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

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

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

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

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

\frac{2}{2} cos (t) = 0

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

5 - 20 cos^{2} (t) = 0

10 tan (B) + 12 = tan (B) + 6

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