What is the general solution for 15cos(x)+8=-cos(x) ?

The general solution for 15cos(x)+8=-cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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Popular Trigonometry >

15cos(x)+8=-cos(x)

Solution

15 cos (x) + 8 = - 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

15 cos (x) + 8 = - cos (x)

Solve by substitution

15 cos (x) + 8 = - cos (x)

Let:

cos (x) = u

15 u + 8 = - u

15 u + 8 = - u : u = - \frac{1}{2}

15 u + 8 = - u

Move

8

to the right side

15 u + 8 = - u

Subtract

8

from both sides

15 u + 8 - 8 = - u - 8

Simplify

15 u = - u - 8

15 u = - u - 8

Move

u

to the left side

15 u = - u - 8

Add

u

to both sides

15 u + u = - u - 8 + u

Simplify

16 u = - 8

16 u = - 8

Divide both sides by

16

16 u = - 8

Divide both sides by

16

\frac{16 u}{16} = \frac{- 8}{16}

Simplify

\frac{16 u}{16} = \frac{- 8}{16}

Simplify

\frac{16 u}{16} : u

\frac{16 u}{16}

Divide the numbers:

\frac{16}{16} = 1

= u

Simplify

\frac{- 8}{16} : - \frac{1}{2}

\frac{- 8}{16}

Apply the fraction rule:

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

= - \frac{8}{16}

Cancel the common factor:

8

= - \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

Graph

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tan(3x)+1=0

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Frequently Asked Questions (FAQ)

What is the general solution for 15cos(x)+8=-cos(x) ?
The general solution for 15cos(x)+8=-cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin