What is the general solution for cos(x+pi/6)-cos(x-pi/6)=1 ?

The general solution for cos(x+pi/6)-cos(x-pi/6)=1 is x=(3pi)/2+2pin

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cos(x+pi/6)-cos(x-pi/6)=1

Solution

cos (x + \frac{π}{6}) - cos (x - \frac{π}{6}) = 1

Solution

x = \frac{3 π}{2} + 2 πn

Degrees

x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (x + \frac{π}{6}) - cos (x - \frac{π}{6}) = 1

Rewrite using trig identities

cos (x + \frac{π}{6}) - cos (x - \frac{π}{6})

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}{6} + x - \frac{π}{6}}{2}) sin (\frac{x + \frac{π}{6} - ( x - \frac{π}{6} )}{2})

Simplify

- 2 sin (\frac{x + \frac{π}{6} + x - \frac{π}{6}}{2}) sin (\frac{x + \frac{π}{6} - ( x - \frac{π}{6} )}{2}) : - sin (x)

- 2 sin (\frac{x + \frac{π}{6} + x - \frac{π}{6}}{2}) sin (\frac{x + \frac{π}{6} - ( x - \frac{π}{6} )}{2})

\frac{x + \frac{π}{6} + x - \frac{π}{6}}{2} = x

\frac{x + \frac{π}{6} + x - \frac{π}{6}}{2}

x + \frac{π}{6} + x - \frac{π}{6} = 2 x

x + \frac{π}{6} + x - \frac{π}{6}

Group like terms

= x + x + \frac{π}{6} - \frac{π}{6}

Add similar elements:

x + x = 2 x

= 2 x + \frac{π}{6} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} = 0

= 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

= - 2 sin (x) sin (\frac{x - ( x - \frac{π}{6} ) + \frac{π}{6}}{2})

\frac{x + \frac{π}{6} - ( x - \frac{π}{6} )}{2} = \frac{π}{6}

\frac{x + \frac{π}{6} - ( x - \frac{π}{6} )}{2}

Join

x + \frac{π}{6} - (x - \frac{π}{6}) : \frac{π}{3}

x + \frac{π}{6} - (x - \frac{π}{6})

Convert element to fraction:

x = \frac{x 6}{6}, (x - \frac{π}{6}) = \frac{( x - \frac{π}{6} ) 6}{6}

= \frac{x \cdot 6}{6} + \frac{π}{6} - \frac{( x - \frac{π}{6} ) \cdot 6}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 6 + π - ( x - \frac{π}{6} ) \cdot 6}{6}

Expand

x \cdot 6 + π - (x - \frac{π}{6}) \cdot 6 : 2 π

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

= 6 x + π - 6 (x - \frac{π}{6})

Expand

- 6 (x - \frac{π}{6}) : - 6 x + π

- 6 (x - \frac{π}{6})

Apply the distributive law:

a (b - c) = ab - a c

a = - 6, b = x, c = \frac{π}{6}

= - 6 x - (- 6) \frac{π}{6}

Apply minus-plus rules

- (- a) = a

= - 6 x + 6 \cdot \frac{π}{6}

6 \cdot \frac{π}{6} = π

6 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 6}{6}

Cancel the common factor:

6

= π

= - 6 x + π

= x \cdot 6 + π - 6 x + π

Simplify

x \cdot 6 + π - 6 x + π : 2 π

x \cdot 6 + π - 6 x + π

Group like terms

= 6 x - 6 x + π + π

Add similar elements:

6 x - 6 x = 0

= π + π

Add similar elements:

π + π = 2 π

= 2 π

= 2 π

= \frac{2 π}{6}

Cancel the common factor:

2

= \frac{π}{3}

= \frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

= - 2 sin (\frac{π}{6}) sin (x)

Simplify

sin (\frac{π}{6}) : \frac{1}{2}

sin (\frac{π}{6})

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{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{1}{2}

= - 2 \cdot \frac{1}{2} sin (x)

Multiply fractions:

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

= - \frac{1 \cdot 2}{2} sin (x)

Cancel the common factor:

2

= - sin (x) \cdot 1

Multiply:

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

= - sin (x)

= - sin (x)

- sin (x) = 1

Divide both sides by

- 1

- sin (x) = 1

Divide both sides by

- 1

\frac{- sin ( x )}{- 1} = \frac{1}{- 1}

Simplify

sin (x) = - 1

sin (x) = - 1

General solutions for

sin (x) = - 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}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x+pi/6)-cos(x-pi/6)=1 ?
The general solution for cos(x+pi/6)-cos(x-pi/6)=1 is x=(3pi)/2+2pin