What is the general solution for cos(pi/4-x)=1,0<= x<= 2pi ?

Q: What is the general solution for cos(pi/4-x)=1,0<= x<= 2pi ?

The general solution for cos(pi/4-x)=1,0<= x<= 2pi is x= pi/4

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cos(pi/4-x)=1,0<= x<= 2pi

Solution

cos (\frac{π}{4} - x) = 1, 0 \leq x \leq 2 π

Solution

x = \frac{π}{4}

Degrees

x = 4 5^{\circ}

Solution steps

cos (\frac{π}{4} - x) = 1, 0 \leq x \leq 2 π

General solutions for

cos (\frac{π}{4} - x) = 1

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{π}{4} - x = 0 + 2 πn

\frac{π}{4} - x = 0 + 2 πn

Solve

\frac{π}{4} - x = 0 + 2 πn : x = - 2 πn + \frac{π}{4}

\frac{π}{4} - x = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{π}{4} - x = 2 πn

Move

\frac{π}{4}

to the right side

\frac{π}{4} - x = 2 πn

Subtract

\frac{π}{4}

from both sides

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

Simplify

- x = 2 πn - \frac{π}{4}

- x = 2 πn - \frac{π}{4}

Divide both sides by

- 1

- x = 2 πn - \frac{π}{4}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{2 πn}{- 1} - \frac{\frac{π}{4}}{- 1}

Simplify

\frac{- x}{- 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 πn}{- 1} - \frac{\frac{π}{4}}{- 1} : - 2 πn + \frac{π}{4}

\frac{2 πn}{- 1} - \frac{\frac{π}{4}}{- 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{\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}

= - 2 πn - (- \frac{π}{4})

Apply rule

- (- a) = a

= - 2 πn + \frac{π}{4}

x = - 2 πn + \frac{π}{4}

x = - 2 πn + \frac{π}{4}

x = - 2 πn + \frac{π}{4}

x = - 2 πn + \frac{π}{4}

Solutions for the range

0 \leq x \leq 2 π

x = \frac{π}{4}

Popular Examples

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sec (θ) = 5

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9+9sin(θ)=6cos^2(θ)

9 + 9 sin (θ) = 6 cos^{2} (θ)

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Frequently Asked Questions (FAQ)

What is the general solution for cos(pi/4-x)=1,0<= x<= 2pi ?
The general solution for cos(pi/4-x)=1,0<= x<= 2pi is x= pi/4