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

The general solution for cos(x)=sin(2x)+cos(3x) is x=pin,x= pi/2+pin,x= pi/6+2pin,x=(5pi)/6+2pin

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

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

x = πn, x = \frac{π}{2} + πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

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

Subtract

sin (2 x) + cos (3 x)

from both sides

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

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

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

2 sin (x) - 1 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = πn, x = \frac{π}{2} + πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

tan (x) sec (x) - 2 tan (x) = 0

- 9 cos^{2} (θ) + 9 = 17 sin (θ) - 8

θ, r = 2 cos (θ)

sin (2 x) = \frac{2 2}{3}

arccos (x) = \frac{15}{17}

What is the general solution for cos(x)=sin(2x)+cos(3x) ?
The general solution for cos(x)=sin(2x)+cos(3x) is x=pin,x= pi/2+pin,x= pi/6+2pin,x=(5pi)/6+2pin