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

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

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cos^2(x)= 1/2+1/2 cos^2(x)

Solution

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

Solve by substitution

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

Let:

cos (x) = u

u^{2} = \frac{1}{2} + \frac{1}{2} u^{2}

u^{2} = \frac{1}{2} + \frac{1}{2} u^{2} : u = 1, u = - 1

u^{2} = \frac{1}{2} + \frac{1}{2} u^{2}

Move

\frac{1}{2} u^{2}

to the left side

u^{2} = \frac{1}{2} + \frac{1}{2} u^{2}

Subtract

\frac{1}{2} u^{2}

from both sides

u^{2} - \frac{1}{2} u^{2} = \frac{1}{2} + \frac{1}{2} u^{2} - \frac{1}{2} u^{2}

Simplify

\frac{1}{2} u^{2} = \frac{1}{2}

\frac{1}{2} u^{2} = \frac{1}{2}

Multiply both sides by

2

\frac{1}{2} u^{2} = \frac{1}{2}

Multiply both sides by

2

2 \cdot \frac{1}{2} u^{2} = \frac{1 \cdot 2}{2}

Simplify

2 \cdot \frac{1}{2} u^{2} = \frac{1 \cdot 2}{2}

Simplify

2 \cdot \frac{1}{2} u^{2} : u^{2}

2 \cdot \frac{1}{2} u^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2} u^{2}

Cancel the common factor:

2

= u^{2} \cdot 1

Multiply:

u^{2} \cdot 1 = u^{2}

= u^{2}

Simplify

\frac{1 \cdot 2}{2} : 1

\frac{1 \cdot 2}{2}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

u^{2} = 1

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

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Frequently Asked Questions (FAQ)

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