What is the general solution for cos(x)=cos(3x)+2sin(2x) ?

The general solution for cos(x)=cos(3x)+2sin(2x) is x=pin,x= pi/2+pin

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cos(x)=cos(3x)+2sin(2x)

cos (x) = cos (3 x) + 2 sin (2 x)

x = πn, x = \frac{π}{2} + πn

Solution steps

cos (x) = cos (3 x) + 2 sin (2 x)

Subtract

cos (3 x) + 2 sin (2 x)

from both sides

cos (x) - cos (3 x) - 2 sin (2 x) = 0

Rewrite using trig identities

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

Factor

- 2 sin (2 x) + 2 sin (2 x) sin (x) : 2 sin (2 x) (sin (x) - 1)

2 sin (2 x) (sin (x) - 1) = 0

Solving each part separately

sin (2 x) = 0 or sin (x) - 1 = 0

sin (2 x) = 0 : x = πn, x = \frac{π}{2} + πn

sin (x) - 1 = 0 : x = \frac{π}{2} + 2 πn

Combine all the solutions

x = πn, x = \frac{π}{2} + πn, x = \frac{π}{2} + 2 πn

Merge Overlapping Intervals

x = πn, x = \frac{π}{2} + πn

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

\frac{700}{sin ( 9 0 ^{\circ} )} = \frac{236.3277}{sin ( θ )}

cos (x) + cos (2 x) = - 0.75

x, 2 y = sin (2 x)

- csc (θ) + 5 = cot (θ) + 6

What is the general solution for cos(x)=cos(3x)+2sin(2x) ?
The general solution for cos(x)=cos(3x)+2sin(2x) is x=pin,x= pi/2+pin