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

The general solution for sin(x+pi/6)+cos(x+pi/3)=cos(2x) is x=(4pin)/3 ,x=(2pi)/3+(4pin)/3

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sin(x+pi/6)+cos(x+pi/3)=cos(2x)

sin (x + \frac{π}{6}) + cos (x + \frac{π}{3}) = cos (2 x)

x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

Solution steps

sin (x + \frac{π}{6}) + cos (x + \frac{π}{3}) = cos (2 x)

Rewrite using trig identities

cos (x) = cos (2 x)

Subtract

cos (2 x)

from both sides

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

Rewrite using trig identities

2 sin (\frac{3 x}{2}) sin (\frac{x}{2}) = 0

Solving each part separately

sin (\frac{3 x}{2}) = 0 or sin (\frac{x}{2}) = 0

sin (\frac{3 x}{2}) = 0 : x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

sin (\frac{x}{2}) = 0 : x = 4 πn, x = 2 π + 4 πn

Combine all the solutions

x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}, x = 4 πn, x = 2 π + 4 πn

Merge Overlapping Intervals

x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

24 sin (2 t) - 24 cos (t) = 0

8^{s i n^{2} (x)} = 4^{s i n (x) - \frac{1}{8}}

2 cos (2 x) = \frac{1}{2}

\frac{sin ( 7 3 ^{\circ} )}{34} = \frac{sin ( d )}{29}

θ, \frac{9 0 ^{2}}{250 ( 32.2 )} = tan (θ)

What is the general solution for sin(x+pi/6)+cos(x+pi/3)=cos(2x) ?
The general solution for sin(x+pi/6)+cos(x+pi/3)=cos(2x) is x=(4pin)/3 ,x=(2pi)/3+(4pin)/3