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

The general solution for cos(x+pi/4)=0 is x=2pin+pi/4 ,x=2pin+(5pi)/4

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cos(x+pi/4)=0

Solution

cos (x + \frac{π}{4}) = 0

Solution

x = 2 πn + \frac{π}{4}, x = 2 πn + \frac{5 π}{4}

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n

Solution steps

cos (x + \frac{π}{4}) = 0

General solutions for

cos (x + \frac{π}{4}) = 0

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}

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

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

Solve

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

x + \frac{π}{4} = \frac{π}{2} + 2 πn

Move

\frac{π}{4}

to the right side

x + \frac{π}{4} = \frac{π}{2} + 2 πn

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= x

Simplify

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

\frac{π}{2} + 2 πn - \frac{π}{4}

Group like terms

= 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 = π

= 2 πn + \frac{π}{4}

x = 2 πn + \frac{π}{4}

x = 2 πn + \frac{π}{4}

x = 2 πn + \frac{π}{4}

Solve

x + \frac{π}{4} = \frac{3 π}{2} + 2 πn : x = 2 πn + \frac{5 π}{4}

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= x

Simplify

\frac{3 π}{2} + 2 πn - \frac{π}{4} : 2 πn + \frac{5 π}{4}

\frac{3 π}{2} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{3 π}{2}

Least Common Multiplier of

4, 2 : 4

4, 2

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

4

2

= 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{3 π}{2} :

multiply the denominator and numerator by

2

\frac{3 π}{2} = \frac{3 π 2}{2 \cdot 2} = \frac{6 π}{4}

= - \frac{π}{4} + \frac{6 π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{- π + 6 π}{4}

Add similar elements:

- π + 6 π = 5 π

= 2 πn + \frac{5 π}{4}

x = 2 πn + \frac{5 π}{4}

x = 2 πn + \frac{5 π}{4}

x = 2 πn + \frac{5 π}{4}

x = 2 πn + \frac{π}{4}, x = 2 πn + \frac{5 π}{4}

Popular Examples

sin^2(θ)cos(θ)+cos^3(θ)=-1/2

sin^{2} (θ) cos (θ) + cos^{3} (θ) = - \frac{1}{2}

csc(x)=cot(x)+1

csc (x) = cot (x) + 1

2sin^2(x)+7sin(x)+3=0

2 sin^{2} (x) + 7 sin (x) + 3 = 0

2sin(3x-pi/3)=-sqrt(3)

2 sin (3 x - \frac{π}{3}) = - 3

2cos(2x-pi/2)=sqrt(3)

2 cos (2 x - \frac{π}{2}) = 3

Frequently Asked Questions (FAQ)

What is the general solution for cos(x+pi/4)=0 ?
The general solution for cos(x+pi/4)=0 is x=2pin+pi/4 ,x=2pin+(5pi)/4