What is the general solution for cos(x)-sqrt(3)=3cos(x) ?

The general solution for cos(x)-sqrt(3)=3cos(x) is x=(5pi)/6+2pin,x=(7pi)/6+2pin

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

cos (x) - 3 = 3 cos (x)

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Degrees

x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (x) - 3 = 3 cos (x)

Solve by substitution

cos (x) - 3 = 3 cos (x)

Let:

cos (x) = u

u - 3 = 3 u

u - 3 = 3 u : u = - \frac{3}{2}

u - 3 = 3 u

Move

3

to the right side

u - 3 = 3 u

Add

3

to both sides

u - 3 + 3 = 3 u + 3

Simplify

u = 3 u + 3

u = 3 u + 3

Move

3 u

to the left side

u = 3 u + 3

Subtract

3 u

from both sides

u - 3 u = 3 u + 3 - 3 u

Simplify

- 2 u = 3

- 2 u = 3

Divide both sides by

- 2

- 2 u = 3

Divide both sides by

- 2

\frac{- 2 u}{- 2} = \frac{3}{- 2}

Simplify

u = - \frac{3}{2}

u = - \frac{3}{2}

Substitute back

u = cos (x)

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{2} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

cos (x) = - \frac{3}{2}

General solutions for

cos (x) = - \frac{3}{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{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Combine all the solutions

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

2 (1 - cos^{2} (x)) = 1 + cos (x)

sec (x) = 1.86798

2 tan^{2} (x) - 7 sec (x) + 5 = 0

cos (θ) = - \frac{3}{7}

\frac{1}{tan ( x )} + \frac{cos ^{2} ( x )}{sin ( x )} = 0

What is the general solution for cos(x)-sqrt(3)=3cos(x) ?
The general solution for cos(x)-sqrt(3)=3cos(x) is x=(5pi)/6+2pin,x=(7pi)/6+2pin