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

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

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

Solution

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

Solution

x = \frac{π}{3} + πn

Degrees

x = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Rewrite using trig identities

sin (x - \frac{π}{6})

Use the Angle Difference identity:

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

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

Simplify

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

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

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

Use the Angle Difference identity:

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

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

Simplify

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

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

Simplify

cos (\frac{π}{3}) : \frac{1}{2}

cos (\frac{π}{3})

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{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{1}{2}

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

Simplify

sin (\frac{π}{3}) : \frac{3}{2}

sin (\frac{π}{3})

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{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{3}{2}

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

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

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

Simplify

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = \frac{3}{2} sin (x), c = \frac{1}{2} cos (x)

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= sin (x) 3

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= cos (x) \cdot 1

Multiply:

cos (x) \cdot 1 = cos (x)

= cos (x)

= 3 sin (x) - cos (x)

= 3 sin (x) - cos (x)

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

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

Subtract

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

from both sides

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

Simplify

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

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

Multiply

\frac{3}{2} cos (x) : \frac{3 cos ( x )}{2}

\frac{3}{2} cos (x)

Multiply fractions:

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

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Combine the fractions

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

Apply rule

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

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

3 sin (x) \cdot 2 - 3 cos (x) - 3 sin (x) = 3 sin (x) - 3 cos (x)

3 sin (x) \cdot 2 - 3 cos (x) - 3 sin (x)

Group like terms

= 23 sin (x) - 3 cos (x) - 3 sin (x)

Add similar elements:

23 sin (x) - 3 sin (x) = 3 sin (x)

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

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

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

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

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

3 tan (x) - 3 = 0

3 tan (x) - 3 = 0

Move

3

to the right side

3 tan (x) - 3 = 0

Add

3

to both sides

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

Simplify

3 tan (x) = 3

3 tan (x) = 3

Divide both sides by

3

3 tan (x) = 3

Divide both sides by

3

\frac{3 tan ( x )}{3} = \frac{3}{3}

Simplify

\frac{3 tan ( x )}{3} = \frac{3}{3}

Simplify

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

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

Cancel the common factor:

3

= tan (x)

Simplify

\frac{3}{3} : 3

\frac{3}{3}

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

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

Apply radical rule:

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

= 3

tan (x) = 3

tan (x) = 3

tan (x) = 3

General solutions for

tan (x) = 3

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

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cot (θ) = - 5

Frequently Asked Questions (FAQ)

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