What is the general solution for sin(x-pi/4)-sin(x+pi/4)=0.25 ?

The general solution for sin(x-pi/4)-sin(x+pi/4)=0.25 is x=1.74850…+2pin,x=-1.74850…+2pin

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sin(x-pi/4)-sin(x+pi/4)=0.25

Solution

sin (x - \frac{π}{4}) - sin (x + \frac{π}{4}) = 0.25

Solution

x = 1.74850 \dots + 2 πn, x = - 1.74850 \dots + 2 πn

Degrees

x = 100.18206 \dots^{\circ} + 36 0^{\circ} n, x = - 100.18206 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (x - \frac{π}{4}) - sin (x + \frac{π}{4}) = 0.25

Rewrite using trig identities

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

Simplify

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

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

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

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

Join

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

x - \frac{π}{4} - (x + \frac{π}{4})

Convert element to fraction:

x = \frac{x 4}{4}, (x + \frac{π}{4}) = \frac{( x + \frac{π}{4} ) 4}{4}

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

Expand

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

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

= 4 x - π - 4 (x + \frac{π}{4})

Expand

- 4 (x + \frac{π}{4}) : - 4 x - π

- 4 (x + \frac{π}{4})

Apply the distributive law:

a (b + c) = ab + a c

a = - 4, b = x, c = \frac{π}{4}

= - 4 x + (- 4) \frac{π}{4}

Apply minus-plus rules

+ (- a) = - a

= - 4 x - 4 \cdot \frac{π}{4}

4 \cdot \frac{π}{4} = π

4 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 4}{4}

Cancel the common factor:

4

= π

= - 4 x - π

= x \cdot 4 - π - 4 x - π

Simplify

x \cdot 4 - π - 4 x - π : - 2 π

x \cdot 4 - π - 4 x - π

Group like terms

= 4 x - 4 x - π - π

Add similar elements:

4 x - 4 x = 0

= - π - π

Add similar elements:

- π - π = - 2 π

= - 2 π

= - 2 π

= \frac{- 2 π}{4}

Apply the fraction rule:

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

= - \frac{2 π}{4}

Cancel the common factor:

2

= - \frac{π}{2}

= \frac{- \frac{π}{2}}{2}

Apply the fraction rule:

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

= - \frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

\frac{\frac{π}{2}}{2} = \frac{π}{2 \cdot 2}

= - \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= - \frac{π}{4}

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

sin (- \frac{π}{4}) = - \frac{2}{2}

sin (- \frac{π}{4})

Use the following property:

sin (- x) = - sin (x)

sin (- \frac{π}{4}) = - sin (\frac{π}{4})

= - sin (\frac{π}{4})

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

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}

= - \frac{2}{2}

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

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

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

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

x - \frac{π}{4} + x + \frac{π}{4}

Group like terms

= x + x - \frac{π}{4} + \frac{π}{4}

Add similar elements:

x + x = 2 x

= 2 x - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

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

Remove parentheses:

(- a) = - a

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

Multiply fractions:

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

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

Cancel the common factor:

2

= - 2 cos (x)

= - 2 cos (x)

- 2 cos (x) = 0.25

Divide both sides by

- 2

- 2 cos (x) = 0.25

Divide both sides by

- 2

\frac{- 2 cos ( x )}{- 2} = \frac{0.25}{- 2}

Simplify

\frac{- 2 cos ( x )}{- 2} = \frac{0.25}{- 2}

Simplify

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

2

= cos (x)

Simplify

\frac{0.25}{- 2} : - 0.17677 \dots

\frac{0.25}{- 2}

Apply the fraction rule:

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

= - \frac{0.25}{2}

Simplify

\frac{0.25}{2} : 0.17677 \dots

\frac{0.25}{2}

Convert element to a decimal form

2 = 1.41421 \dots

= \frac{0.25}{1.41421 \dots}

Divide the numbers:

\frac{0.25}{1.41421 \dots} = 0.17677 \dots

= 0.17677 \dots

= - 0.17677 \dots

cos (x) = - 0.17677 \dots

cos (x) = - 0.17677 \dots

cos (x) = - 0.17677 \dots

Apply trig inverse properties

cos (x) = - 0.17677 \dots

General solutions for

cos (x) = - 0.17677 \dots

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- 0.17677 \dots) + 2 πn, x = - arccos (- 0.17677 \dots) + 2 πn

x = arccos (- 0.17677 \dots) + 2 πn, x = - arccos (- 0.17677 \dots) + 2 πn

Show solutions in decimal form

x = 1.74850 \dots + 2 πn, x = - 1.74850 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x-pi/4)-sin(x+pi/4)=0.25 ?
The general solution for sin(x-pi/4)-sin(x+pi/4)=0.25 is x=1.74850…+2pin,x=-1.74850…+2pin