What is the general solution for 5cos(x)+2=cos(x) ?

The general solution for 5cos(x)+2=cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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Popular >

5cos(x)+2=cos(x)

Solution

5 cos (x) + 2 = 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

5 cos (x) + 2 = cos (x)

Solve by substitution

5 cos (x) + 2 = cos (x)

Let:

cos (x) = u

5 u + 2 = u

5 u + 2 = u : u = - \frac{1}{2}

5 u + 2 = u

Move

2

to the right side

5 u + 2 = u

Subtract

2

from both sides

5 u + 2 - 2 = u - 2

Simplify

5 u = u - 2

5 u = u - 2

Move

u

to the left side

5 u = u - 2

Subtract

u

from both sides

5 u - u = u - 2 - u

Simplify

4 u = - 2

4 u = - 2

Divide both sides by

4

4 u = - 2

Divide both sides by

4

\frac{4 u}{4} = \frac{- 2}{4}

Simplify

\frac{4 u}{4} = \frac{- 2}{4}

Simplify

\frac{4 u}{4} : u

\frac{4 u}{4}

Divide the numbers:

\frac{4}{4} = 1

= u

Simplify

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

\frac{- 2}{4}

Apply the fraction rule:

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

= - \frac{2}{4}

Cancel the common factor:

2

= - \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

2tan(4x)+1=3tan(4x)

2 tan (4 x) + 1 = 3 tan (4 x)

sec(x)=-5,pi<x<(3pi)/2

sec (x) = - 5, π < x < \frac{3 π}{2}

sin(θ/2)= 3/5

sin (\frac{θ}{2}) = \frac{3}{5}

2sin^3(t)+sin^2(t)-2sin(t)-1=0

2 sin^{3} (t) + sin^{2} (t) - 2 sin (t) - 1 = 0

sin(x)= 1/5 ,cos(x)>0

sin (x) = \frac{1}{5}, cos (x) > 0

Frequently Asked Questions (FAQ)

What is the general solution for 5cos(x)+2=cos(x) ?
The general solution for 5cos(x)+2=cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin