To solve this trigonometric equation, we will use the trigonometric identity:
sin(A)sin(B) = 1/2[cos(A-B) - cos(A+B)]
Applying this identity to the equation 2sin(10x)sin(5x) + cos(15x) = 0, we get:
2[1/2(cos(10x-5x) - cos(10x+5x)] + cos(15x) = 0
Simplifying further, we get:
cos(5x) - cos(15x) + cos(15x) = 0
cos(5x) = 0
To find the values of x that satisfy this equation, we need to find the values of x for which cos(5x) = 0. Cosine is equal to zero at certain specific angles. The general form for these angles is:
To solve this trigonometric equation, we will use the trigonometric identity:
sin(A)sin(B) = 1/2[cos(A-B) - cos(A+B)]
Applying this identity to the equation 2sin(10x)sin(5x) + cos(15x) = 0, we get:
2[1/2(cos(10x-5x) - cos(10x+5x)] + cos(15x) = 0
Simplifying further, we get:
cos(5x) - cos(15x) + cos(15x) = 0
cos(5x) = 0
To find the values of x that satisfy this equation, we need to find the values of x for which cos(5x) = 0. Cosine is equal to zero at certain specific angles. The general form for these angles is:
5x = (2n + 1/2)π, where n is an integer.
Therefore, the solutions for x are:
x = [(2n + 1/2)π] / 5, where n is an integer.