Finding Solutions to the Equation
To solve the equation cos(x) * cos(3x) * sin(x) * sin(3x) = 0, we need to find the values of x that make the product equal to zero. Since we’re dealing with a product of several trigonometric functions, we can apply the zero product property, which states that if a product of several factors equals zero, at least one of the factors must be zero.
Analyzing Each Factor
We have four factors in our equation:
- cos(x)
- cos(3x)
- sin(x)
- sin(3x)
1. Solving cos(x) = 0
We know that cosine is zero at odd multiples of π/2:
- x = (2n + 1) * π/2, where n is any integer.
2. Solving cos(3x) = 0
For cosine of three times x:
- 3x = (2m + 1) * π/2, where m is any integer.
- Thus, x = (2m + 1) * π/6.
3. Solving sin(x) = 0
Sine is zero at integer multiples of π:
- x = n * π, where n is any integer.
4. Solving sin(3x) = 0
For sine of three times x:
- 3x = k * π, where k is any integer.
- Thus, x = k * π/3.
Combining All Solutions
After analyzing all factors, we can combine the solutions:
- x = (2n + 1) * π/2 (from cos(x) = 0)
- x = (2m + 1) * π/6 (from cos(3x) = 0)
- x = n * π (from sin(x) = 0)
- x = k * π/3 (from sin(3x) = 0)
Conclusion
In conclusion, the equation has infinite solutions represented by the integer multiples derived from each component, thus demonstrating the rich interplay between angles in trigonometric functions. Make sure to consider the integer parameters when calculating the specific values!