What is the general solution for cos(pi/4-x)= 12/13 ?

The general solution for cos(pi/4-x)= 12/13 is x=-2pin+pi/4-0.39479…,x=-2pi-2pin+pi/4+0.39479…

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cos(pi/4-x)= 12/13

Solution

cos (\frac{π}{4} - x) = \frac{12}{13}

Solution

x = - 2 πn + \frac{π}{4} - 0.39479 \dots, x = - 2 π - 2 πn + \frac{π}{4} + 0.39479 \dots

Degrees

x = 22.38013 \dots^{\circ} - 36 0^{\circ} n, x = - 292.38013 \dots^{\circ} - 36 0^{\circ} n

Solution steps

cos (\frac{π}{4} - x) = \frac{12}{13}

Apply trig inverse properties

cos (\frac{π}{4} - x) = \frac{12}{13}

General solutions for

cos (\frac{π}{4} - x) = \frac{12}{13}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

\frac{π}{4} - x = arccos (\frac{12}{13}) + 2 πn, \frac{π}{4} - x = 2 π - arccos (\frac{12}{13}) + 2 πn

\frac{π}{4} - x = arccos (\frac{12}{13}) + 2 πn, \frac{π}{4} - x = 2 π - arccos (\frac{12}{13}) + 2 πn

Solve

\frac{π}{4} - x = arccos (\frac{12}{13}) + 2 πn : x = - 2 πn + \frac{π}{4} - arccos (\frac{12}{13})

\frac{π}{4} - x = arccos (\frac{12}{13}) + 2 πn

Move

\frac{π}{4}

to the right side

\frac{π}{4} - x = arccos (\frac{12}{13}) + 2 πn

Subtract

\frac{π}{4}

from both sides

\frac{π}{4} - x - \frac{π}{4} = arccos (\frac{12}{13}) + 2 πn - \frac{π}{4}

Simplify

- x = arccos (\frac{12}{13}) + 2 πn - \frac{π}{4}

- x = arccos (\frac{12}{13}) + 2 πn - \frac{π}{4}

Divide both sides by

- 1

- x = arccos (\frac{12}{13}) + 2 πn - \frac{π}{4}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{arccos ( \frac{12}{13} )}{- 1} + \frac{2 πn}{- 1} - \frac{\frac{π}{4}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{arccos ( \frac{12}{13} )}{- 1} + \frac{2 πn}{- 1} - \frac{\frac{π}{4}}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{arccos ( \frac{12}{13} )}{- 1} + \frac{2 πn}{- 1} - \frac{\frac{π}{4}}{- 1} : - 2 πn + \frac{π}{4} - arccos (\frac{12}{13})

\frac{arccos ( \frac{12}{13} )}{- 1} + \frac{2 πn}{- 1} - \frac{\frac{π}{4}}{- 1}

Group like terms

= \frac{2 πn}{- 1} - \frac{\frac{π}{4}}{- 1} + \frac{arccos ( \frac{12}{13} )}{- 1}

\frac{2 πn}{- 1} = - 2 πn

\frac{2 πn}{- 1}

Apply the fraction rule:

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

= - \frac{2 πn}{1}

Apply rule

\frac{a}{1} = a

= - 2 πn

= - 2 πn - \frac{\frac{π}{4}}{- 1} + \frac{arccos ( \frac{12}{13} )}{- 1}

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

\frac{\frac{π}{4}}{- 1}

Apply the fraction rule:

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

= - \frac{\frac{π}{4}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{\frac{π}{4}}{1} = \frac{π}{4}

= - \frac{π}{4}

\frac{arccos ( \frac{12}{13} )}{- 1} = - arccos (\frac{12}{13})

\frac{arccos ( \frac{12}{13} )}{- 1}

Apply the fraction rule:

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

= - \frac{arccos ( \frac{12}{13} )}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{arccos ( \frac{12}{13} )}{1} = arccos (\frac{12}{13})

= - arccos (\frac{12}{13})

= - 2 πn - (- \frac{π}{4}) - arccos (\frac{12}{13})

Apply rule

- (- a) = a

= - 2 πn + \frac{π}{4} - arccos (\frac{12}{13})

x = - 2 πn + \frac{π}{4} - arccos (\frac{12}{13})

x = - 2 πn + \frac{π}{4} - arccos (\frac{12}{13})

x = - 2 πn + \frac{π}{4} - arccos (\frac{12}{13})

Solve

\frac{π}{4} - x = 2 π - arccos (\frac{12}{13}) + 2 πn : x = - 2 π - 2 πn + \frac{π}{4} + arccos (\frac{12}{13})

\frac{π}{4} - x = 2 π - arccos (\frac{12}{13}) + 2 πn

Move

\frac{π}{4}

to the right side

\frac{π}{4} - x = 2 π - arccos (\frac{12}{13}) + 2 πn

Subtract

\frac{π}{4}

from both sides

\frac{π}{4} - x - \frac{π}{4} = 2 π - arccos (\frac{12}{13}) + 2 πn - \frac{π}{4}

Simplify

- x = 2 π - arccos (\frac{12}{13}) + 2 πn - \frac{π}{4}

- x = 2 π - arccos (\frac{12}{13}) + 2 πn - \frac{π}{4}

Divide both sides by

- 1

- x = 2 π - arccos (\frac{12}{13}) + 2 πn - \frac{π}{4}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{2 π}{- 1} - \frac{arccos ( \frac{12}{13} )}{- 1} + \frac{2 πn}{- 1} - \frac{\frac{π}{4}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{2 π}{- 1} - \frac{arccos ( \frac{12}{13} )}{- 1} + \frac{2 πn}{- 1} - \frac{\frac{π}{4}}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{2 π}{- 1} - \frac{arccos ( \frac{12}{13} )}{- 1} + \frac{2 πn}{- 1} - \frac{\frac{π}{4}}{- 1} : - 2 π - 2 πn + \frac{π}{4} + arccos (\frac{12}{13})

\frac{2 π}{- 1} - \frac{arccos ( \frac{12}{13} )}{- 1} + \frac{2 πn}{- 1} - \frac{\frac{π}{4}}{- 1}

Group like terms

= \frac{2 π}{- 1} + \frac{2 πn}{- 1} - \frac{\frac{π}{4}}{- 1} - \frac{arccos ( \frac{12}{13} )}{- 1}

\frac{2 π}{- 1} = - 2 π

\frac{2 π}{- 1}

Apply the fraction rule:

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

= - \frac{2 π}{1}

Apply rule

\frac{a}{1} = a

= - 2 π

= - 2 π + \frac{2 πn}{- 1} - \frac{\frac{π}{4}}{- 1} - \frac{arccos ( \frac{12}{13} )}{- 1}

\frac{2 πn}{- 1} = - 2 πn

\frac{2 πn}{- 1}

Apply the fraction rule:

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

= - \frac{2 πn}{1}

Apply rule

\frac{a}{1} = a

= - 2 πn

= - 2 π - 2 πn - \frac{\frac{π}{4}}{- 1} - \frac{arccos ( \frac{12}{13} )}{- 1}

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

\frac{\frac{π}{4}}{- 1}

Apply the fraction rule:

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

= - \frac{\frac{π}{4}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{\frac{π}{4}}{1} = \frac{π}{4}

= - \frac{π}{4}

\frac{arccos ( \frac{12}{13} )}{- 1} = - arccos (\frac{12}{13})

\frac{arccos ( \frac{12}{13} )}{- 1}

Apply the fraction rule:

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

= - \frac{arccos ( \frac{12}{13} )}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{arccos ( \frac{12}{13} )}{1} = arccos (\frac{12}{13})

= - arccos (\frac{12}{13})

= - 2 π - 2 πn - (- \frac{π}{4}) - (- arccos (\frac{12}{13}))

Apply rule

- (- a) = a

= - 2 π - 2 πn + \frac{π}{4} + arccos (\frac{12}{13})

x = - 2 π - 2 πn + \frac{π}{4} + arccos (\frac{12}{13})

x = - 2 π - 2 πn + \frac{π}{4} + arccos (\frac{12}{13})

x = - 2 π - 2 πn + \frac{π}{4} + arccos (\frac{12}{13})

x = - 2 πn + \frac{π}{4} - arccos (\frac{12}{13}), x = - 2 π - 2 πn + \frac{π}{4} + arccos (\frac{12}{13})

Show solutions in decimal form

x = - 2 πn + \frac{π}{4} - 0.39479 \dots, x = - 2 π - 2 πn + \frac{π}{4} + 0.39479 \dots

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Frequently Asked Questions (FAQ)

What is the general solution for cos(pi/4-x)= 12/13 ?
The general solution for cos(pi/4-x)= 12/13 is x=-2pin+pi/4-0.39479…,x=-2pi-2pin+pi/4+0.39479…