What is the general solution for cos(x+pi/2)-cos(x-pi/2)=sqrt(2) ?

The general solution for cos(x+pi/2)-cos(x-pi/2)=sqrt(2) is x=(5pi)/4+2pin,x=(7pi)/4+2pin

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cos(x+pi/2)-cos(x-pi/2)=sqrt(2)

Solution

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

Solution

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Use the Sum to Product identity:

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

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

Simplify

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

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

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

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

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

x + \frac{π}{2} + x - \frac{π}{2}

Group like terms

= x + x + \frac{π}{2} - \frac{π}{2}

Add similar elements:

x + x = 2 x

= 2 x + \frac{π}{2} - \frac{π}{2}

Add similar elements:

\frac{π}{2} - \frac{π}{2} = 0

= 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

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

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

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

Join

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

x + \frac{π}{2} - (x - \frac{π}{2})

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Expand

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

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

= 2 x + π - 2 (x - \frac{π}{2})

Expand

- 2 (x - \frac{π}{2}) : - 2 x + π

- 2 (x - \frac{π}{2})

Apply the distributive law:

a (b - c) = ab - a c

a = - 2, b = x, c = \frac{π}{2}

= - 2 x - (- 2) \frac{π}{2}

Apply minus-plus rules

- (- a) = a

= - 2 x + 2 \cdot \frac{π}{2}

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 x + π

= x \cdot 2 + π - 2 x + π

Simplify

x \cdot 2 + π - 2 x + π : 2 π

x \cdot 2 + π - 2 x + π

Group like terms

= 2 x - 2 x + π + π

Add similar elements:

2 x - 2 x = 0

= π + π

Add similar elements:

π + π = 2 π

= 2 π

= 2 π

= \frac{2 π}{2}

Divide the numbers:

\frac{2}{2} = 1

= π

= \frac{π}{2}

= - 2 sin (\frac{π}{2}) sin (x)

Simplify

sin (\frac{π}{2}) : 1

sin (\frac{π}{2})

Use the following trivial identity:

sin (\frac{π}{2}) = 1

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}

= 1

= - 2 \cdot 1 \cdot sin (x)

Multiply the numbers:

2 \cdot 1 = 2

= - 2 sin (x)

= - 2 sin (x)

- 2 sin (x) = 2

Divide both sides by

- 2

- 2 sin (x) = 2

Divide both sides by

- 2

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

Simplify

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

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

General solutions for

sin (x) = - \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}

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x+pi/2)-cos(x-pi/2)=sqrt(2) ?
The general solution for cos(x+pi/2)-cos(x-pi/2)=sqrt(2) is x=(5pi)/4+2pin,x=(7pi)/4+2pin