What is the general solution for 13cos(x)+7=-cos(x) ?

The general solution for 13cos(x)+7=-cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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13cos(x)+7=-cos(x)

Solution

13 cos (x) + 7 = - 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

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

Solve by substitution

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

Let:

cos (x) = u

13 u + 7 = - u

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

13 u + 7 = - u

Move

7

to the right side

13 u + 7 = - u

Subtract

7

from both sides

13 u + 7 - 7 = - u - 7

Simplify

13 u = - u - 7

13 u = - u - 7

Move

u

to the left side

13 u = - u - 7

Add

u

to both sides

13 u + u = - u - 7 + u

Simplify

14 u = - 7

14 u = - 7

Divide both sides by

14

14 u = - 7

Divide both sides by

14

\frac{14 u}{14} = \frac{- 7}{14}

Simplify

\frac{14 u}{14} = \frac{- 7}{14}

Simplify

\frac{14 u}{14} : u

\frac{14 u}{14}

Divide the numbers:

\frac{14}{14} = 1

= u

Simplify

\frac{- 7}{14} : - \frac{1}{2}

\frac{- 7}{14}

Apply the fraction rule:

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

= - \frac{7}{14}

Cancel the common factor:

7

= - \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} (x) = 0.25

2sin(x)+(sqrt(3))/2 =sin(x)

2 sin (x) + \frac{3}{2} = sin (x)

3tan(2x)=sqrt(3)

3 tan (2 x) = 3

12cos^2(x)=3

12 cos^{2} (x) = 3

sqrt(2)cos(x)-sqrt(2)sin(x)=1,0<= x<2pi

2 cos (x) - 2 sin (x) = 1, 0 \leq x < 2 π

Frequently Asked Questions (FAQ)

What is the general solution for 13cos(x)+7=-cos(x) ?
The general solution for 13cos(x)+7=-cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin