What is the general solution for 2cos(x)=cos^2(x)-1 ?

The general solution for 2cos(x)=cos^2(x)-1 is x=1.99787…+2pin,x=-1.99787…+2pin

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

Solution

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

Solution

x = 1.99787 \dots + 2 πn, x = - 1.99787 \dots + 2 πn

Degrees

x = 114.46980 \dots^{\circ} + 36 0^{\circ} n, x = - 114.46980 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

cos (x) = u

2 u = u^{2} - 1

2 u = u^{2} - 1 : u = 1 + 2, u = 1 - 2

2 u = u^{2} - 1

Switch sides

u^{2} - 1 = 2 u

Move

2 u

to the left side

u^{2} - 1 = 2 u

Subtract

2 u

from both sides

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

Simplify

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 1, b = - 2, c = - 1

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

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

(- 2)^{2} - 4 \cdot 1 \cdot (- 1) = 22

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

Apply rule

- (- a) = a

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 2)^{2} = 2^{2}

= 2^{2} + 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 2^{2} + 4

2^{2} = 4

= 4 + 4

Add the numbers:

4 + 4 = 8

= 8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

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

Separate the solutions

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

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

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

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 + 2 2}{2}

Factor

2 + 22 : 2 (1 + 2)

2 + 22

Rewrite as

= 2 \cdot 1 + 22

Factor out common term

2

= 2 (1 + 2)

= \frac{2 ( 1 + 2 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 1 + 2

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

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

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 - 2 2}{2}

Factor

2 - 22 : 2 (1 - 2)

2 - 22

Rewrite as

= 2 \cdot 1 - 22

Factor out common term

2

= 2 (1 - 2)

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

Divide the numbers:

\frac{2}{2} = 1

= 1 - 2

The solutions to the quadratic equation are:

u = 1 + 2, u = 1 - 2

Substitute back

u = cos (x)

cos (x) = 1 + 2, cos (x) = 1 - 2

cos (x) = 1 + 2, cos (x) = 1 - 2

cos (x) = 1 + 2 :

No Solution

cos (x) = 1 + 2

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = 1 - 2 : x = arccos (1 - 2) + 2 πn, x = - arccos (1 - 2) + 2 πn

cos (x) = 1 - 2

Apply trig inverse properties

cos (x) = 1 - 2

General solutions for

cos (x) = 1 - 2

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

x = arccos (1 - 2) + 2 πn, x = - arccos (1 - 2) + 2 πn

x = arccos (1 - 2) + 2 πn, x = - arccos (1 - 2) + 2 πn

Combine all the solutions

x = arccos (1 - 2) + 2 πn, x = - arccos (1 - 2) + 2 πn

Show solutions in decimal form

x = 1.99787 \dots + 2 πn, x = - 1.99787 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2cos(x)=cos^2(x)-1 ?
The general solution for 2cos(x)=cos^2(x)-1 is x=1.99787…+2pin,x=-1.99787…+2pin