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

The general solution for sqrt(2)cos(x)+1=0,pi<x<4pi is x=(5pi)/4 ,x=(11pi)/4 ,x=(13pi)/4

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sqrt(2)cos(x)+1=0,pi<x<4pi

2 cos (x) + 1 = 0, π < x < 4 π

x = \frac{5 π}{4}, x = \frac{11 π}{4}, x = \frac{13 π}{4}

Degrees

x = 22 5^{\circ}, x = 49 5^{\circ}, x = 58 5^{\circ}

Solution steps

2 cos (x) + 1 = 0, π < x < 4 π

Move

1

to the right side

2 cos (x) + 1 = 0

Subtract

1

from both sides

2 cos (x) + 1 - 1 = 0 - 1

Simplify

2 cos (x) = - 1

2 cos (x) = - 1

Divide both sides by

2

2 cos (x) = - 1

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{- 1}{2}

Simplify

\frac{2 cos ( x )}{2} = \frac{- 1}{2}

Simplify

\frac{2 cos ( x )}{2} : cos (x)

\frac{2 cos ( x )}{2}

Cancel the common factor:

2

= cos (x)

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

cos (x) = - \frac{2}{2}

cos (x) = - \frac{2}{2}

cos (x) = - \frac{2}{2}

General solutions for

cos (x) = - \frac{2}{2}

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

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

Solutions for the range

π < x < 4 π

x = \frac{5 π}{4}, x = \frac{11 π}{4}, x = \frac{13 π}{4}

cos (3 x) = 3

4 cos^{2} (θ) - 5 cos (θ) = - 4 cos (θ) + 3

x, cos (x) = - 0.5

- 2 cos (x) = 3 sin (x)

3 sin (x) - cos (2 x) = 1

What is the general solution for sqrt(2)cos(x)+1=0,pi<x<4pi ?
The general solution for sqrt(2)cos(x)+1=0,pi<x<4pi is x=(5pi)/4 ,x=(11pi)/4 ,x=(13pi)/4