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

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

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5cos(x)=5-5cos(x)

5 cos (x) = 5 - 5 cos (x)

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

5 cos (x) = 5 - 5 cos (x)

Solve by substitution

5 cos (x) = 5 - 5 cos (x)

Let:

cos (x) = u

5 u = 5 - 5 u

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

5 u = 5 - 5 u

Move

5 u

to the left side

5 u = 5 - 5 u

Add

5 u

to both sides

5 u + 5 u = 5 - 5 u + 5 u

Simplify

10 u = 5

10 u = 5

Divide both sides by

10

10 u = 5

Divide both sides by

10

\frac{10 u}{10} = \frac{5}{10}

Simplify

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{π}{3} + 2 πn, x = \frac{5 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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

cos (2 A) + cos (A) = 1

2 \cdot 9.19 \cdot sin (x) = 1.5406

csc^{3} (x) - 4 csc (x) = 3 cot^{2} (x)

1 + tan^{2} (θ) = 4 tan^{2} (θ)

(tan (θ) + 1) (cos (θ) + 1) (sin (θ)) = 0

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