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

The general solution for sin(3x-pi/6)=-cos(3x-pi/6) is x=(-0.26179…)/3+(pin)/3

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sin(3x-pi/6)=-cos(3x-pi/6)

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

x = \frac{- 0.26179 \dots}{3} + \frac{πn}{3}

Solution steps

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

Rewrite using trig identities

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

Subtract

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

from both sides

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

Simplify

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

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

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

(- 1 + 3) cos (3 x) + (1 + 3) sin (3 x) = 0

Rewrite using trig identities

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

Move

1

to the right side

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

Move

3

to the right side

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

Divide both sides by

1 + 3

tan (3 x) = - 2 + 3

Apply trig inverse properties

3 x = arctan (- 2 + 3) + πn

Solve

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

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

Show solutions in decimal form

x = \frac{- 0.26179 \dots}{3} + \frac{πn}{3}

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

cos^{2} (x) - 2 cos (x) = 0

sin (3 x) cos (x) + cos (3 x) sin (x) = 1

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

sin (θ) tan^{2} (θ) - sin (θ) = 0

What is the general solution for sin(3x-pi/6)=-cos(3x-pi/6) ?
The general solution for sin(3x-pi/6)=-cos(3x-pi/6) is x=(-0.26179…)/3+(pin)/3