trigonometry!!!!!!!!!!!!!!!!!!!!!!

Question

solve for x such that    0 degrees is less than or equal to x which is less than or equal to 360 degrees

cosx + cos 3x = sin x + sin 3x

Answer 1

The trigonometric equation is cos x + cos 3x = sin x + sin 3x

Apply formulas: cos C + cos D = 2 cos {(C+D)/2}cos {(C-D)/2}  and

sin C + sin D = 2 sin {(C+D)/2}cos {(C-D)/2}

2cos{(x+3x)/2}cos{(x-3x)/2} = 2sin{(x+3x)/2}cos{(x-3x)/2}

2cos(4x/2)cos(-2x/2) = 2sin(4x/2)cos(-2x/2)

2cos 2x cos(-x) = 2sin 2x cos(-x)

2cos 2x cos x = 2sin 2x cos x

Subtract 2sin 2x cos x from each side

2cos 2x cos x - 2sin 2x cos x = 2sin 2x cos x - 2sin 2x cos x

2cos 2x cos x - 2sin 2x cos x = 0

Take out common factor

2cos x(cos 2x - 2sin 2x) = 0

Apply zero product property

2cos x = 0 or cos 2x - 2sin 2x = 0

cos x = 0/2                  cos 2x = sin 2x

x = cos^-1(0)                 2x = 45⁰ (because the angles are same)

x = 90⁰ or  x = 22.5⁰

The values of the x = 90⁰, 22.5⁰.