What is the general solution for cos^2(x)+cos(x)=cos(2x) ?

The general solution for cos^2(x)+cos(x)=cos(2x) is x=2.23703…+2pin,x=-2.23703…+2pin

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cos^2(x)+cos(x)=cos(2x)

Solution

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

Solution

x = 2.23703 \dots + 2 πn, x = - 2.23703 \dots + 2 πn

Degrees

x = 128.17270 \dots^{\circ} + 36 0^{\circ} n, x = - 128.17270 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

cos (2 x)

from both sides

cos^{2} (x) + cos (x) - cos (2 x) = 0

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Simplify

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

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

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

1 + cos (x) - cos^{2} (x) = 0

Solve by substitution

1 + cos (x) - cos^{2} (x) = 0

Let:

cos (x) = u

1 + u - u^{2} = 0

1 + u - u^{2} = 0 : u = - \frac{- 1 + 5}{2}, u = \frac{1 + 5}{2}

1 + u - u^{2} = 0

Write in the standard form

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

- u^{2} + u + 1 = 0

Solve with the quadratic formula

- u^{2} + u + 1 = 0

Quadratic Equation Formula:

For

a = - 1, b = 1, c = 1

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

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

1^{2} - 4 (- 1) \cdot 1 = 5

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

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 1) \cdot 1

Apply rule

- (- a) = a

= 1 + 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 1 + 4

Add the numbers:

1 + 4 = 5

= 5

u_{1, 2} = \frac{- 1 \pm 5}{2 ( - 1 )}

Separate the solutions

u_{1} = \frac{- 1 + 5}{2 ( - 1 )}, u_{2} = \frac{- 1 - 5}{2 ( - 1 )}

u = \frac{- 1 + 5}{2 ( - 1 )} : - \frac{- 1 + 5}{2}

\frac{- 1 + 5}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 + 5}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 1 + 5}{- 2}

Apply the fraction rule:

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

= - \frac{- 1 + 5}{2}

u = \frac{- 1 - 5}{2 ( - 1 )} : \frac{1 + 5}{2}

\frac{- 1 - 5}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 - 5}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

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

Apply the fraction rule:

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

- 1 - 5 = - (1 + 5)

= \frac{1 + 5}{2}

The solutions to the quadratic equation are:

u = - \frac{- 1 + 5}{2}, u = \frac{1 + 5}{2}

Substitute back

u = cos (x)

cos (x) = - \frac{- 1 + 5}{2}, cos (x) = \frac{1 + 5}{2}

cos (x) = - \frac{- 1 + 5}{2}, cos (x) = \frac{1 + 5}{2}

cos (x) = - \frac{- 1 + 5}{2} : x = arccos (- \frac{- 1 + 5}{2}) + 2 πn, x = - arccos (- \frac{- 1 + 5}{2}) + 2 πn

cos (x) = - \frac{- 1 + 5}{2}

Apply trig inverse properties

cos (x) = - \frac{- 1 + 5}{2}

General solutions for

cos (x) = - \frac{- 1 + 5}{2}

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

x = arccos (- \frac{- 1 + 5}{2}) + 2 πn, x = - arccos (- \frac{- 1 + 5}{2}) + 2 πn

x = arccos (- \frac{- 1 + 5}{2}) + 2 πn, x = - arccos (- \frac{- 1 + 5}{2}) + 2 πn

cos (x) = \frac{1 + 5}{2} :

No Solution

cos (x) = \frac{1 + 5}{2}

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = arccos (- \frac{- 1 + 5}{2}) + 2 πn, x = - arccos (- \frac{- 1 + 5}{2}) + 2 πn

Show solutions in decimal form

x = 2.23703 \dots + 2 πn, x = - 2.23703 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x)+cos(x)=cos(2x) ?
The general solution for cos^2(x)+cos(x)=cos(2x) is x=2.23703…+2pin,x=-2.23703…+2pin