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

The general solution for cos^3(x)-2cos^2(x)+cos(x)-2=0 is No Solution for x\in\mathbb{R}

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Popular >

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

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Solve by substitution

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

Let:

cos (x) = u

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

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

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

Factor

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

u^{3} - 2 u^{2} + u - 2

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

Factor out

u^{2}

from

u^{3} - 2 u^{2} : u^{2} (u - 2)

u^{3} - 2 u^{2}

Apply exponent rule:

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

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

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

Factor out common term

u^{2}

= u^{2} (u - 2)

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

Factor out common term

u - 2

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

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

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

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

Solve

u - 2 = 0 : u = 2

u - 2 = 0

Move

2

to the right side

u - 2 = 0

Add

2

to both sides

u - 2 + 2 = 0 + 2

Simplify

u = 2

u = 2

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 = 2, u = i, u = - i

Substitute back

u = cos (x)

cos (x) = 2, cos (x) = i, cos (x) = - i

cos (x) = 2, cos (x) = i, cos (x) = - i

cos (x) = 2 :

No Solution

cos (x) = 2

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = i :

No Solution

cos (x) = i

No Solution

cos (x) = - i :

No Solution

cos (x) = - i

No Solution

Combine all the solutions

No Solution for x \in R

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cos(5x-21)=sin(13-3x)

cos (5 x - 21) = sin (13 - 3 x)

sin(θ)= 5/3

sin (θ) = \frac{5}{3}

Frequently Asked Questions (FAQ)

What is the general solution for cos^3(x)-2cos^2(x)+cos(x)-2=0 ?
The general solution for cos^3(x)-2cos^2(x)+cos(x)-2=0 is No Solution for x\in\mathbb{R}