English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Solution

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

Solution

x = \frac{πn}{2} - 1, x = \frac{π}{5} + \frac{4 πn}{5} + \frac{1}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5} + \frac{1}{5}, x = - 4 πn + 1, x = 1 - 2 π - 4 πn

Degrees

x = - 57.29577 \dots^{\circ} + 9 0^{\circ} n, x = 47.45915 \dots^{\circ} + 14 4^{\circ} n, x = 119.45915 \dots^{\circ} + 14 4^{\circ} n, x = 57.29577 \dots^{\circ} - 72 0^{\circ} n, x = - 302.70422 \dots^{\circ} - 72 0^{\circ} n

Solution steps

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

Subtract

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

from both sides

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

Factor

sin (2 x) tan (2 x + 2) - sin (3 x - 1) tan (2 x + 2) : tan (2 (x + 1)) (sin (2 x) - sin (- 1 + 3 x))

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

Factor out common term

tan (2 + 2 x)

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

Factor

2 x + 2 : 2 (x + 1)

2 x + 2

Factor out common term

2

= 2 (x + 1)

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

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

Solving each part separately

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

tan (2 (x + 1)) = 0 : x = \frac{πn}{2} - 1

tan (2 (x + 1)) = 0

General solutions for

tan (2 (x + 1)) = 0

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}

2 (x + 1) = 0 + πn

2 (x + 1) = 0 + πn

Solve

2 (x + 1) = 0 + πn : x = \frac{πn}{2} - 1

2 (x + 1) = 0 + πn

0 + πn = πn

2 (x + 1) = πn

Divide both sides by

2

2 (x + 1) = πn

Divide both sides by

2

\frac{2 ( x + 1 )}{2} = \frac{πn}{2}

Simplify

x + 1 = \frac{πn}{2}

x + 1 = \frac{πn}{2}

Move

1

to the right side

x + 1 = \frac{πn}{2}

Subtract

1

from both sides

x + 1 - 1 = \frac{πn}{2} - 1

Simplify

x = \frac{πn}{2} - 1

x = \frac{πn}{2} - 1

x = \frac{πn}{2} - 1

sin (2 x) - sin (- 1 + 3 x) = 0 : x = \frac{π}{5} + \frac{4 πn}{5} + \frac{1}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5} + \frac{1}{5}, x = - 4 πn + 1, x = 1 - 2 π - 4 πn

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

Rewrite using trig identities

- sin (- 1 + 3 x) + sin (2 x)

Use the Sum to Product identity:

sin (s) - sin (t) = 2 sin (\frac{s - t}{2}) cos (\frac{s + t}{2})

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

Simplify

2 sin (\frac{2 x - ( - 1 + 3 x )}{2}) cos (\frac{2 x - 1 + 3 x}{2}) : 2 sin (\frac{- x + 1}{2}) cos (\frac{5 x - 1}{2})

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

Expand

2 x - (- 1 + 3 x) : - x + 1

2 x - (- 1 + 3 x)

- (- 1 + 3 x) : 1 - 3 x

- (- 1 + 3 x)

Distribute parentheses

= - (- 1) - (3 x)

Apply minus-plus rules

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

= 1 - 3 x

= 2 x + 1 - 3 x

Simplify

2 x + 1 - 3 x : - x + 1

2 x + 1 - 3 x

Group like terms

= 2 x - 3 x + 1

Add similar elements:

2 x - 3 x = - x

= - x + 1

= - x + 1

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

2 x - 1 + 3 x = 5 x - 1

2 x - 1 + 3 x

Group like terms

= 2 x + 3 x - 1

Add similar elements:

2 x + 3 x = 5 x

= 5 x - 1

= 2 sin (\frac{- x + 1}{2}) cos (\frac{5 x - 1}{2})

= 2 sin (\frac{- x + 1}{2}) cos (\frac{5 x - 1}{2})

2 cos (\frac{- 1 + 5 x}{2}) sin (\frac{1 - x}{2}) = 0

Solving each part separately

cos (\frac{- 1 + 5 x}{2}) = 0 or sin (\frac{1 - x}{2}) = 0

cos (\frac{- 1 + 5 x}{2}) = 0 : x = \frac{π}{5} + \frac{4 πn}{5} + \frac{1}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5} + \frac{1}{5}

cos (\frac{- 1 + 5 x}{2}) = 0

General solutions for

cos (\frac{- 1 + 5 x}{2}) = 0

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 + 5 x}{2} = \frac{π}{2} + 2 πn, \frac{- 1 + 5 x}{2} = \frac{3 π}{2} + 2 πn

\frac{- 1 + 5 x}{2} = \frac{π}{2} + 2 πn, \frac{- 1 + 5 x}{2} = \frac{3 π}{2} + 2 πn

Solve

\frac{- 1 + 5 x}{2} = \frac{π}{2} + 2 πn : x = \frac{π}{5} + \frac{4 πn}{5} + \frac{1}{5}

\frac{- 1 + 5 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{- 1 + 5 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 ( - 1 + 5 x )}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( - 1 + 5 x )}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( - 1 + 5 x )}{2} : - 1 + 5 x

\frac{2 ( - 1 + 5 x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 1 + 5 x

Simplify

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

- 1 + 5 x = π + 4 πn

- 1 + 5 x = π + 4 πn

- 1 + 5 x = π + 4 πn

Move

1

to the right side

- 1 + 5 x = π + 4 πn

Add

1

to both sides

- 1 + 5 x + 1 = π + 4 πn + 1

Simplify

5 x = π + 4 πn + 1

5 x = π + 4 πn + 1

Divide both sides by

5

5 x = π + 4 πn + 1

Divide both sides by

5

\frac{5 x}{5} = \frac{π}{5} + \frac{4 πn}{5} + \frac{1}{5}

Simplify

x = \frac{π}{5} + \frac{4 πn}{5} + \frac{1}{5}

x = \frac{π}{5} + \frac{4 πn}{5} + \frac{1}{5}

Solve

\frac{- 1 + 5 x}{2} = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{5} + \frac{4 πn}{5} + \frac{1}{5}

\frac{- 1 + 5 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{- 1 + 5 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{2 ( - 1 + 5 x )}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( - 1 + 5 x )}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( - 1 + 5 x )}{2} : - 1 + 5 x

\frac{2 ( - 1 + 5 x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 1 + 5 x

Simplify

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn : 3 π + 4 πn

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

- 1 + 5 x = 3 π + 4 πn

- 1 + 5 x = 3 π + 4 πn

- 1 + 5 x = 3 π + 4 πn

Move

1

to the right side

- 1 + 5 x = 3 π + 4 πn

Add

1

to both sides

- 1 + 5 x + 1 = 3 π + 4 πn + 1

Simplify

5 x = 3 π + 4 πn + 1

5 x = 3 π + 4 πn + 1

Divide both sides by

5

5 x = 3 π + 4 πn + 1

Divide both sides by

5

\frac{5 x}{5} = \frac{3 π}{5} + \frac{4 πn}{5} + \frac{1}{5}

Simplify

x = \frac{3 π}{5} + \frac{4 πn}{5} + \frac{1}{5}

x = \frac{3 π}{5} + \frac{4 πn}{5} + \frac{1}{5}

x = \frac{π}{5} + \frac{4 πn}{5} + \frac{1}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5} + \frac{1}{5}

sin (\frac{1 - x}{2}) = 0 : x = - 4 πn + 1, x = 1 - 2 π - 4 πn

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

General solutions for

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

\frac{1 - x}{2} = 0 + 2 πn, \frac{1 - x}{2} = π + 2 πn

\frac{1 - x}{2} = 0 + 2 πn, \frac{1 - x}{2} = π + 2 πn

Solve

\frac{1 - x}{2} = 0 + 2 πn : x = - 4 πn + 1

\frac{1 - x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{1 - x}{2} = 2 πn

Multiply both sides by

2

\frac{1 - x}{2} = 2 πn

Multiply both sides by

2

\frac{2 ( 1 - x )}{2} = 2 \cdot 2 πn

Simplify

1 - x = 4 πn

1 - x = 4 πn

Move

1

to the right side

1 - x = 4 πn

Subtract

1

from both sides

1 - x - 1 = 4 πn - 1

Simplify

- x = 4 πn - 1

- x = 4 πn - 1

Divide both sides by

- 1

- x = 4 πn - 1

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{4 πn}{- 1} - \frac{1}{- 1}

Simplify

\frac{- x}{- 1} = \frac{4 πn}{- 1} - \frac{1}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{4 πn}{- 1} - \frac{1}{- 1} : - 4 πn + 1

\frac{4 πn}{- 1} - \frac{1}{- 1}

\frac{4 πn}{- 1} = - 4 πn

\frac{4 πn}{- 1}

Apply the fraction rule:

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

= - \frac{4 πn}{1}

Apply rule

\frac{a}{1} = a

= - 4 πn

= - 4 πn - \frac{1}{- 1}

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

\frac{1}{- 1}

Apply the fraction rule:

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

= - \frac{1}{1}

Apply rule

\frac{a}{1} = a

= - 1

= - 4 πn - (- 1)

Apply rule

- (- a) = a

= - 4 πn + 1

x = - 4 πn + 1

x = - 4 πn + 1

x = - 4 πn + 1

Solve

\frac{1 - x}{2} = π + 2 πn : x = 1 - 2 π - 4 πn

\frac{1 - x}{2} = π + 2 πn

Multiply both sides by

2

\frac{1 - x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 ( 1 - x )}{2} = 2 π + 2 \cdot 2 πn

Simplify

1 - x = 2 π + 4 πn

1 - x = 2 π + 4 πn

Move

1

to the right side

1 - x = 2 π + 4 πn

Subtract

1

from both sides

1 - x - 1 = 2 π + 4 πn - 1

Simplify

- x = 2 π + 4 πn - 1

- x = 2 π + 4 πn - 1

Divide both sides by

- 1

- x = 2 π + 4 πn - 1

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{2 π}{- 1} + \frac{4 πn}{- 1} - \frac{1}{- 1}

Simplify

\frac{- x}{- 1} = \frac{2 π}{- 1} + \frac{4 πn}{- 1} - \frac{1}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{2 π}{- 1} + \frac{4 πn}{- 1} - \frac{1}{- 1} : 1 - 2 π - 4 πn

\frac{2 π}{- 1} + \frac{4 πn}{- 1} - \frac{1}{- 1}

Group like terms

= - \frac{1}{- 1} + \frac{2 π}{- 1} + \frac{4 πn}{- 1}

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

\frac{1}{- 1}

Apply the fraction rule:

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

= - \frac{1}{1}

Apply rule

\frac{a}{1} = a

= - 1

\frac{2 π}{- 1} = - 2 π

\frac{2 π}{- 1}

Apply the fraction rule:

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

= - \frac{2 π}{1}

Apply rule

\frac{a}{1} = a

= - 2 π

= - (- 1) - 2 π + \frac{4 πn}{- 1}

Apply rule

- (- a) = a

= 1 - 2 π + \frac{4 πn}{- 1}

\frac{4 πn}{- 1} = - 4 πn

\frac{4 πn}{- 1}

Apply the fraction rule:

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

= - \frac{4 πn}{1}

Apply rule

\frac{a}{1} = a

= - 4 πn

= 1 - 2 π - 4 πn

x = 1 - 2 π - 4 πn

x = 1 - 2 π - 4 πn

x = 1 - 2 π - 4 πn

x = - 4 πn + 1, x = 1 - 2 π - 4 πn

Combine all the solutions

x = \frac{π}{5} + \frac{4 πn}{5} + \frac{1}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5} + \frac{1}{5}, x = - 4 πn + 1, x = 1 - 2 π - 4 πn

Combine all the solutions

x = \frac{πn}{2} - 1, x = \frac{π}{5} + \frac{4 πn}{5} + \frac{1}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5} + \frac{1}{5}, x = - 4 πn + 1, x = 1 - 2 π - 4 πn

Popular Examples

2sin^2(θ)-7sin(θ)-4=0

2 sin^{2} (θ) - 7 sin (θ) - 4 = 0

2tan(x)-2sqrt(3)=0

2 tan (x) - 23 = 0

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

sin^{2} (x) + cos (x) = - 1

2sin^2(x/4)-3cos(x/4)=0

2 sin^{2} (\frac{x}{4}) - 3 cos (\frac{x}{4}) = 0

cos^2(A)+sin^2(A)= 1/(sin^2(A))

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