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

The general solution for sin^2(x)= 1/2+1/2 cos(2x) is x= pi/4+pin,x=(3pi)/4+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Solution

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

Solution

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

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

from both sides

sin^{2} (x) - \frac{1}{2} - \frac{1}{2} cos (2 x) = 0

Simplify

sin^{2} (x) - \frac{1}{2} - \frac{1}{2} cos (2 x) : \frac{2 sin ^{2} ( x ) - 1 - cos ( 2 x )}{2}

sin^{2} (x) - \frac{1}{2} - \frac{1}{2} cos (2 x)

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

\frac{1}{2} cos (2 x)

Multiply fractions:

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

= \frac{1 \cdot cos ( 2 x )}{2}

Multiply:

1 \cdot cos (2 x) = cos (2 x)

= \frac{cos ( 2 x )}{2}

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

Combine the fractions

- \frac{1}{2} - \frac{cos ( 2 x )}{2} : \frac{- 1 - cos ( 2 x )}{2}

Apply rule

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

= \frac{- 1 - cos ( 2 x )}{2}

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

Convert element to fraction:

sin^{2} (x) = \frac{sin ^{2} ( x ) 2}{2}

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

Since the denominators are equal, combine the fractions:

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

= \frac{sin ^{2} ( x ) \cdot 2 - 1 - cos ( 2 x )}{2}

\frac{2 sin ^{2} ( x ) - 1 - cos ( 2 x )}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 sin^{2} (x) - 1 - cos (2 x) = 0

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

= - cos (2 x) - cos (2 x)

Simplify

= - 2 cos (2 x)

- 2 cos (2 x) = 0

Divide both sides by

- 2

- 2 cos (2 x) = 0

Divide both sides by

- 2

\frac{- 2 cos ( 2 x )}{- 2} = \frac{0}{- 2}

Simplify

cos (2 x) = 0

cos (2 x) = 0

General solutions for

cos (2 x) = 0

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}

2 x = \frac{π}{2} + 2 πn, 2 x = \frac{3 π}{2} + 2 πn

2 x = \frac{π}{2} + 2 πn, 2 x = \frac{3 π}{2} + 2 πn

Solve

2 x = \frac{π}{2} + 2 πn : x = \frac{π}{4} + πn

2 x = \frac{π}{2} + 2 πn

Divide both sides by

2

2 x = \frac{π}{2} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2} : \frac{π}{4} + πn

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Solve

2 x = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{4} + πn

2 x = \frac{3 π}{2} + 2 πn

Divide both sides by

2

2 x = \frac{3 π}{2} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{3 π}{2}}{2} + \frac{2 πn}{2} : \frac{3 π}{4} + πn

\frac{\frac{3 π}{2}}{2} + \frac{2 πn}{2}

\frac{\frac{3 π}{2}}{2} = \frac{3 π}{4}

\frac{\frac{3 π}{2}}{2}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

Graph

Popular Examples

tan(x)= 8/12

tan (x) = \frac{8}{12}

tan(x)= 8/10

tan (x) = \frac{8}{10}

3cot(x)-1=0

3 cot (x) - 1 = 0

4sin(3θ)=0

4 sin (3 θ) = 0

tan(2θ)+2cos(θ)=0,0<= θ<2pi

tan (2 θ) + 2 cos (θ) = 0, 0 \leq θ < 2 π

Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)= 1/2+1/2 cos(2x) ?
The general solution for sin^2(x)= 1/2+1/2 cos(2x) is x= pi/4+pin,x=(3pi)/4+pin