What is the general solution for cos(40-x)=cos(x) ?

The general solution for cos(40-x)=cos(x) is x=-360n+20,x=-160-360n

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Popular Trigonometry >

cos(40-x)=cos(x)

Solution

cos (4 0^{\circ} - x) = cos (x)

Solution

x = - 36 0^{\circ} n + 2 0^{\circ}, x = - 16 0^{\circ} - 36 0^{\circ} n

Radians

x = \frac{π}{9} - 2 πn, x = - \frac{8 π}{9} - 2 πn

Solution steps

cos (4 0^{\circ} - x) = cos (x)

Subtract

cos (x)

from both sides

cos (4 0^{\circ} - x) - cos (x) = 0

Rewrite using trig identities

cos (4 0^{\circ} - x) - cos (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{4 0 ^{\circ} - x + x}{2}) sin (\frac{4 0 ^{\circ} - x - x}{2})

Simplify

- 2 sin (\frac{4 0 ^{\circ} - x + x}{2}) sin (\frac{4 0 ^{\circ} - x - x}{2}) : - 2 sin (2 0^{\circ}) sin (\frac{18 0 ^{\circ} - 9 x}{9})

- 2 sin (\frac{4 0 ^{\circ} - x + x}{2}) sin (\frac{4 0 ^{\circ} - x - x}{2})

\frac{4 0 ^{\circ} - x + x}{2} = 2 0^{\circ}

\frac{4 0 ^{\circ} - x + x}{2}

Add similar elements:

- x + x = 0

= \frac{4 0 ^{\circ}}{2}

Apply the fraction rule:

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

= \frac{36 0 ^{\circ}}{9 \cdot 2}

Multiply the numbers:

9 \cdot 2 = 18

= 2 0^{\circ}

Cancel the common factor:

2

= 2 0^{\circ}

= - 2 sin (2 0^{\circ}) sin (\frac{- x - x + 4 0 ^{\circ}}{2})

\frac{4 0 ^{\circ} - x - x}{2} = \frac{18 0 ^{\circ} - 9 x}{9}

\frac{4 0 ^{\circ} - x - x}{2}

Add similar elements:

- x - x = - 2 x

= \frac{4 0 ^{\circ} - 2 x}{2}

Join

4 0^{\circ} - 2 x : \frac{36 0 ^{\circ} - 18 x}{9}

4 0^{\circ} - 2 x

Convert element to fraction:

2 x = \frac{2 x 9}{9}

= 4 0^{\circ} - \frac{2 x \cdot 9}{9}

Since the denominators are equal, combine the fractions:

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

= \frac{36 0 ^{\circ} - 2 x \cdot 9}{9}

Multiply the numbers:

2 \cdot 9 = 18

= \frac{36 0 ^{\circ} - 18 x}{9}

= \frac{\frac{36 0 ^{\circ} - 18 x}{9}}{2}

Apply the fraction rule:

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

= \frac{36 0 ^{\circ} - 18 x}{9 \cdot 2}

Multiply the numbers:

9 \cdot 2 = 18

= \frac{36 0 ^{\circ} - 18 x}{18}

Factor

36 0^{\circ} - 18 x : 2 (18 0^{\circ} - 9 x)

36 0^{\circ} - 18 x

Rewrite as

= 36 0^{\circ} - 2 \cdot 9 x

Factor out common term

2

= 2 (18 0^{\circ} - 9 x)

= \frac{2 ( 18 0 ^{\circ} - 9 x )}{18}

Cancel the common factor:

2

= \frac{18 0 ^{\circ} - 9 x}{9}

= - 2 sin (2 0^{\circ}) sin (\frac{- 9 x + 18 0 ^{\circ}}{9})

= - 2 sin (2 0^{\circ}) sin (\frac{18 0 ^{\circ} - 9 x}{9})

- 2 sin (2 0^{\circ}) sin (\frac{18 0 ^{\circ} - 9 x}{9}) = 0

Divide both sides by

- 2 sin (2 0^{\circ})

- 2 sin (2 0^{\circ}) sin (\frac{18 0 ^{\circ} - 9 x}{9}) = 0

Divide both sides by

- 2 sin (2 0^{\circ})

\frac{- 2 sin ( 2 0 ^{\circ} ) sin ( \frac{18 0 ^{\circ} - 9 x}{9} )}{- 2 sin ( 2 0 ^{\circ} )} = \frac{0}{- 2 sin ( 2 0 ^{\circ} )}

Simplify

sin (\frac{18 0 ^{\circ} - 9 x}{9}) = 0

sin (\frac{18 0 ^{\circ} - 9 x}{9}) = 0

General solutions for

sin (\frac{18 0 ^{\circ} - 9 x}{9}) = 0

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{18 0 ^{\circ} - 9 x}{9} = 0 + 36 0^{\circ} n, \frac{18 0 ^{\circ} - 9 x}{9} = 18 0^{\circ} + 36 0^{\circ} n

\frac{18 0 ^{\circ} - 9 x}{9} = 0 + 36 0^{\circ} n, \frac{18 0 ^{\circ} - 9 x}{9} = 18 0^{\circ} + 36 0^{\circ} n

Solve

\frac{18 0 ^{\circ} - 9 x}{9} = 0 + 36 0^{\circ} n : x = - 36 0^{\circ} n + 2 0^{\circ}

\frac{18 0 ^{\circ} - 9 x}{9} = 0 + 36 0^{\circ} n

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

\frac{18 0 ^{\circ} - 9 x}{9} = 36 0^{\circ} n

Multiply both sides by

9

\frac{18 0 ^{\circ} - 9 x}{9} = 36 0^{\circ} n

Multiply both sides by

9

\frac{9 ( 18 0 ^{\circ} - 9 x )}{9} = 9 \cdot 36 0^{\circ} n

Simplify

18 0^{\circ} - 9 x = 324 0^{\circ} n

18 0^{\circ} - 9 x = 324 0^{\circ} n

Move

18 0^{\circ}

to the right side

18 0^{\circ} - 9 x = 324 0^{\circ} n

Subtract

18 0^{\circ}

from both sides

18 0^{\circ} - 9 x - 18 0^{\circ} = 324 0^{\circ} n - 18 0^{\circ}

Simplify

- 9 x = 324 0^{\circ} n - 18 0^{\circ}

- 9 x = 324 0^{\circ} n - 18 0^{\circ}

Divide both sides by

- 9

- 9 x = 324 0^{\circ} n - 18 0^{\circ}

Divide both sides by

- 9

\frac{- 9 x}{- 9} = \frac{324 0 ^{\circ} n}{- 9} - \frac{18 0 ^{\circ}}{- 9}

Simplify

\frac{- 9 x}{- 9} = \frac{324 0 ^{\circ} n}{- 9} - \frac{18 0 ^{\circ}}{- 9}

Simplify

\frac{- 9 x}{- 9} : x

\frac{- 9 x}{- 9}

Apply the fraction rule:

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

= \frac{9 x}{9}

Divide the numbers:

\frac{9}{9} = 1

= x

Simplify

\frac{324 0 ^{\circ} n}{- 9} - \frac{18 0 ^{\circ}}{- 9} : - 36 0^{\circ} n + 2 0^{\circ}

\frac{324 0 ^{\circ} n}{- 9} - \frac{18 0 ^{\circ}}{- 9}

\frac{324 0 ^{\circ} n}{- 9} = - 36 0^{\circ} n

\frac{324 0 ^{\circ} n}{- 9}

Apply the fraction rule:

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

= - \frac{324 0 ^{\circ} n}{9}

Divide the numbers:

\frac{18}{9} = 2

= - 36 0^{\circ} n

= - 36 0^{\circ} n - \frac{18 0 ^{\circ}}{- 9}

Apply the fraction rule:

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

= - 36 0^{\circ} n - (- 2 0^{\circ})

Apply rule

- (- a) = a

= - 36 0^{\circ} n + 2 0^{\circ}

x = - 36 0^{\circ} n + 2 0^{\circ}

x = - 36 0^{\circ} n + 2 0^{\circ}

x = - 36 0^{\circ} n + 2 0^{\circ}

Solve

\frac{18 0 ^{\circ} - 9 x}{9} = 18 0^{\circ} + 36 0^{\circ} n : x = - 16 0^{\circ} - 36 0^{\circ} n

\frac{18 0 ^{\circ} - 9 x}{9} = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

9

\frac{18 0 ^{\circ} - 9 x}{9} = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

9

\frac{9 ( 18 0 ^{\circ} - 9 x )}{9} = 162 0^{\circ} + 9 \cdot 36 0^{\circ} n

Simplify

18 0^{\circ} - 9 x = 162 0^{\circ} + 324 0^{\circ} n

18 0^{\circ} - 9 x = 162 0^{\circ} + 324 0^{\circ} n

Move

18 0^{\circ}

to the right side

18 0^{\circ} - 9 x = 162 0^{\circ} + 324 0^{\circ} n

Subtract

18 0^{\circ}

from both sides

18 0^{\circ} - 9 x - 18 0^{\circ} = 162 0^{\circ} + 324 0^{\circ} n - 18 0^{\circ}

Simplify

- 9 x = 144 0^{\circ} + 324 0^{\circ} n

- 9 x = 144 0^{\circ} + 324 0^{\circ} n

Divide both sides by

- 9

- 9 x = 144 0^{\circ} + 324 0^{\circ} n

Divide both sides by

- 9

\frac{- 9 x}{- 9} = \frac{144 0 ^{\circ}}{- 9} + \frac{324 0 ^{\circ} n}{- 9}

Simplify

\frac{- 9 x}{- 9} = \frac{144 0 ^{\circ}}{- 9} + \frac{324 0 ^{\circ} n}{- 9}

Simplify

\frac{- 9 x}{- 9} : x

\frac{- 9 x}{- 9}

Apply the fraction rule:

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

= \frac{9 x}{9}

Divide the numbers:

\frac{9}{9} = 1

= x

Simplify

\frac{144 0 ^{\circ}}{- 9} + \frac{324 0 ^{\circ} n}{- 9} : - 16 0^{\circ} - 36 0^{\circ} n

\frac{144 0 ^{\circ}}{- 9} + \frac{324 0 ^{\circ} n}{- 9}

Apply the fraction rule:

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

= - 16 0^{\circ} + \frac{324 0 ^{\circ} n}{- 9}

\frac{324 0 ^{\circ} n}{- 9} = - 36 0^{\circ} n

\frac{324 0 ^{\circ} n}{- 9}

Apply the fraction rule:

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

= - \frac{324 0 ^{\circ} n}{9}

Divide the numbers:

\frac{18}{9} = 2

= - 36 0^{\circ} n

= - 16 0^{\circ} - 36 0^{\circ} n

x = - 16 0^{\circ} - 36 0^{\circ} n

x = - 16 0^{\circ} - 36 0^{\circ} n

x = - 16 0^{\circ} - 36 0^{\circ} n

x = - 36 0^{\circ} n + 2 0^{\circ}, x = - 16 0^{\circ} - 36 0^{\circ} n

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos(40-x)=cos(x) ?
The general solution for cos(40-x)=cos(x) is x=-360n+20,x=-160-360n