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

The general solution for cos(2x)+cos^2(x)=2 is x=2pin,x=pi+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos(2x)+cos^2(x)=2

Solution

cos (2 x) + cos^{2} (x) = 2

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 (2 x) + cos^{2} (x) = 2

Subtract

2

from both sides

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

Rewrite using trig identities

- 2 + cos (2 x) + cos^{2} (x)

Use the Double Angle identity:

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

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

Simplify

- 2 + 2 cos^{2} (x) - 1 + cos^{2} (x) : 3 cos^{2} (x) - 3

- 2 + 2 cos^{2} (x) - 1 + cos^{2} (x)

Group like terms

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

Add similar elements:

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

= 3 cos^{2} (x) - 2 - 1

Subtract the numbers:

- 2 - 1 = - 3

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

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

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

Solve by substitution

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

Let:

cos (x) = u

- 3 + 3 u^{2} = 0

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

- 3 + 3 u^{2} = 0

Move

3

to the right side

- 3 + 3 u^{2} = 0

Add

3

to both sides

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

Simplify

3 u^{2} = 3

3 u^{2} = 3

Divide both sides by

3

3 u^{2} = 3

Divide both sides by

3

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

Simplify

u^{2} = 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

Substitute back

u = cos (x)

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

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

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

Graph

Popular Examples

tan^2(x)+5tan(x)-7=0

tan^{2} (x) + 5 tan (x) - 7 = 0

sin(x)=(-3)/4

sin (x) = \frac{- 3}{4}

cot^2(x)-sqrt(3)cot(x)=0

cot^{2} (x) - 3 cot (x) = 0

solvefor x,y=sec(x)solve for

x, y = sec (x)

tan(θ)=sqrt(2)

tan (θ) = 2

Frequently Asked Questions (FAQ)

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