What is the general solution for cos(pi/2+x)-cos(pi/2-x)=-sqrt(3) ?

The general solution for cos(pi/2+x)-cos(pi/2-x)=-sqrt(3) is x= pi/3+2pin,x=(2pi)/3+2pin

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cos(pi/2+x)-cos(pi/2-x)=-sqrt(3)

Solution

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

Solution

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Use the Sum to Product identity:

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

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

Simplify

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

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

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

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

Combine the fractions

\frac{π}{2} + \frac{π}{2} : π

Apply rule

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

= \frac{π + π}{2}

Add similar elements:

π + π = 2 π

= \frac{2 π}{2}

Divide the numbers:

\frac{2}{2} = 1

= π

= \frac{π + x - x}{2}

Add similar elements:

x - x = 0

= \frac{π}{2}

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

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

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

Join

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

\frac{π}{2} + x - (\frac{π}{2} - x)

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Expand

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

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

= π + 2 x - 2 (\frac{π}{2} - x)

Expand

- 2 (\frac{π}{2} - x) : - π + 2 x

- 2 (\frac{π}{2} - x)

Apply the distributive law:

a (b - c) = ab - a c

a = - 2, b = \frac{π}{2}, c = x

= - 2 \cdot \frac{π}{2} - (- 2) x

Apply minus-plus rules

- (- a) = a

= - 2 \cdot \frac{π}{2} + 2 x

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

= - π + 2 x

= π + x \cdot 2 - π + 2 x

Simplify

π + x \cdot 2 - π + 2 x : 4 x

π + x \cdot 2 - π + 2 x

Group like terms

= 2 x + 2 x + π - π

Add similar elements:

2 x + 2 x = 4 x

= 4 x + π - π

Add similar elements:

π - π = 0

= 4 x

= 4 x

= \frac{4 x}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

= - 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) = - 3

Divide both sides by

- 2

- 2 sin (x) = - 3

Divide both sides by

- 2

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

Simplify

sin (x) = \frac{3}{2}

sin (x) = \frac{3}{2}

General solutions for

sin (x) = \frac{3}{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{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

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

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Frequently Asked Questions (FAQ)

What is the general solution for cos(pi/2+x)-cos(pi/2-x)=-sqrt(3) ?
The general solution for cos(pi/2+x)-cos(pi/2-x)=-sqrt(3) is x= pi/3+2pin,x=(2pi)/3+2pin