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Popular Trigonometry >

cos(x/2)-2cos(x)-2=0

Solution

cos (\frac{x}{2}) - 2 cos (x) - 2 = 0

Solution

x = π + 4 πn, x = 3 π + 4 πn, x = 2 \cdot 1.31811 \dots + 4 πn, x = 4 π - 2 \cdot 1.31811 \dots + 4 πn

Degrees

x = 18 0^{\circ} + 72 0^{\circ} n, x = 54 0^{\circ} + 72 0^{\circ} n, x = 151.04497 \dots^{\circ} + 72 0^{\circ} n, x = 568.95502 \dots^{\circ} + 72 0^{\circ} n

Solution steps

cos (\frac{x}{2}) - 2 cos (x) - 2 = 0

Let:

u = \frac{x}{2}

cos (u) - 2 cos (2 u) - 2 = 0

Rewrite using trig identities

- 2 + cos (u) - 2 cos (2 u)

Use the Double Angle identity:

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

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

Simplify

- 2 + cos (u) - 2 (2 cos^{2} (u) - 1) : cos (u) - 4 cos^{2} (u)

- 2 + cos (u) - 2 (2 cos^{2} (u) - 1)

Expand

- 2 (2 cos^{2} (u) - 1) : - 4 cos^{2} (u) + 2

- 2 (2 cos^{2} (u) - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = - 2, b = 2 cos^{2} (u), c = 1

= - 2 \cdot 2 cos^{2} (u) - (- 2) \cdot 1

Apply minus-plus rules

- (- a) = a

= - 2 \cdot 2 cos^{2} (u) + 2 \cdot 1

Simplify

- 2 \cdot 2 cos^{2} (u) + 2 \cdot 1 : - 4 cos^{2} (u) + 2

- 2 \cdot 2 cos^{2} (u) + 2 \cdot 1

Multiply the numbers:

2 \cdot 2 = 4

= - 4 cos^{2} (u) + 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= - 4 cos^{2} (u) + 2

= - 4 cos^{2} (u) + 2

= - 2 + cos (u) - 4 cos^{2} (u) + 2

Simplify

- 2 + cos (u) - 4 cos^{2} (u) + 2 : cos (u) - 4 cos^{2} (u)

- 2 + cos (u) - 4 cos^{2} (u) + 2

Group like terms

= cos (u) - 4 cos^{2} (u) - 2 + 2

- 2 + 2 = 0

= cos (u) - 4 cos^{2} (u)

= cos (u) - 4 cos^{2} (u)

= cos (u) - 4 cos^{2} (u)

cos (u) - 4 cos^{2} (u) = 0

Solve by substitution

cos (u) - 4 cos^{2} (u) = 0

Let:

cos (u) = u

u - 4 u^{2} = 0

u - 4 u^{2} = 0 : u = 0, u = \frac{1}{4}

u - 4 u^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

- 4 u^{2} + u = 0

Solve with the quadratic formula

- 4 u^{2} + u = 0

Quadratic Equation Formula:

For

a = - 4, b = 1, c = 0

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 4 ) \cdot 0}{2 ( - 4 )}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 4 ) \cdot 0}{2 ( - 4 )}

1^{2} - 4 (- 4) \cdot 0 = 1

1^{2} - 4 (- 4) \cdot 0

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 4) \cdot 0

Apply rule

- (- a) = a

= 1 + 4 \cdot 4 \cdot 0

Apply rule

0 \cdot a = 0

= 1 + 0

Add the numbers:

1 + 0 = 1

= 1

Apply rule

1 = 1

= 1

u_{1, 2} = \frac{- 1 \pm 1}{2 ( - 4 )}

Separate the solutions

u_{1} = \frac{- 1 + 1}{2 ( - 4 )}, u_{2} = \frac{- 1 - 1}{2 ( - 4 )}

u = \frac{- 1 + 1}{2 ( - 4 )} : 0

\frac{- 1 + 1}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 + 1}{- 2 \cdot 4}

Add/Subtract the numbers:

- 1 + 1 = 0

= \frac{0}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{0}{- 8}

Apply the fraction rule:

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

= - \frac{0}{8}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= - 0

= 0

u = \frac{- 1 - 1}{2 ( - 4 )} : \frac{1}{4}

\frac{- 1 - 1}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 - 1}{- 2 \cdot 4}

Subtract the numbers:

- 1 - 1 = - 2

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 2}{- 8}

Apply the fraction rule:

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

= \frac{2}{8}

Cancel the common factor:

2

= \frac{1}{4}

The solutions to the quadratic equation are:

u = 0, u = \frac{1}{4}

Substitute back

u = cos (u)

cos (u) = 0, cos (u) = \frac{1}{4}

cos (u) = 0, cos (u) = \frac{1}{4}

cos (u) = 0 : u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

cos (u) = 0

General solutions for

cos (u) = 0

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}

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

cos (u) = \frac{1}{4} : u = arccos (\frac{1}{4}) + 2 πn, u = 2 π - arccos (\frac{1}{4}) + 2 πn

cos (u) = \frac{1}{4}

Apply trig inverse properties

cos (u) = \frac{1}{4}

General solutions for

cos (u) = \frac{1}{4}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

u = arccos (\frac{1}{4}) + 2 πn, u = 2 π - arccos (\frac{1}{4}) + 2 πn

u = arccos (\frac{1}{4}) + 2 πn, u = 2 π - arccos (\frac{1}{4}) + 2 πn

Combine all the solutions

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn, u = arccos (\frac{1}{4}) + 2 πn, u = 2 π - arccos (\frac{1}{4}) + 2 πn

Substitute back

u = \frac{x}{2}

\frac{x}{2} = \frac{π}{2} + 2 πn : x = π + 4 πn

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

x = π + 4 πn

x = π + 4 πn

x = π + 4 πn

\frac{x}{2} = \frac{3 π}{2} + 2 πn : x = 3 π + 4 πn

\frac{x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn : 3 π + 4 πn

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

x = 3 π + 4 πn

x = 3 π + 4 πn

x = 3 π + 4 πn

\frac{x}{2} = arccos (\frac{1}{4}) + 2 πn : x = 2 arccos (\frac{1}{4}) + 4 πn

\frac{x}{2} = arccos (\frac{1}{4}) + 2 πn

Multiply both sides by

2

\frac{x}{2} = arccos (\frac{1}{4}) + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 arccos (\frac{1}{4}) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 arccos (\frac{1}{4}) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 arccos (\frac{1}{4}) + 2 \cdot 2 πn : 2 arccos (\frac{1}{4}) + 4 πn

2 arccos (\frac{1}{4}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 arccos (\frac{1}{4}) + 4 πn

x = 2 arccos (\frac{1}{4}) + 4 πn

x = 2 arccos (\frac{1}{4}) + 4 πn

x = 2 arccos (\frac{1}{4}) + 4 πn

\frac{x}{2} = 2 π - arccos (\frac{1}{4}) + 2 πn : x = 4 π - 2 arccos (\frac{1}{4}) + 4 πn

\frac{x}{2} = 2 π - arccos (\frac{1}{4}) + 2 πn

Multiply both sides by

2

\frac{x}{2} = 2 π - arccos (\frac{1}{4}) + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot 2 π - 2 arccos (\frac{1}{4}) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot 2 π - 2 arccos (\frac{1}{4}) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot 2 π - 2 arccos (\frac{1}{4}) + 2 \cdot 2 πn : 4 π - 2 arccos (\frac{1}{4}) + 4 πn

2 \cdot 2 π - 2 arccos (\frac{1}{4}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 π - 2 arccos (\frac{1}{4}) + 4 πn

x = 4 π - 2 arccos (\frac{1}{4}) + 4 πn

x = 4 π - 2 arccos (\frac{1}{4}) + 4 πn

x = 4 π - 2 arccos (\frac{1}{4}) + 4 πn

x = π + 4 πn, x = 3 π + 4 πn, x = 2 arccos (\frac{1}{4}) + 4 πn, x = 4 π - 2 arccos (\frac{1}{4}) + 4 πn

Show solutions in decimal form

x = π + 4 πn, x = 3 π + 4 πn, x = 2 \cdot 1.31811 \dots + 4 πn, x = 4 π - 2 \cdot 1.31811 \dots + 4 πn

Graph

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