Are the functions f g and h given below linearly independent

Are the functions f, g, and h given below linearly independent? f(x) = 0, g(x) = cos(5x), h(x) = sin(5x) If they are independent, enter all zeroes. If they are not linearly independent, find a nontrivial solution to the equation below. Be sure you can justify your answer. (0) + (cos (5x)) + (sin(5x)) = 0.

Solution

Let there exists a solution of the above equation as

=> c₁( 0 ) + c₂.cos(5x) + c₃.sin(5x) = 0

=> c₂.cos(5x) + c₃.sin(5x) = 0

=> tan(5x) = (-c₂/c₃)

=> x = tan^-1(-c₂/c₃)

Hence , there are infinitely many values of c_iwe can have a non-trivial solution of the equation . Therefore , the equation not lineraly independent.