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)

Solution

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

Solution

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

Degrees

x = - 5^{\circ} + 6 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Rewrite using trig identities

sin (3 x - \frac{π}{6})

Use the Angle Difference identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

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

Simplify

sin (3 x) cos (\frac{π}{6}) - cos (3 x) sin (\frac{π}{6}) : \frac{3}{2} sin (3 x) - \frac{1}{2} cos (3 x)

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

Simplify

cos (\frac{π}{6}) : \frac{3}{2}

cos (\frac{π}{6})

Use the following trivial identity:

cos (\frac{π}{6}) = \frac{3}{2}

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

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

Simplify

sin (\frac{π}{6}) : \frac{1}{2}

sin (\frac{π}{6})

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

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

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

Use the Angle Difference identity:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

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

Simplify

cos (3 x) cos (\frac{π}{6}) + sin (3 x) sin (\frac{π}{6}) : \frac{3}{2} cos (3 x) + \frac{1}{2} sin (3 x)

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

Simplify

cos (\frac{π}{6}) : \frac{3}{2}

cos (\frac{π}{6})

Use the following trivial identity:

cos (\frac{π}{6}) = \frac{3}{2}

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

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

Simplify

sin (\frac{π}{6}) : \frac{1}{2}

sin (\frac{π}{6})

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

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

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

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

Simplify

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

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

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

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

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

Multiply

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

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

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

Multiply

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

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

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

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

Divide both sides by

cos (3 x), cos (3 x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

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

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

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

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

Move

3

to the right side

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

Subtract

3

from both sides

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

Simplify

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

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

Divide both sides by

1 + 3

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

Divide both sides by

1 + 3

\frac{( 1 + 3 ) tan ( 3 x )}{1 + 3} = \frac{1}{1 + 3} - \frac{3}{1 + 3}

Simplify

\frac{( 1 + 3 ) tan ( 3 x )}{1 + 3} = \frac{1}{1 + 3} - \frac{3}{1 + 3}

Simplify

\frac{( 1 + 3 ) tan ( 3 x )}{1 + 3} : tan (3 x)

\frac{( 1 + 3 ) tan ( 3 x )}{1 + 3}

Cancel the common factor:

1 + 3

= tan (3 x)

Simplify

\frac{1}{1 + 3} - \frac{3}{1 + 3} : - 2 + 3

\frac{1}{1 + 3} - \frac{3}{1 + 3}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 - 3}{1 + 3}

Multiply by the conjugate

\frac{1 - 3}{1 - 3}

= \frac{( 1 - 3 ) ( 1 - 3 )}{( 1 + 3 ) ( 1 - 3 )}

(1 - 3) (1 - 3) = 4 - 23

(1 - 3) (1 - 3)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

(1 - 3) (1 - 3) = (1 - 3)^{1 + 1}

= (1 - 3)^{1 + 1}

Add the numbers:

1 + 1 = 2

= (1 - 3)^{2}

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 1, b = 3

= 1^{2} - 2 \cdot 1 \cdot 3 + (3)^{2}

Simplify

1^{2} - 2 \cdot 1 \cdot 3 + (3)^{2} : 4 - 23

1^{2} - 2 \cdot 1 \cdot 3 + (3)^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 2 \cdot 1 \cdot 3 + (3)^{2}

2 \cdot 1 \cdot 3 = 23

2 \cdot 1 \cdot 3

Multiply the numbers:

2 \cdot 1 = 2

= 23

(3)^{2} = 3

(3)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

= 1 - 23 + 3

Add the numbers:

1 + 3 = 4

= 4 - 23

= 4 - 23

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

(1 + 3) (1 - 3)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = 3

= 1^{2} - (3)^{2}

Simplify

1^{2} - (3)^{2} : - 2

1^{2} - (3)^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - (3)^{2}

(3)^{2} = 3

(3)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

= 1 - 3

Subtract the numbers:

1 - 3 = - 2

= - 2

= - 2

= \frac{4 - 2 3}{- 2}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{4 - 2 3}{2}

Cancel

\frac{4 - 2 3}{2} : 2 - 3

\frac{4 - 2 3}{2}

Factor

4 - 23 : 2 (2 - 3)

4 - 23

Rewrite as

= 2 \cdot 2 - 23

Factor out common term

2

= 2 (2 - 3)

= \frac{2 ( 2 - 3 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 - 3

= - (2 - 3)

Distribute parentheses

= - (2) - (- 3)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 2 + 3

tan (3 x) = - 2 + 3

tan (3 x) = - 2 + 3

tan (3 x) = - 2 + 3

Apply trig inverse properties

tan (3 x) = - 2 + 3

General solutions for

tan (3 x) = - 2 + 3

tan (x) = - a \Rightarrow x = arctan (- a) + πn

3 x = arctan (- 2 + 3) + πn

3 x = arctan (- 2 + 3) + πn

Solve

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

3 x = arctan (- 2 + 3) + πn

Divide both sides by

3

3 x = arctan (- 2 + 3) + πn

Divide both sides by

3

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

Simplify

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

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}

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Frequently Asked Questions (FAQ)

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