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

The general solution for cos(2x+pi/3)=sin(2x) is x=(0.26179…)/2+(pin)/2

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

cos (2 x + \frac{π}{3}) = sin (2 x)

x = \frac{0.26179 \dots}{2} + \frac{πn}{2}

Solution steps

cos (2 x + \frac{π}{3}) = sin (2 x)

Rewrite using trig identities

\frac{1}{2} cos (2 x) - \frac{3}{2} sin (2 x) = sin (2 x)

Subtract

sin (2 x)

from both sides

\frac{1}{2} cos (2 x) + \frac{- 3 - 2}{2} sin (2 x) = 0

Simplify

\frac{1}{2} cos (2 x) + \frac{- 3 - 2}{2} sin (2 x) : \frac{cos ( 2 x ) + ( - 3 - 2 ) sin ( 2 x )}{2}

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Rewrite using trig identities

1 - (2 + 3) tan (2 x) = 0

Move

1

to the right side

- (2 + 3) tan (2 x) = - 1

Simplify

- (2 + 3) : - 2 - 3

(- 2 - 3) tan (2 x) = - 1

Divide both sides by

- 2 - 3

tan (2 x) = 2 - 3

Apply trig inverse properties

2 x = arctan (2 - 3) + πn

Solve

2 x = arctan (2 - 3) + πn : x = \frac{arctan ( 2 - 3 )}{2} + \frac{πn}{2}

x = \frac{arctan ( 2 - 3 )}{2} + \frac{πn}{2}

Show solutions in decimal form

x = \frac{0.26179 \dots}{2} + \frac{πn}{2}

cot (2 t) = 0

tan (θ) = cot (θ), 0 \leq θ < 2 π

sec (θ) = 1.984

2 sin (\frac{1}{2} x) - 3 = 0

sin (2 θ) = 2 cos (θ)

What is the general solution for cos(2x+pi/3)=sin(2x) ?
The general solution for cos(2x+pi/3)=sin(2x) is x=(0.26179…)/2+(pin)/2