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Popular >

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

Solution

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

Solution

x = \frac{π}{4} + πn, x = 2 π - 2 πn - \frac{7 π}{6}, x = 2 π - 2 πn - \frac{11 π}{6}

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} - 36 0^{\circ} n, x = 3 0^{\circ} - 36 0^{\circ} n

Solution steps

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

Subtract

tan (x) + 2 sin (x)

from both sides

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

Rewrite using trig identities

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

sin (x) = - sin (2 π - x)

sin (x)

Use the following property:

sin (θ) = - sin (- θ)

sin (x) = - sin (- x)

= - sin (- x)

Apply the periodicity of

sin

sin (2 π + θ) = sin (θ)

- sin (- x) = - sin (2 π - x)

= - sin (2 π - x)

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

Simplify

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

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

Factor

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

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

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

Factor out

- 2 sin (- x + 2 π)

from

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

2 sin (- x + 2 π) - 2 tan (x) sin (- x + 2 π)

Factor out common term

- 2 sin (- x + 2 π)

= - 2 sin (- x + 2 π) (- 1 + tan (x))

Factor out

- 1

from

- tan (x) + 1 : - (tan (x) - 1)

- tan (x) + 1

Factor out common term

- 1

= - (tan (x) - 1)

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

Factor out common term

tan (x) - 1

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

Factor

- 2 sin (- x + 2 π) - 1 : - (2 sin (- x + 2 π) + 1)

- 2 sin (- x + 2 π) - 1

Factor out common term

- 1

= - (2 sin (- x + 2 π) + 1)

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

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

Solving each part separately

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

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

tan (x) - 1 = 0

Move

1

to the right side

tan (x) - 1 = 0

Add

1

to both sides

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

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

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}{4} + πn

x = \frac{π}{4} + πn

2 sin (- x + 2 π) + 1 = 0 : x = 2 π - 2 πn - \frac{7 π}{6}, x = 2 π - 2 πn - \frac{11 π}{6}

2 sin (- x + 2 π) + 1 = 0

Move

1

to the right side

2 sin (- x + 2 π) + 1 = 0

Subtract

1

from both sides

2 sin (- x + 2 π) + 1 - 1 = 0 - 1

Simplify

2 sin (- x + 2 π) = - 1

2 sin (- x + 2 π) = - 1

Divide both sides by

2

2 sin (- x + 2 π) = - 1

Divide both sides by

2

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

Simplify

sin (- x + 2 π) = - \frac{1}{2}

sin (- x + 2 π) = - \frac{1}{2}

General solutions for

sin (- x + 2 π) = - \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}

- x + 2 π = \frac{7 π}{6} + 2 πn, - x + 2 π = \frac{11 π}{6} + 2 πn

- x + 2 π = \frac{7 π}{6} + 2 πn, - x + 2 π = \frac{11 π}{6} + 2 πn

Solve

- x + 2 π = \frac{7 π}{6} + 2 πn : x = 2 π - 2 πn - \frac{7 π}{6}

- x + 2 π = \frac{7 π}{6} + 2 πn

Move

2 π

to the right side

- x + 2 π = \frac{7 π}{6} + 2 πn

Subtract

2 π

from both sides

- x + 2 π - 2 π = \frac{7 π}{6} + 2 πn - 2 π

Simplify

- x = \frac{7 π}{6} + 2 πn - 2 π

- x = \frac{7 π}{6} + 2 πn - 2 π

Divide both sides by

- 1

- x = \frac{7 π}{6} + 2 πn - 2 π

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{\frac{7 π}{6}}{- 1} + \frac{2 πn}{- 1} - \frac{2 π}{- 1}

Simplify

\frac{- x}{- 1} = \frac{\frac{7 π}{6}}{- 1} + \frac{2 πn}{- 1} - \frac{2 π}{- 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{\frac{7 π}{6}}{- 1} + \frac{2 πn}{- 1} - \frac{2 π}{- 1} : 2 π - 2 πn - \frac{7 π}{6}

\frac{\frac{7 π}{6}}{- 1} + \frac{2 πn}{- 1} - \frac{2 π}{- 1}

Group like terms

= - \frac{2 π}{- 1} + \frac{2 πn}{- 1} + \frac{\frac{7 π}{6}}{- 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

\frac{2 πn}{- 1} = - 2 πn

\frac{2 πn}{- 1}

Apply the fraction rule:

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

= - \frac{2 πn}{1}

Apply rule

\frac{a}{1} = a

= - 2 πn

= - (- 2 π) - 2 πn + \frac{\frac{7 π}{6}}{- 1}

Apply rule

- (- a) = a

= 2 π - 2 πn + \frac{\frac{7 π}{6}}{- 1}

\frac{\frac{7 π}{6}}{- 1} = - \frac{7 π}{6}

\frac{\frac{7 π}{6}}{- 1}

Apply the fraction rule:

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

= - \frac{\frac{7 π}{6}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{\frac{7 π}{6}}{1} = \frac{7 π}{6}

= - \frac{7 π}{6}

= 2 π - 2 πn - \frac{7 π}{6}

x = 2 π - 2 πn - \frac{7 π}{6}

x = 2 π - 2 πn - \frac{7 π}{6}

x = 2 π - 2 πn - \frac{7 π}{6}

Solve

- x + 2 π = \frac{11 π}{6} + 2 πn : x = 2 π - 2 πn - \frac{11 π}{6}

- x + 2 π = \frac{11 π}{6} + 2 πn

Move

2 π

to the right side

- x + 2 π = \frac{11 π}{6} + 2 πn

Subtract

2 π

from both sides

- x + 2 π - 2 π = \frac{11 π}{6} + 2 πn - 2 π

Simplify

- x = \frac{11 π}{6} + 2 πn - 2 π

- x = \frac{11 π}{6} + 2 πn - 2 π

Divide both sides by

- 1

- x = \frac{11 π}{6} + 2 πn - 2 π

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{\frac{11 π}{6}}{- 1} + \frac{2 πn}{- 1} - \frac{2 π}{- 1}

Simplify

\frac{- x}{- 1} = \frac{\frac{11 π}{6}}{- 1} + \frac{2 πn}{- 1} - \frac{2 π}{- 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{\frac{11 π}{6}}{- 1} + \frac{2 πn}{- 1} - \frac{2 π}{- 1} : 2 π - 2 πn - \frac{11 π}{6}

\frac{\frac{11 π}{6}}{- 1} + \frac{2 πn}{- 1} - \frac{2 π}{- 1}

Group like terms

= - \frac{2 π}{- 1} + \frac{2 πn}{- 1} + \frac{\frac{11 π}{6}}{- 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

\frac{2 πn}{- 1} = - 2 πn

\frac{2 πn}{- 1}

Apply the fraction rule:

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

= - \frac{2 πn}{1}

Apply rule

\frac{a}{1} = a

= - 2 πn

= - (- 2 π) - 2 πn + \frac{\frac{11 π}{6}}{- 1}

Apply rule

- (- a) = a

= 2 π - 2 πn + \frac{\frac{11 π}{6}}{- 1}

\frac{\frac{11 π}{6}}{- 1} = - \frac{11 π}{6}

\frac{\frac{11 π}{6}}{- 1}

Apply the fraction rule:

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

= - \frac{\frac{11 π}{6}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{\frac{11 π}{6}}{1} = \frac{11 π}{6}

= - \frac{11 π}{6}

= 2 π - 2 πn - \frac{11 π}{6}

x = 2 π - 2 πn - \frac{11 π}{6}

x = 2 π - 2 πn - \frac{11 π}{6}

x = 2 π - 2 πn - \frac{11 π}{6}

x = 2 π - 2 πn - \frac{7 π}{6}, x = 2 π - 2 πn - \frac{11 π}{6}

Combine all the solutions

x = \frac{π}{4} + πn, x = 2 π - 2 πn - \frac{7 π}{6}, x = 2 π - 2 πn - \frac{11 π}{6}

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