What is the general solution for cos^4(x)= 3/8+1/2 cos^2(x)+1/8 cos^4(x) ?

The general solution for cos^4(x)= 3/8+1/2 cos^2(x)+1/8 cos^4(x) is x=2pin,x=pi+2pin

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cos^4(x)= 3/8+1/2 cos^2(x)+1/8 cos^4(x)

Solution

cos^{4} (x) = \frac{3}{8} + \frac{1}{2} cos^{2} (x) + \frac{1}{8} cos^{4} (x)

Solution

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

cos^{4} (x) = \frac{3}{8} + \frac{1}{2} cos^{2} (x) + \frac{1}{8} cos^{4} (x)

Solve by substitution

cos^{4} (x) = \frac{3}{8} + \frac{1}{2} cos^{2} (x) + \frac{1}{8} cos^{4} (x)

Let:

cos (x) = u

u^{4} = \frac{3}{8} + \frac{1}{2} u^{2} + \frac{1}{8} u^{4}

u^{4} = \frac{3}{8} + \frac{1}{2} u^{2} + \frac{1}{8} u^{4} : u = i \frac{3}{7}, u = - i \frac{3}{7}, u = 1, u = - 1

u^{4} = \frac{3}{8} + \frac{1}{2} u^{2} + \frac{1}{8} u^{4}

Find Least Common Multiplier of

8, 2 : 8

8, 2

Least Common Multiplier (LCM)

Prime factorization of

8 : 2 \cdot 2 \cdot 2

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

8

2

= 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 8

Multiply by LCM=

8

u^{4} \cdot 8 = \frac{3}{8} \cdot 8 + \frac{1}{2} u^{2} \cdot 8 + \frac{1}{8} u^{4} \cdot 8

Simplify

8 u^{4} = 3 + 4 u^{2} + u^{4}

Switch sides

3 + 4 u^{2} + u^{4} = 8 u^{4}

Move

8 u^{4}

to the left side

3 + 4 u^{2} + u^{4} = 8 u^{4}

Subtract

8 u^{4}

from both sides

3 + 4 u^{2} + u^{4} - 8 u^{4} = 8 u^{4} - 8 u^{4}

Simplify

3 + 4 u^{2} - 7 u^{4} = 0

3 + 4 u^{2} - 7 u^{4} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 7 u^{4} + 4 u^{2} + 3 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

- 7 v^{2} + 4 v + 3 = 0

Solve

- 7 v^{2} + 4 v + 3 = 0 : v = - \frac{3}{7}, v = 1

- 7 v^{2} + 4 v + 3 = 0

Solve with the quadratic formula

- 7 v^{2} + 4 v + 3 = 0

Quadratic Equation Formula:

For

a = - 7, b = 4, c = 3

v_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 7 ) \cdot 3}{2 ( - 7 )}

v_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 7 ) \cdot 3}{2 ( - 7 )}

4^{2} - 4 (- 7) \cdot 3 = 10

4^{2} - 4 (- 7) \cdot 3

Apply rule

- (- a) = a

= 4^{2} + 4 \cdot 7 \cdot 3

Multiply the numbers:

4 \cdot 7 \cdot 3 = 84

= 4^{2} + 84

4^{2} = 16

= 16 + 84

Add the numbers:

16 + 84 = 100

= 100

Factor the number:

100 = 1 0^{2}

= 1 0^{2}

Apply radical rule:

1 0^{2} = 10

= 10

v_{1, 2} = \frac{- 4 \pm 10}{2 ( - 7 )}

Separate the solutions

v_{1} = \frac{- 4 + 10}{2 ( - 7 )}, v_{2} = \frac{- 4 - 10}{2 ( - 7 )}

v = \frac{- 4 + 10}{2 ( - 7 )} : - \frac{3}{7}

\frac{- 4 + 10}{2 ( - 7 )}

Remove parentheses:

(- a) = - a

= \frac{- 4 + 10}{- 2 \cdot 7}

Add/Subtract the numbers:

- 4 + 10 = 6

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

Multiply the numbers:

2 \cdot 7 = 14

= \frac{6}{- 14}

Apply the fraction rule:

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

= - \frac{6}{14}

Cancel the common factor:

2

= - \frac{3}{7}

v = \frac{- 4 - 10}{2 ( - 7 )} : 1

\frac{- 4 - 10}{2 ( - 7 )}

Remove parentheses:

(- a) = - a

= \frac{- 4 - 10}{- 2 \cdot 7}

Subtract the numbers:

- 4 - 10 = - 14

= \frac{- 14}{- 2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{- 14}{- 14}

Apply the fraction rule:

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

= \frac{14}{14}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

v = - \frac{3}{7}, v = 1

v = - \frac{3}{7}, v = 1

Substitute back

v = u^{2},

solve for

u

Solve

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

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

For

x^{2} = f (a)

the solutions are

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

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

Simplify

- \frac{3}{7} : i \frac{3}{7}

- \frac{3}{7}

Apply radical rule:

- a = - 1 a

- \frac{3}{7} = - 1 \frac{3}{7}

= - 1 \frac{3}{7}

Apply imaginary number rule:

- 1 = i

= i \frac{3}{7}

Simplify

- - \frac{3}{7} : - i \frac{3}{7}

- - \frac{3}{7}

Simplify

- \frac{3}{7} : i \frac{3}{7}

- \frac{3}{7}

Apply radical rule:

- a = - 1 a

- \frac{3}{7} = - 1 \frac{3}{7}

= - 1 \frac{3}{7}

Apply imaginary number rule:

- 1 = i

= i \frac{3}{7}

= - i \frac{3}{7}

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

Solve

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

u^{2} = 1

For

x^{2} = f (a)

the solutions are

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

u = 1, u = - 1

1 = 1

1

Apply rule

1 = 1

= 1

- 1 = - 1

- 1

Apply rule

1 = 1

= - 1

u = 1, u = - 1

The solutions are

u = i \frac{3}{7}, u = - i \frac{3}{7}, u = 1, u = - 1

Substitute back

u = cos (x)

cos (x) = i \frac{3}{7}, cos (x) = - i \frac{3}{7}, cos (x) = 1, cos (x) = - 1

cos (x) = i \frac{3}{7}, cos (x) = - i \frac{3}{7}, cos (x) = 1, cos (x) = - 1

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

No Solution

cos (x) = i \frac{3}{7}

No Solution

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

No Solution

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

No Solution

cos (x) = 1 : x = 2 πn

cos (x) = 1

General solutions for

cos (x) = 1

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}

x = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

General solutions for

cos (x) = - 1

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}

x = π + 2 πn

x = π + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos^4(x)= 3/8+1/2 cos^2(x)+1/8 cos^4(x) ?
The general solution for cos^4(x)= 3/8+1/2 cos^2(x)+1/8 cos^4(x) is x=2pin,x=pi+2pin