Cos(x)cos(3x) - sin(x)sin(3x) = 0,

Question

how do I find the 4 solutions in interval [0,pi]

Answer 1

Given equation : cosx cos3x - sinx sin3x = 0

Apply formula : cosA cosB - sinA sinB = cos(A+B)

cos (x+3x) = 0

cos (4x) = 0

General solution for cos(x) = 0 is (2n+1)(π/2)

4x = (2n+1)(π/2)

x = (2n+1)(π/2)/4 = (2n+1)(π/8)

 For n = 0 ⇒ x = (2*0+1)(π/8) = π/8

 For n = 1 ⇒ x = (2*1+1)(π/8) = 3π/8

 For n = 2 ⇒ x = (2*2+1)(π/8) = 5π/8

 For n = 3 ⇒ x = (2*3+1)(π/8) = 7π/8

 For n = 4 ⇒ x = (2*4+1)(π/8) = 9π/8 is out of given range [0,π].

So, for n > 3, x will be out of range [0,π].

Solution :

x = { π/8 , 3π/8 , 5π/8 , 7π/8 }