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

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

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2-2cos^2(x)=0

Solution

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

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

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

Solve by substitution

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

Let:

cos (x) = u

2 - 2 u^{2} = 0

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

2 - 2 u^{2} = 0

Move

2

to the right side

2 - 2 u^{2} = 0

Subtract

2

from both sides

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

Simplify

- 2 u^{2} = - 2

- 2 u^{2} = - 2

Divide both sides by

- 2

- 2 u^{2} = - 2

Divide both sides by

- 2

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

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

Popular Examples

3csc^3(x)-4csc(x)=0

3 csc^{3} (x) - 4 csc (x) = 0

tan(θ)=(6.06)/(-33.5)

tan (θ) = \frac{6.06}{- 33.5}

csc(θ)=(sqrt(13))/2 ,sec(θ)=(sqrt(13))/3

csc (θ) = \frac{13}{2}, sec (θ) = \frac{13}{3}

4cos^2(x)+cos(2x)-7cos(x)=-2,0<= x<= 360

4 cos^{2} (x) + cos (2 x) - 7 cos (x) = - 2, 0^{\circ} \leq x \leq 36 0^{\circ}

(16)/(sin(a))=(12)/(sin(31))

\frac{16}{sin ( a )} = \frac{12}{sin ( 3 1 ^{\circ} )}

Frequently Asked Questions (FAQ)

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