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sin(x+pi/4)-sin(x-(5pi)/4)=(sqrt(2))/2

Solution

sin (x + \frac{π}{4}) - sin (x - \frac{5 π}{4}) = \frac{2}{2}

Solution

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

Degrees

x = 15 0^{\circ} + 36 0^{\circ} n, x = 39 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (x + \frac{π}{4}) - sin (x - \frac{5 π}{4}) = \frac{2}{2}

Rewrite using trig identities

sin (x + \frac{π}{4}) - sin (x - \frac{5 π}{4})

Use the Sum to Product identity:

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

= 2 sin (\frac{x + \frac{π}{4} - ( x - \frac{5 π}{4} )}{2}) cos (\frac{x + \frac{π}{4} + x - \frac{5 π}{4}}{2})

Simplify

2 sin (\frac{x + \frac{π}{4} - ( x - \frac{5 π}{4} )}{2}) cos (\frac{x + \frac{π}{4} + x - \frac{5 π}{4}}{2}) : 2 cos (\frac{2 x - π}{2})

2 sin (\frac{x + \frac{π}{4} - ( x - \frac{5 π}{4} )}{2}) cos (\frac{x + \frac{π}{4} + x - \frac{5 π}{4}}{2})

\frac{x + \frac{π}{4} - ( x - \frac{5 π}{4} )}{2} = \frac{3 π}{4}

\frac{x + \frac{π}{4} - ( x - \frac{5 π}{4} )}{2}

Join

x + \frac{π}{4} - (x - \frac{5 π}{4}) : \frac{3 π}{2}

x + \frac{π}{4} - (x - \frac{5 π}{4})

Convert element to fraction:

x = \frac{x 4}{4}, (x - \frac{5 π}{4}) = \frac{( x - \frac{5 π}{4} ) 4}{4}

= \frac{x \cdot 4}{4} + \frac{π}{4} - \frac{( x - \frac{5 π}{4} ) \cdot 4}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 4 + π - ( x - \frac{5 π}{4} ) \cdot 4}{4}

Expand

x \cdot 4 + π - (x - \frac{5 π}{4}) \cdot 4 : 6 π

x \cdot 4 + π - (x - \frac{5 π}{4}) \cdot 4

= 4 x + π - 4 (x - \frac{5 π}{4})

Expand

- 4 (x - \frac{5 π}{4}) : - 4 x + 5 π

- 4 (x - \frac{5 π}{4})

Apply the distributive law:

a (b - c) = ab - a c

a = - 4, b = x, c = \frac{5 π}{4}

= - 4 x - (- 4) \frac{5 π}{4}

Apply minus-plus rules

- (- a) = a

= - 4 x + 4 \cdot \frac{5 π}{4}

4 \cdot \frac{5 π}{4} = 5 π

4 \cdot \frac{5 π}{4}

Multiply fractions:

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

= \frac{5 π 4}{4}

Cancel the common factor:

4

= 5 π

= - 4 x + 5 π

= x \cdot 4 + π - 4 x + 5 π

Simplify

x \cdot 4 + π - 4 x + 5 π : 6 π

x \cdot 4 + π - 4 x + 5 π

Group like terms

= 4 x - 4 x + π + 5 π

Add similar elements:

4 x - 4 x = 0

= π + 5 π

Add similar elements:

π + 5 π = 6 π

= 6 π

= 6 π

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

= \frac{\frac{3 π}{2}}{2}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π}{4}

= 2 sin (\frac{3 π}{4}) cos (\frac{x + x + \frac{π}{4} - \frac{5 π}{4}}{2})

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

\frac{x + \frac{π}{4} + x - \frac{5 π}{4}}{2}

Combine the fractions

\frac{π}{4} - \frac{5 π}{4} : - π

Apply rule

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

= \frac{π - 5 π}{4}

Add similar elements:

π - 5 π = - 4 π

= \frac{- 4 π}{4}

Apply the fraction rule:

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

= - \frac{4 π}{4}

Divide the numbers:

\frac{4}{4} = 1

= - π

= \frac{x - π + x}{2}

x - π + x = 2 x - π

x - π + x

Group like terms

= x + x - π

Add similar elements:

x + x = 2 x

= 2 x - π

= \frac{2 x - π}{2}

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

Simplify

sin (\frac{3 π}{4}) : \frac{2}{2}

sin (\frac{3 π}{4})

Use the following trivial identity:

sin (\frac{3 π}{4}) = \frac{2}{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{2}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 2 cos (\frac{2 x - π}{2})

= 2 cos (\frac{2 x - π}{2})

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 cos ( \frac{2 x - π}{2} )}{2} : cos (\frac{2 x - π}{2})

\frac{2 cos ( \frac{2 x - π}{2} )}{2}

Cancel the common factor:

2

= cos (\frac{2 x - π}{2})

Simplify

\frac{\frac{2}{2}}{2} : \frac{1}{2}

\frac{\frac{2}{2}}{2}

Apply the fraction rule:

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

= \frac{2}{2 2}

Rationalize

\frac{2}{2 2} : \frac{1}{2}

\frac{2}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{2 2}{2 2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

= \frac{1}{2}

cos (\frac{2 x - π}{2}) = \frac{1}{2}

cos (\frac{2 x - π}{2}) = \frac{1}{2}

cos (\frac{2 x - π}{2}) = \frac{1}{2}

General solutions for

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

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

Solve

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 ( 2 x - π )}{2} : 2 x - π

\frac{2 ( 2 x - π )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 x - π

Simplify

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

2 \cdot \frac{π}{3} + 2 \cdot 2 πn

Multiply

2 \cdot \frac{π}{3} : \frac{2 π}{3}

2 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 2}{3}

= \frac{2 π}{3} + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 π}{3} + 4 πn

2 x - π = \frac{2 π}{3} + 4 πn

2 x - π = \frac{2 π}{3} + 4 πn

2 x - π = \frac{2 π}{3} + 4 πn

Move

π

to the right side

2 x - π = \frac{2 π}{3} + 4 πn

Add

π

to both sides

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

Simplify

2 x = \frac{2 π}{3} + 4 πn + π

2 x = \frac{2 π}{3} + 4 πn + π

Divide both sides by

2

2 x = \frac{2 π}{3} + 4 πn + π

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{2 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{2 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

Group like terms

= \frac{π}{2} + \frac{4 πn}{2} + \frac{\frac{2 π}{3}}{2}

\frac{4 πn}{2} = 2 πn

\frac{4 πn}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 πn

\frac{\frac{2 π}{3}}{2} = \frac{π}{3}

\frac{\frac{2 π}{3}}{2}

Apply the fraction rule:

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

= \frac{2 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{2 π}{6}

Cancel the common factor:

2

= \frac{π}{3}

= \frac{π}{2} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{2} + \frac{π}{3}

x = 2 πn + \frac{π}{2} + \frac{π}{3}

x = 2 πn + \frac{π}{2} + \frac{π}{3}

x = 2 πn + \frac{π}{2} + \frac{π}{3}

Solve

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 ( 2 x - π )}{2} : 2 x - π

\frac{2 ( 2 x - π )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 x - π

Simplify

2 \cdot \frac{5 π}{3} + 2 \cdot 2 πn : \frac{10 π}{3} + 4 πn

2 \cdot \frac{5 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{3} = \frac{10 π}{3}

2 \cdot \frac{5 π}{3}

Multiply fractions:

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

= \frac{5 π 2}{3}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{10 π}{3} + 4 πn

2 x - π = \frac{10 π}{3} + 4 πn

2 x - π = \frac{10 π}{3} + 4 πn

2 x - π = \frac{10 π}{3} + 4 πn

Move

π

to the right side

2 x - π = \frac{10 π}{3} + 4 πn

Add

π

to both sides

2 x - π + π = \frac{10 π}{3} + 4 πn + π

Simplify

2 x = \frac{10 π}{3} + 4 πn + π

2 x = \frac{10 π}{3} + 4 πn + π

Divide both sides by

2

2 x = \frac{10 π}{3} + 4 πn + π

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{10 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{10 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{10 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2} : 2 πn + \frac{π}{2} + \frac{5 π}{3}

\frac{\frac{10 π}{3}}{2} + \frac{4 πn}{2} + \frac{π}{2}

Group like terms

= \frac{π}{2} + \frac{4 πn}{2} + \frac{\frac{10 π}{3}}{2}

\frac{4 πn}{2} = 2 πn

\frac{4 πn}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 πn

\frac{\frac{10 π}{3}}{2} = \frac{5 π}{3}

\frac{\frac{10 π}{3}}{2}

Apply the fraction rule:

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

= \frac{10 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{10 π}{6}

Cancel the common factor:

2

= \frac{5 π}{3}

= \frac{π}{2} + 2 πn + \frac{5 π}{3}

Group like terms

= 2 πn + \frac{π}{2} + \frac{5 π}{3}

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

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

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

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

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