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

The general solution for cos(x-pi/4)=-sin(x) is x=-(pi+8pin)/8

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cos(x-pi/4)=-sin(x)

Solution

cos (x - \frac{π}{4}) = - sin (x)

Solution

x = - \frac{π + 8 πn}{8}

Degrees

x = - 22. 5^{\circ} - 18 0^{\circ} n

Solution steps

cos (x - \frac{π}{4}) = - sin (x)

Rewrite using trig identities

cos (x - \frac{π}{4}) = - sin (x)

Use the following identity:

- sin (x) = sin (- x)

cos (x - \frac{π}{4}) = sin (- (x))

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

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

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

Apply trig inverse properties

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

sin (x) = sin (y) \Rightarrow x = y + 2 π n, x = π - y + 2 π n

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

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

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

True for all

x; 0 = 2 πn + \frac{3 π}{4}

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

Expand

- (x) : - x

- (x)

Remove parentheses:

(a) = a

= - x

Expand

\frac{π}{2} - (x - \frac{π}{4}) + 2 πn : - x + 2 πn + \frac{3 π}{4}

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

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

- (x - \frac{π}{4})

Distribute parentheses

= - (x) - (- \frac{π}{4})

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - x + \frac{π}{4}

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

Simplify

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

\frac{π}{2} - x + \frac{π}{4} + 2 πn

Group like terms

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

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

4

For

\frac{π}{2} :

multiply the denominator and numerator by

2

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

= \frac{π 2}{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{π 2 + π}{4}

Add similar elements:

2 π + π = 3 π

= - x + 2 πn + \frac{3 π}{4}

= - x + 2 πn + \frac{3 π}{4}

- x = - x + 2 πn + \frac{3 π}{4}

Move

x

to the left side

- x = - x + 2 πn + \frac{3 π}{4}

Add

x

to both sides

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

Simplify

0 = 2 πn + \frac{3 π}{4}

0 = 2 πn + \frac{3 π}{4}

Both sides are equal

True for all x; 0 = 2 πn + \frac{3 π}{4}

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

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

Expand

- (x) : - x

- (x)

Remove parentheses:

(a) = a

= - x

Expand

π - (\frac{π}{2} - (x - \frac{π}{4})) + 2 πn : π + x - \frac{3 π}{4} + 2 πn

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

Expand

\frac{π}{2} - (x - \frac{π}{4}) : - x + \frac{3 π}{4}

\frac{π}{2} - (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, - (a) = - a

= - x + \frac{π}{4}

= \frac{π}{2} - x + \frac{π}{4}

Simplify

\frac{π}{2} - x + \frac{π}{4} : - x + \frac{3 π}{4}

\frac{π}{2} - x + \frac{π}{4}

Group like terms

= - x + \frac{π}{2} + \frac{π}{4}

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

4

For

\frac{π}{2} :

multiply the denominator and numerator by

2

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

= \frac{π 2}{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{π 2 + π}{4}

Add similar elements:

2 π + π = 3 π

= - x + \frac{3 π}{4}

= - x + \frac{3 π}{4}

= π - (- x + \frac{3 π}{4}) + 2 πn

- (- x + \frac{3 π}{4}) : x - \frac{3 π}{4}

- (- x + \frac{3 π}{4})

Distribute parentheses

= - (- x) - (\frac{3 π}{4})

Apply minus-plus rules

- (- a) = a, - (a) = - a

= x - \frac{3 π}{4}

= π + x - \frac{3 π}{4} + 2 πn

- x = π + x - \frac{3 π}{4} + 2 πn

Move

x

to the left side

- x = π + x - \frac{3 π}{4} + 2 πn

Subtract

x

from both sides

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

Simplify

- 2 x = π - \frac{3 π}{4} + 2 πn

- 2 x = π - \frac{3 π}{4} + 2 πn

Divide both sides by

- 2

- 2 x = π - \frac{3 π}{4} + 2 πn

Divide both sides by

- 2

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

Simplify

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

Simplify

\frac{- 2 x}{- 2} : x

\frac{- 2 x}{- 2}

Apply the fraction rule:

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

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{π}{- 2} - \frac{\frac{3 π}{4}}{- 2} + \frac{2 πn}{- 2} : - \frac{π + 8 πn}{8}

\frac{π}{- 2} - \frac{\frac{3 π}{4}}{- 2} + \frac{2 πn}{- 2}

Apply rule

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

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

Apply the fraction rule:

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

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

Join

π - \frac{3 π}{4} + 2 πn : \frac{π + 8 πn}{4}

π - \frac{3 π}{4} + 2 πn

Convert element to fraction:

π = \frac{π 4}{4}, 2 πn = \frac{2 πn 4}{4}

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

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 - 3 π + 2 πn \cdot 4}{4}

π 4 - 3 π + 2 πn \cdot 4 = π + 8 πn

π 4 - 3 π + 2 πn \cdot 4

Add similar elements:

4 π - 3 π = π

= π + 2 \cdot 4 πn

Multiply the numbers:

2 \cdot 4 = 8

= π + 8 πn

= \frac{π + 8 πn}{4}

= - \frac{\frac{π + 8 πn}{4}}{2}

Simplify

\frac{\frac{π + 8 πn}{4}}{2} : \frac{π + 8 πn}{8}

\frac{\frac{π + 8 πn}{4}}{2}

Apply the fraction rule:

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

= \frac{π + 8 πn}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{π + 8 πn}{8}

= - \frac{π + 8 πn}{8}

x = - \frac{π + 8 πn}{8}

x = - \frac{π + 8 πn}{8}

x = - \frac{π + 8 πn}{8}

Since the equation is undefined for:

True for all

x

x = - \frac{π + 8 πn}{8}

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Frequently Asked Questions (FAQ)

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