What is the general solution for sin(135-x)=sin(x) ?

The general solution for sin(135-x)=sin(x) is x=-2pin+135/2 ,x=-pi+135/2-2pin

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Popular Trigonometry >

sin(135-x)=sin(x)

Solution

sin (135 - x) = sin (x)

Solution

x = - 2 πn + \frac{135}{2}, x = - π + \frac{135}{2} - 2 πn

Degrees

x = 3867.46511 \dots^{\circ} - 36 0^{\circ} n, x = 3687.46511 \dots^{\circ} - 36 0^{\circ} n

Solution steps

sin (135 - x) = sin (x)

Subtract

sin (x)

from both sides

sin (135 - x) - sin (x) = 0

Rewrite using trig identities

sin (135 - x) - sin (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{135 - x - x}{2}) cos (\frac{135 - x + x}{2})

Simplify

2 sin (\frac{135 - x - x}{2}) cos (\frac{135 - x + x}{2}) : 2 cos (\frac{135}{2}) sin (\frac{135 - 2 x}{2})

2 sin (\frac{135 - x - x}{2}) cos (\frac{135 - x + x}{2})

Add similar elements:

- x - x = - 2 x

= 2 sin (\frac{- 2 x + 135}{2}) cos (\frac{x - x + 135}{2})

Add similar elements:

- x + x = 0

= 2 cos (\frac{135}{2}) sin (\frac{- 2 x + 135}{2})

= 2 cos (\frac{135}{2}) sin (\frac{135 - 2 x}{2})

2 cos (\frac{135}{2}) sin (\frac{135 - 2 x}{2}) = 0

Divide both sides by

2 cos (\frac{135}{2})

2 cos (\frac{135}{2}) sin (\frac{135 - 2 x}{2}) = 0

Divide both sides by

2 cos (\frac{135}{2})

\frac{2 cos ( \frac{135}{2} ) sin ( \frac{135 - 2 x}{2} )}{2 cos ( \frac{135}{2} )} = \frac{0}{2 cos ( \frac{135}{2} )}

Simplify

sin (\frac{135 - 2 x}{2}) = 0

sin (\frac{135 - 2 x}{2}) = 0

General solutions for

sin (\frac{135 - 2 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{135 - 2 x}{2} = 0 + 2 πn, \frac{135 - 2 x}{2} = π + 2 πn

\frac{135 - 2 x}{2} = 0 + 2 πn, \frac{135 - 2 x}{2} = π + 2 πn

Solve

\frac{135 - 2 x}{2} = 0 + 2 πn : x = - 2 πn + \frac{135}{2}

\frac{135 - 2 x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{135 - 2 x}{2} = 2 πn

Multiply both sides by

2

\frac{135 - 2 x}{2} = 2 πn

Multiply both sides by

2

\frac{2 ( 135 - 2 x )}{2} = 2 \cdot 2 πn

Simplify

135 - 2 x = 4 πn

135 - 2 x = 4 πn

Move

135

to the right side

135 - 2 x = 4 πn

Subtract

135

from both sides

135 - 2 x - 135 = 4 πn - 135

Simplify

- 2 x = 4 πn - 135

- 2 x = 4 πn - 135

Divide both sides by

- 2

- 2 x = 4 πn - 135

Divide both sides by

- 2

\frac{- 2 x}{- 2} = \frac{4 πn}{- 2} - \frac{135}{- 2}

Simplify

\frac{- 2 x}{- 2} = \frac{4 πn}{- 2} - \frac{135}{- 2}

Simplify

\frac{- 2 x}{- 2} : x

\frac{- 2 x}{- 2}

Apply the fraction rule:

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

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{4 πn}{- 2} - \frac{135}{- 2} : - 2 πn + \frac{135}{2}

\frac{4 πn}{- 2} - \frac{135}{- 2}

\frac{4 πn}{- 2} = - 2 πn

\frac{4 πn}{- 2}

Apply the fraction rule:

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

= - \frac{4 πn}{2}

Divide the numbers:

\frac{4}{2} = 2

= - 2 πn

= - 2 πn - \frac{135}{- 2}

Apply the fraction rule:

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

= - 2 πn - (- \frac{135}{2})

Apply rule

- (- a) = a

= - 2 πn + \frac{135}{2}

x = - 2 πn + \frac{135}{2}

x = - 2 πn + \frac{135}{2}

x = - 2 πn + \frac{135}{2}

Solve

\frac{135 - 2 x}{2} = π + 2 πn : x = - π + \frac{135}{2} - 2 πn

\frac{135 - 2 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{135 - 2 x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 ( 135 - 2 x )}{2} = 2 π + 2 \cdot 2 πn

Simplify

135 - 2 x = 2 π + 4 πn

135 - 2 x = 2 π + 4 πn

Move

135

to the right side

135 - 2 x = 2 π + 4 πn

Subtract

135

from both sides

135 - 2 x - 135 = 2 π + 4 πn - 135

Simplify

- 2 x = 2 π + 4 πn - 135

- 2 x = 2 π + 4 πn - 135

Divide both sides by

- 2

- 2 x = 2 π + 4 πn - 135

Divide both sides by

- 2

\frac{- 2 x}{- 2} = \frac{2 π}{- 2} + \frac{4 πn}{- 2} - \frac{135}{- 2}

Simplify

\frac{- 2 x}{- 2} = \frac{2 π}{- 2} + \frac{4 πn}{- 2} - \frac{135}{- 2}

Simplify

\frac{- 2 x}{- 2} : x

\frac{- 2 x}{- 2}

Apply the fraction rule:

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

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 π}{- 2} + \frac{4 πn}{- 2} - \frac{135}{- 2} : - π + \frac{135}{2} - 2 πn

\frac{2 π}{- 2} + \frac{4 πn}{- 2} - \frac{135}{- 2}

Group like terms

= \frac{2 π}{- 2} - \frac{135}{- 2} + \frac{4 πn}{- 2}

\frac{2 π}{- 2} = - π

\frac{2 π}{- 2}

Apply the fraction rule:

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

= - \frac{2 π}{2}

Divide the numbers:

\frac{2}{2} = 1

= - π

= - π - \frac{135}{- 2} + \frac{4 πn}{- 2}

\frac{135}{- 2} = - \frac{135}{2}

\frac{135}{- 2}

Apply the fraction rule:

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

= - \frac{135}{2}

\frac{4 πn}{- 2} = - 2 πn

\frac{4 πn}{- 2}

Apply the fraction rule:

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

= - \frac{4 πn}{2}

Divide the numbers:

\frac{4}{2} = 2

= - 2 πn

= - π - (- \frac{135}{2}) - 2 πn

Apply rule

- (- a) = a

= - π + \frac{135}{2} - 2 πn

x = - π + \frac{135}{2} - 2 πn

x = - π + \frac{135}{2} - 2 πn

x = - π + \frac{135}{2} - 2 πn

x = - 2 πn + \frac{135}{2}, x = - π + \frac{135}{2} - 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(135-x)=sin(x) ?
The general solution for sin(135-x)=sin(x) is x=-2pin+135/2 ,x=-pi+135/2-2pin