What is the general solution for 11cos(x)+6=-cos(x) ?

The general solution for 11cos(x)+6=-cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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11cos(x)+6=-cos(x)

Solution

11 cos (x) + 6 = - cos (x)

Solution

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

Degrees

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

11 cos (x) + 6 = - cos (x)

Solve by substitution

11 cos (x) + 6 = - cos (x)

Let:

cos (x) = u

11 u + 6 = - u

11 u + 6 = - u : u = - \frac{1}{2}

11 u + 6 = - u

Move

6

to the right side

11 u + 6 = - u

Subtract

6

from both sides

11 u + 6 - 6 = - u - 6

Simplify

11 u = - u - 6

11 u = - u - 6

Move

u

to the left side

11 u = - u - 6

Add

u

to both sides

11 u + u = - u - 6 + u

Simplify

12 u = - 6

12 u = - 6

Divide both sides by

12

12 u = - 6

Divide both sides by

12

\frac{12 u}{12} = \frac{- 6}{12}

Simplify

\frac{12 u}{12} = \frac{- 6}{12}

Simplify

\frac{12 u}{12} : u

\frac{12 u}{12}

Divide the numbers:

\frac{12}{12} = 1

= u

Simplify

\frac{- 6}{12} : - \frac{1}{2}

\frac{- 6}{12}

Apply the fraction rule:

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

= - \frac{6}{12}

Cancel the common factor:

6

= - \frac{1}{2}

u = - \frac{1}{2}

u = - \frac{1}{2}

u = - \frac{1}{2}

Substitute back

u = cos (x)

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

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

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

General solutions for

cos (x) = - \frac{1}{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{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

Combine all the solutions

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

Popular Examples

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

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(csc(θ)-2)(cot(θ)+1)=0

(csc (θ) - 2) (cot (θ) + 1) = 0

sin(x)=csc(x)

sin (x) = csc (x)

Frequently Asked Questions (FAQ)

What is the general solution for 11cos(x)+6=-cos(x) ?
The general solution for 11cos(x)+6=-cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin