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

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

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Popular Trigonometry >

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

Solution

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

Solution

x = πn, x = \frac{π}{2} + πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Rewrite using trig identities

cos (2 x - \frac{π}{3})

Use the Angle Difference identity:

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

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

Simplify

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

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

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

Use the Angle Difference identity:

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

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

Simplify

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

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

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

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

Simplify

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

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

Apply the distributive law:

a (b + c) = ab + a c

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

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

Apply minus-plus rules

+ (- a) = - a

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= cos (2 x) \cdot 1

Multiply:

cos (2 x) \cdot 1 = cos (2 x)

= cos (2 x)

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= sin (2 x) 3

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

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

Simplify

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= sin (2 x) 3

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= cos (2 x) \cdot 1

Multiply:

cos (2 x) \cdot 1 = cos (2 x)

= cos (2 x)

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

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

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

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

Subtract

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

from both sides

- 23 sin (2 x) = 0

Divide both sides by

- 23

- 23 sin (2 x) = 0

Divide both sides by

- 23

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

Simplify

sin (2 x) = 0

sin (2 x) = 0

General solutions for

sin (2 x) = 0

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}

2 x = 0 + 2 πn, 2 x = π + 2 πn

2 x = 0 + 2 πn, 2 x = π + 2 πn

Solve

2 x = 0 + 2 πn : x = πn

2 x = 0 + 2 πn

0 + 2 πn = 2 πn

2 x = 2 πn

Divide both sides by

2

2 x = 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2}

Simplify

x = πn

x = πn

Solve

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

Divide both sides by

2

2 x = π + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = πn, x = \frac{π}{2} + πn

Graph

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Frequently Asked Questions (FAQ)

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