What is the general solution for cos^2(x)-2cos(x)-2=0 ?

The general solution for cos^2(x)-2cos(x)-2=0 is x=2.39212…+2pin,x=-2.39212…+2pin

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

Solution

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

Solution

x = 2.39212 \dots + 2 πn, x = - 2.39212 \dots + 2 πn

Degrees

x = 137.05859 \dots^{\circ} + 36 0^{\circ} n, x = - 137.05859 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

cos (x) = u

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

u^{2} - 2 u - 2 = 0 : u = 1 + 3, u = 1 - 3

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

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

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

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

(- 2)^{2} - 4 \cdot 1 \cdot (- 2) = 23

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

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

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 2^{2} + 8

2^{2} = 4

= 4 + 8

Add the numbers:

4 + 8 = 12

= 12

Prime factorization of

12 : 2^{2} \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

= 2^{2} \cdot 3

= 2^{2} \cdot 3

Apply radical rule:

= 3 2^{2}

Apply radical rule:

2^{2} = 2

= 23

u_{1, 2} = \frac{- ( - 2 ) \pm 2 3}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 2 ) + 2 3}{2 \cdot 1}, u_{2} = \frac{- ( - 2 ) - 2 3}{2 \cdot 1}

u = \frac{- ( - 2 ) + 2 3}{2 \cdot 1} : 1 + 3

\frac{- ( - 2 ) + 2 3}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{2 + 2 3}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 + 2 3}{2}

Factor

2 + 23 : 2 (1 + 3)

2 + 23

Rewrite as

= 2 \cdot 1 + 23

Factor out common term

2

= 2 (1 + 3)

= \frac{2 ( 1 + 3 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 1 + 3

u = \frac{- ( - 2 ) - 2 3}{2 \cdot 1} : 1 - 3

\frac{- ( - 2 ) - 2 3}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{2 - 2 3}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

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

Factor

2 - 23 : 2 (1 - 3)

2 - 23

Rewrite as

= 2 \cdot 1 - 23

Factor out common term

2

= 2 (1 - 3)

= \frac{2 ( 1 - 3 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 1 - 3

The solutions to the quadratic equation are:

u = 1 + 3, u = 1 - 3

Substitute back

u = cos (x)

cos (x) = 1 + 3, cos (x) = 1 - 3

cos (x) = 1 + 3, cos (x) = 1 - 3

cos (x) = 1 + 3 :

No Solution

cos (x) = 1 + 3

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = 1 - 3 : x = arccos (1 - 3) + 2 πn, x = - arccos (1 - 3) + 2 πn

cos (x) = 1 - 3

Apply trig inverse properties

cos (x) = 1 - 3

General solutions for

cos (x) = 1 - 3

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

x = arccos (1 - 3) + 2 πn, x = - arccos (1 - 3) + 2 πn

x = arccos (1 - 3) + 2 πn, x = - arccos (1 - 3) + 2 πn

Combine all the solutions

x = arccos (1 - 3) + 2 πn, x = - arccos (1 - 3) + 2 πn

Show solutions in decimal form

x = 2.39212 \dots + 2 πn, x = - 2.39212 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x)-2cos(x)-2=0 ?
The general solution for cos^2(x)-2cos(x)-2=0 is x=2.39212…+2pin,x=-2.39212…+2pin