What is the general solution for 7cos(x)+4=-cos(x) ?

The general solution for 7cos(x)+4=-cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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7cos(x)+4=-cos(x)

Solution

7 cos (x) + 4 = - 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

7 cos (x) + 4 = - cos (x)

Solve by substitution

7 cos (x) + 4 = - cos (x)

Let:

cos (x) = u

7 u + 4 = - u

7 u + 4 = - u : u = - \frac{1}{2}

7 u + 4 = - u

Move

4

to the right side

7 u + 4 = - u

Subtract

4

from both sides

7 u + 4 - 4 = - u - 4

Simplify

7 u = - u - 4

7 u = - u - 4

Move

u

to the left side

7 u = - u - 4

Add

u

to both sides

7 u + u = - u - 4 + u

Simplify

8 u = - 4

8 u = - 4

Divide both sides by

8

8 u = - 4

Divide both sides by

8

\frac{8 u}{8} = \frac{- 4}{8}

Simplify

\frac{8 u}{8} = \frac{- 4}{8}

Simplify

\frac{8 u}{8} : u

\frac{8 u}{8}

Divide the numbers:

\frac{8}{8} = 1

= u

Simplify

\frac{- 4}{8} : - \frac{1}{2}

\frac{- 4}{8}

Apply the fraction rule:

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

= - \frac{4}{8}

Cancel the common factor:

4

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

cos(4x)=(sqrt(3))/2

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2sin^2(x)=sqrt(2)sin(x)

2 sin^{2} (x) = 2 sin (x)

Frequently Asked Questions (FAQ)

What is the general solution for 7cos(x)+4=-cos(x) ?
The general solution for 7cos(x)+4=-cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin