What is the general solution for solvefor x,cos(x)+cos(x+y)=0 ?

The general solution for solvefor x,cos(x)+cos(x+y)=0 is x= pi/2+2pin-y/2 ,x=(3pi)/2+2pin-y/2

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solvefor x,cos(x)+cos(x+y)=0

Solution

solve for

x, cos (x) + cos (x + y) = 0

Solution

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

Solution steps

cos (x) + cos (x + y) = 0

Rewrite using trig identities

cos (x) + cos (x + y)

Use the Sum to Product identity:

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

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

Simplify

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

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

Add similar elements:

x + x = 2 x

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

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

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

Expand

x - (x + y) : - y

x - (x + y)

- (x + y) : - x - y

- (x + y)

Distribute parentheses

= - x - y

Apply minus-plus rules

+ (- a) = - a

= - x - y

= x - x - y

Add similar elements:

x - x = 0

= - y

= \frac{- y}{2}

Apply the fraction rule:

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

= - \frac{y}{2}

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

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

2 cos (- \frac{y}{2}) cos (\frac{2 x + y}{2}) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

cos (\frac{2 x + y}{2}) = 0

General solutions for

cos (\frac{2 x + y}{2}) = 0

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

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

Solve

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

\frac{2 x + y}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 x + y}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 ( 2 x + y )}{2} : 2 x + y

\frac{2 ( 2 x + y )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 x + y

Simplify

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

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 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

2 x + y = π + 4 πn

2 x + y = π + 4 πn

2 x + y = π + 4 πn

Move

y

to the right side

2 x + y = π + 4 πn

Subtract

y

from both sides

2 x + y - y = π + 4 πn - y

Simplify

2 x = π + 4 πn - y

2 x = π + 4 πn - y

Divide both sides by

2

2 x = π + 4 πn - y

Divide both sides by

2

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

Simplify

x = \frac{π}{2} + 2 πn - \frac{y}{2}

x = \frac{π}{2} + 2 πn - \frac{y}{2}

Solve

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 ( 2 x + y )}{2} : 2 x + y

\frac{2 ( 2 x + y )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 x + y

Simplify

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

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

2 x + y = 3 π + 4 πn

2 x + y = 3 π + 4 πn

2 x + y = 3 π + 4 πn

Move

y

to the right side

2 x + y = 3 π + 4 πn

Subtract

y

from both sides

2 x + y - y = 3 π + 4 πn - y

Simplify

2 x = 3 π + 4 πn - y

2 x = 3 π + 4 πn - y

Divide both sides by

2

2 x = 3 π + 4 πn - y

Divide both sides by

2

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

Simplify

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

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

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

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor x,cos(x)+cos(x+y)=0 ?
The general solution for solvefor x,cos(x)+cos(x+y)=0 is x= pi/2+2pin-y/2 ,x=(3pi)/2+2pin-y/2