What is the general solution for cos^3(x)+cos^2(x)+cos(x)+1=0 ?

The general solution for cos^3(x)+cos^2(x)+cos(x)+1=0 is x=pi+2pin

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Popular >

cos^3(x)+cos^2(x)+cos(x)+1=0

Solution

cos^{3} (x) + cos^{2} (x) + cos (x) + 1 = 0

Solution

x = π + 2 πn

Degrees

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

Solution steps

cos^{3} (x) + cos^{2} (x) + cos (x) + 1 = 0

Solve by substitution

cos^{3} (x) + cos^{2} (x) + cos (x) + 1 = 0

Let:

cos (x) = u

u^{3} + u^{2} + u + 1 = 0

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

u^{3} + u^{2} + u + 1 = 0

Factor

u^{3} + u^{2} + u + 1 : (u + 1) (u^{2} + 1)

u^{3} + u^{2} + u + 1

= (u^{3} + u^{2}) + (u + 1)

Factor out

u^{2}

from

u^{3} + u^{2} : u^{2} (u + 1)

u^{3} + u^{2}

Apply exponent rule:

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

u^{3} = u u^{2}

= u u^{2} + u^{2}

Factor out common term

u^{2}

= u^{2} (u + 1)

= (u + 1) + u^{2} (u + 1)

Factor out common term

u + 1

= (u + 1) (u^{2} + 1)

(u + 1) (u^{2} + 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

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

Solve

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

Solve

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

u^{2} + 1 = 0

Move

1

to the right side

u^{2} + 1 = 0

Subtract

1

from both sides

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

Simplify

u^{2} = - 1

u^{2} = - 1

For

x^{2} = f (a)

the solutions are

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

u = - 1, u = - - 1

Simplify

- 1 : i

- 1

Apply imaginary number rule:

- 1 = i

= i

Simplify

- - 1 : - i

- - 1

Apply imaginary number rule:

- 1 = i

= - i

u = i, u = - i

The solutions are

u = - 1, u = i, u = - i

Substitute back

u = cos (x)

cos (x) = - 1, cos (x) = i, cos (x) = - i

cos (x) = - 1, cos (x) = i, cos (x) = - i

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

cos (x) = i :

No Solution

cos (x) = i

No Solution

cos (x) = - i :

No Solution

cos (x) = - i

No Solution

Combine all the solutions

x = π + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos^3(x)+cos^2(x)+cos(x)+1=0 ?
The general solution for cos^3(x)+cos^2(x)+cos(x)+1=0 is x=pi+2pin