What is the general solution for cos(x)-cos(x+pi/4)=0 ?

The general solution for cos(x)-cos(x+pi/4)=0 is x=2pin-pi/8 ,x=(7pi)/8+2pin

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cos(x)-cos(x+pi/4)=0

Solution

cos (x) - cos (x + \frac{π}{4}) = 0

Solution

x = 2 πn - \frac{π}{8}, x = \frac{7 π}{8} + 2 πn

Degrees

x = - 22. 5^{\circ} + 36 0^{\circ} n, x = 157. 5^{\circ} + 36 0^{\circ} n

Solution steps

cos (x) - cos (x + \frac{π}{4}) = 0

Rewrite using trig identities

cos (x) - cos (x + \frac{π}{4})

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

Simplify

- 2 sin (\frac{x + x + \frac{π}{4}}{2}) sin (\frac{x - ( x + \frac{π}{4} )}{2}) : 2 - 2 sin (\frac{8 x + π}{8})

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

\frac{x + x + \frac{π}{4}}{2} = \frac{8 x + π}{8}

\frac{x + x + \frac{π}{4}}{2}

Add similar elements:

x + x = 2 x

= \frac{2 x + \frac{π}{4}}{2}

Join

2 x + \frac{π}{4} : \frac{8 x + π}{4}

2 x + \frac{π}{4}

Convert element to fraction:

2 x = \frac{2 x 4}{4}

= \frac{2 x \cdot 4}{4} + \frac{π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{2 x \cdot 4 + π}{4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{8 x + π}{4}

= \frac{\frac{8 x + π}{4}}{2}

Apply the fraction rule:

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

= \frac{8 x + π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{8 x + π}{8}

= - 2 sin (\frac{8 x + π}{8}) sin (\frac{x - ( x + \frac{π}{4} )}{2})

\frac{x - ( x + \frac{π}{4} )}{2} = - \frac{π}{8}

\frac{x - ( x + \frac{π}{4} )}{2}

Expand

x - (x + \frac{π}{4}) : - \frac{π}{4}

x - (x + \frac{π}{4})

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

- (x + \frac{π}{4})

Distribute parentheses

= - (x) - (\frac{π}{4})

Apply minus-plus rules

+ (- a) = - a

= - x - \frac{π}{4}

= x - x - \frac{π}{4}

Add similar elements:

x - x = 0

= - \frac{π}{4}

= \frac{- \frac{π}{4}}{2}

Apply the fraction rule:

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

= - \frac{\frac{π}{4}}{2}

Apply the fraction rule:

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

\frac{\frac{π}{4}}{2} = \frac{π}{4 \cdot 2}

= - \frac{π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= - \frac{π}{8}

= - 2 sin (- \frac{π}{8}) sin (\frac{8 x + π}{8})

sin (- \frac{π}{8}) = - \frac{2 - 2}{2}

sin (- \frac{π}{8})

Use the following property:

sin (- x) = - sin (x)

sin (- \frac{π}{8}) = - sin (\frac{π}{8})

= - sin (\frac{π}{8})

Rewrite using trig identities:

sin (\frac{π}{8}) = \frac{2 - 2}{2}

sin (\frac{π}{8})

Rewrite using trig identities:

\frac{1 - cos ( \frac{π}{4} )}{2}

sin (\frac{π}{8})

Write

sin (\frac{π}{8})

sin (\frac{\frac{π}{4}}{2})

= sin (\frac{\frac{π}{4}}{2})

Use the Half Angle identity:

sin (\frac{θ}{2}) = \frac{1 - cos ( θ )}{2}

Use the Double Angle identity

cos (2 θ) = 1 - 2 sin^{2} (θ)

Substitute

θ

with

\frac{θ}{2}

cos (θ) = 1 - 2 sin^{2} (\frac{θ}{2})

Switch sides

2 sin^{2} (\frac{θ}{2}) = 1 - cos (θ)

Divide both sides by

2

sin^{2} (\frac{θ}{2}) = \frac{( 1 - cos ( θ ) )}{2}

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

range [0, \frac{π}{2}] [\frac{π}{2}, π] [π, \frac{3 π}{2}] [\frac{3 π}{2}, 2 π] quadrant I II III I V sin positive positive negative negative cos positive negative negative positive

sin (\frac{θ}{2}) = \frac{( 1 - cos ( θ ) )}{2}

= \frac{1 - cos ( \frac{π}{4} )}{2}

= \frac{1 - cos ( \frac{π}{4} )}{2}

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

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}{2}

= \frac{1 - \frac{2}{2}}{2}

Simplify

\frac{1 - \frac{2}{2}}{2} : \frac{2 - 2}{2}

\frac{1 - \frac{2}{2}}{2}

\frac{1 - \frac{2}{2}}{2} = \frac{2 - 2}{4}

\frac{1 - \frac{2}{2}}{2}

Join

1 - \frac{2}{2} : \frac{2 - 2}{2}

1 - \frac{2}{2}

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

= \frac{1 \cdot 2}{2} - \frac{2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 2 - 2}{2}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 - 2}{2}

= \frac{\frac{2 - 2}{2}}{2}

Apply the fraction rule:

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

= \frac{2 - 2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 - 2}{4}

= \frac{2 - 2}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2 - 2}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2 - 2}{2}

= \frac{2 - 2}{2}

= - \frac{2 - 2}{2}

= - 2 (- \frac{2 - 2}{2}) sin (\frac{8 x + π}{8})

Apply rule

- (- a) = a

= 2 sin (\frac{8 x + π}{8}) \frac{2 - 2}{2}

Multiply fractions:

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

= \frac{2 2 - 2}{2} sin (\frac{8 x + π}{8})

Cancel the common factor:

2

= sin (\frac{8 x + π}{8}) 2 - 2

= 2 - 2 sin (\frac{8 x + π}{8})

2 - 2 sin (\frac{8 x + π}{8}) = 0

Divide both sides by

2 - 2

2 - 2 sin (\frac{8 x + π}{8}) = 0

Divide both sides by

2 - 2

\frac{2 - 2 sin ( \frac{8 x + π}{8} )}{2 - 2} = \frac{0}{2 - 2}

Simplify

sin (\frac{8 x + π}{8}) = 0

sin (\frac{8 x + π}{8}) = 0

General solutions for

sin (\frac{8 x + π}{8}) = 0

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{8 x + π}{8} = 0 + 2 πn, \frac{8 x + π}{8} = π + 2 πn

\frac{8 x + π}{8} = 0 + 2 πn, \frac{8 x + π}{8} = π + 2 πn

Solve

\frac{8 x + π}{8} = 0 + 2 πn : x = 2 πn - \frac{π}{8}

\frac{8 x + π}{8} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{8 x + π}{8} = 2 πn

Multiply both sides by

8

\frac{8 x + π}{8} = 2 πn

Multiply both sides by

8

\frac{8 ( 8 x + π )}{8} = 8 \cdot 2 πn

Simplify

8 x + π = 16 πn

8 x + π = 16 πn

Move

π

to the right side

8 x + π = 16 πn

Subtract

π

from both sides

8 x + π - π = 16 πn - π

Simplify

8 x = 16 πn - π

8 x = 16 πn - π

Divide both sides by

8

8 x = 16 πn - π

Divide both sides by

8

\frac{8 x}{8} = \frac{16 πn}{8} - \frac{π}{8}

Simplify

x = 2 πn - \frac{π}{8}

x = 2 πn - \frac{π}{8}

Solve

\frac{8 x + π}{8} = π + 2 πn : x = \frac{7 π}{8} + 2 πn

\frac{8 x + π}{8} = π + 2 πn

Multiply both sides by

8

\frac{8 x + π}{8} = π + 2 πn

Multiply both sides by

8

\frac{8 ( 8 x + π )}{8} = 8 π + 8 \cdot 2 πn

Simplify

8 x + π = 8 π + 16 πn

8 x + π = 8 π + 16 πn

Move

π

to the right side

8 x + π = 8 π + 16 πn

Subtract

π

from both sides

8 x + π - π = 8 π + 16 πn - π

Simplify

8 x = 7 π + 16 πn

8 x = 7 π + 16 πn

Divide both sides by

8

8 x = 7 π + 16 πn

Divide both sides by

8

\frac{8 x}{8} = \frac{7 π}{8} + \frac{16 πn}{8}

Simplify

x = \frac{7 π}{8} + 2 πn

x = \frac{7 π}{8} + 2 πn

x = 2 πn - \frac{π}{8}, x = \frac{7 π}{8} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x)-cos(x+pi/4)=0 ?
The general solution for cos(x)-cos(x+pi/4)=0 is x=2pin-pi/8 ,x=(7pi)/8+2pin