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Popular Trigonometry >

2cos^4(x)-2cos^2(x)=4cos^2(x)-1

Solution

2 cos^{4} (x) - 2 cos^{2} (x) = 4 cos^{2} (x) - 1

Solution

x = 1.13640 \dots + 2 πn, x = 2 π - 1.13640 \dots + 2 πn, x = 2.00519 \dots + 2 πn, x = - 2.00519 \dots + 2 πn

Degrees

x = 65.11101 \dots^{\circ} + 36 0^{\circ} n, x = 294.88898 \dots^{\circ} + 36 0^{\circ} n, x = 114.88898 \dots^{\circ} + 36 0^{\circ} n, x = - 114.88898 \dots^{\circ} + 36 0^{\circ} n

Solution steps

2 cos^{4} (x) - 2 cos^{2} (x) = 4 cos^{2} (x) - 1

Solve by substitution

2 cos^{4} (x) - 2 cos^{2} (x) = 4 cos^{2} (x) - 1

Let:

cos (x) = u

2 u^{4} - 2 u^{2} = 4 u^{2} - 1

2 u^{4} - 2 u^{2} = 4 u^{2} - 1 : u = \frac{3 + 7}{2}, u = - \frac{3 + 7}{2}, u = \frac{3 - 7}{2}, u = - \frac{3 - 7}{2}

2 u^{4} - 2 u^{2} = 4 u^{2} - 1

Move

1

to the left side

2 u^{4} - 2 u^{2} = 4 u^{2} - 1

Add

1

to both sides

2 u^{4} - 2 u^{2} + 1 = 4 u^{2} - 1 + 1

Simplify

2 u^{4} - 2 u^{2} + 1 = 4 u^{2}

2 u^{4} - 2 u^{2} + 1 = 4 u^{2}

Move

4 u^{2}

to the left side

2 u^{4} - 2 u^{2} + 1 = 4 u^{2}

Subtract

4 u^{2}

from both sides

2 u^{4} - 2 u^{2} + 1 - 4 u^{2} = 4 u^{2} - 4 u^{2}

Simplify

2 u^{4} - 6 u^{2} + 1 = 0

2 u^{4} - 6 u^{2} + 1 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

2 v^{2} - 6 v + 1 = 0

Solve

2 v^{2} - 6 v + 1 = 0 : v = \frac{3 + 7}{2}, v = \frac{3 - 7}{2}

2 v^{2} - 6 v + 1 = 0

Solve with the quadratic formula

2 v^{2} - 6 v + 1 = 0

Quadratic Equation Formula:

For

a = 2, b = - 6, c = 1

v_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 2 \cdot 1}{2 \cdot 2}

v_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 2 \cdot 1}{2 \cdot 2}

(- 6)^{2} - 4 \cdot 2 \cdot 1 = 27

(- 6)^{2} - 4 \cdot 2 \cdot 1

Apply exponent rule:

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

n

is even

(- 6)^{2} = 6^{2}

= 6^{2} - 4 \cdot 2 \cdot 1

Multiply the numbers:

4 \cdot 2 \cdot 1 = 8

= 6^{2} - 8

6^{2} = 36

= 36 - 8

Subtract the numbers:

36 - 8 = 28

= 28

Prime factorization of

28 : 2^{2} \cdot 7

28

28

divides by

228 = 14 \cdot 2

= 2 \cdot 14

14

divides by

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

Apply radical rule:

= 7 2^{2}

Apply radical rule:

2^{2} = 2

= 27

v_{1, 2} = \frac{- ( - 6 ) \pm 2 7}{2 \cdot 2}

Separate the solutions

v_{1} = \frac{- ( - 6 ) + 2 7}{2 \cdot 2}, v_{2} = \frac{- ( - 6 ) - 2 7}{2 \cdot 2}

v = \frac{- ( - 6 ) + 2 7}{2 \cdot 2} : \frac{3 + 7}{2}

\frac{- ( - 6 ) + 2 7}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{6 + 2 7}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{6 + 2 7}{4}

Factor

6 + 27 : 2 (3 + 7)

6 + 27

Rewrite as

= 2 \cdot 3 + 27

Factor out common term

2

= 2 (3 + 7)

= \frac{2 ( 3 + 7 )}{4}

Cancel the common factor:

2

= \frac{3 + 7}{2}

v = \frac{- ( - 6 ) - 2 7}{2 \cdot 2} : \frac{3 - 7}{2}

\frac{- ( - 6 ) - 2 7}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{6 - 2 7}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{6 - 2 7}{4}

Factor

6 - 27 : 2 (3 - 7)

6 - 27

Rewrite as

= 2 \cdot 3 - 27

Factor out common term

2

= 2 (3 - 7)

= \frac{2 ( 3 - 7 )}{4}

Cancel the common factor:

2

= \frac{3 - 7}{2}

The solutions to the quadratic equation are:

v = \frac{3 + 7}{2}, v = \frac{3 - 7}{2}

v = \frac{3 + 7}{2}, v = \frac{3 - 7}{2}

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = \frac{3 + 7}{2} : u = \frac{3 + 7}{2}, u = - \frac{3 + 7}{2}

u^{2} = \frac{3 + 7}{2}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{3 + 7}{2}, u = - \frac{3 + 7}{2}

Solve

u^{2} = \frac{3 - 7}{2} : u = \frac{3 - 7}{2}, u = - \frac{3 - 7}{2}

u^{2} = \frac{3 - 7}{2}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{3 - 7}{2}, u = - \frac{3 - 7}{2}

The solutions are

u = \frac{3 + 7}{2}, u = - \frac{3 + 7}{2}, u = \frac{3 - 7}{2}, u = - \frac{3 - 7}{2}

Substitute back

u = cos (x)

cos (x) = \frac{3 + 7}{2}, cos (x) = - \frac{3 + 7}{2}, cos (x) = \frac{3 - 7}{2}, cos (x) = - \frac{3 - 7}{2}

cos (x) = \frac{3 + 7}{2}, cos (x) = - \frac{3 + 7}{2}, cos (x) = \frac{3 - 7}{2}, cos (x) = - \frac{3 - 7}{2}

cos (x) = \frac{3 + 7}{2} :

No Solution

cos (x) = \frac{3 + 7}{2}

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = - \frac{3 + 7}{2} :

No Solution

cos (x) = - \frac{3 + 7}{2}

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = \frac{3 - 7}{2} : x = arccos \frac{3 - 7}{2} + 2 πn, x = 2 π - arccos \frac{3 - 7}{2} + 2 πn

cos (x) = \frac{3 - 7}{2}

Apply trig inverse properties

cos (x) = \frac{3 - 7}{2}

General solutions for

cos (x) = \frac{3 - 7}{2}

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

x = arccos \frac{3 - 7}{2} + 2 πn, x = 2 π - arccos \frac{3 - 7}{2} + 2 πn

x = arccos \frac{3 - 7}{2} + 2 πn, x = 2 π - arccos \frac{3 - 7}{2} + 2 πn

cos (x) = - \frac{3 - 7}{2} : x = arccos - \frac{3 - 7}{2} + 2 πn, x = - arccos - \frac{3 - 7}{2} + 2 πn

cos (x) = - \frac{3 - 7}{2}

Apply trig inverse properties

cos (x) = - \frac{3 - 7}{2}

General solutions for

cos (x) = - \frac{3 - 7}{2}

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

x = arccos - \frac{3 - 7}{2} + 2 πn, x = - arccos - \frac{3 - 7}{2} + 2 πn

x = arccos - \frac{3 - 7}{2} + 2 πn, x = - arccos - \frac{3 - 7}{2} + 2 πn

Combine all the solutions

x = arccos \frac{3 - 7}{2} + 2 πn, x = 2 π - arccos \frac{3 - 7}{2} + 2 πn, x = arccos - \frac{3 - 7}{2} + 2 πn, x = - arccos - \frac{3 - 7}{2} + 2 πn

Show solutions in decimal form

x = 1.13640 \dots + 2 πn, x = 2 π - 1.13640 \dots + 2 πn, x = 2.00519 \dots + 2 πn, x = - 2.00519 \dots + 2 πn

Graph

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