext{π}
  • x = rac{15}{8} ext{π}
  • x = rac{7}{4} ext{π}

Conclusion

The solutions to the equation cos(4x) * cos(2x) = 0 in the interval [0, 2π] are:

x = rac{ ext{π}}{8}, rac{ ext{π}}{4}, rac{3}{8} ext{π}, rac{5}{8} ext{π}, rac{7}{8} ext{π}, rac{3}{4} ext{π}, rac{9}{8} ext{π}, rac{11}{8} ext{π}, rac{5}{4} ext{π}, rac{13}{8} ext{π}, rac{15}{8} ext{π}, rac{7}{4} ext{π}.

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