What is the general solution for 4sin^2(x-pi/3)=3 ?

The general solution for 4sin^2(x-pi/3)=3 is x=(2pi)/3+2pin,x=2pin+pi,x=2pin+(5pi)/3 ,x=2pin+2pi

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4sin^2(x-pi/3)=3

Solution

4 sin^{2} (x - \frac{π}{3}) = 3

Solution

x = \frac{2 π}{3} + 2 πn, x = 2 πn + π, x = 2 πn + \frac{5 π}{3}, x = 2 πn + 2 π

Degrees

x = 12 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} + 36 0^{\circ} n

Solution steps

4 sin^{2} (x - \frac{π}{3}) = 3

Solve by substitution

4 sin^{2} (x - \frac{π}{3}) = 3

Let:

sin (x - \frac{π}{3}) = u

4 u^{2} = 3

4 u^{2} = 3 : u = \frac{3}{2}, u = - \frac{3}{2}

4 u^{2} = 3

Divide both sides by

4

4 u^{2} = 3

Divide both sides by

4

\frac{4 u ^{2}}{4} = \frac{3}{4}

Simplify

u^{2} = \frac{3}{4}

u^{2} = \frac{3}{4}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{3}{4}, u = - \frac{3}{4}

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

\frac{3}{4}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

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

- \frac{3}{4}

Simplify

\frac{3}{4} : \frac{3}{2}

\frac{3}{4}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

= - \frac{3}{2}

u = \frac{3}{2}, u = - \frac{3}{2}

Substitute back

u = sin (x - \frac{π}{3})

sin (x - \frac{π}{3}) = \frac{3}{2}, sin (x - \frac{π}{3}) = - \frac{3}{2}

sin (x - \frac{π}{3}) = \frac{3}{2}, sin (x - \frac{π}{3}) = - \frac{3}{2}

sin (x - \frac{π}{3}) = \frac{3}{2} : x = \frac{2 π}{3} + 2 πn, x = 2 πn + π

sin (x - \frac{π}{3}) = \frac{3}{2}

General solutions for

sin (x - \frac{π}{3}) = \frac{3}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

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

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

Solve

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

x - \frac{π}{3} = \frac{π}{3} + 2 πn

Move

\frac{π}{3}

to the right side

x - \frac{π}{3} = \frac{π}{3} + 2 πn

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

x - \frac{π}{3} + \frac{π}{3} : x

x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= x

Simplify

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

\frac{π}{3} + 2 πn + \frac{π}{3}

Group like terms

= \frac{π}{3} + \frac{π}{3} + 2 πn

Combine the fractions

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

Apply rule

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

= \frac{π + π}{3}

Add similar elements:

π + π = 2 π

= \frac{2 π}{3}

= \frac{2 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn

Solve

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

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

x - \frac{π}{3} + \frac{π}{3} : x

x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= x

Simplify

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

\frac{2 π}{3} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{3} + \frac{2 π}{3}

Combine the fractions

\frac{π}{3} + \frac{2 π}{3} : π

Apply rule

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

= \frac{π + 2 π}{3}

Add similar elements:

π + 2 π = 3 π

= \frac{3 π}{3}

Divide the numbers:

\frac{3}{3} = 1

= π

= 2 πn + π

x = 2 πn + π

x = 2 πn + π

x = 2 πn + π

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

sin (x - \frac{π}{3}) = - \frac{3}{2} : x = 2 πn + \frac{5 π}{3}, x = 2 πn + 2 π

sin (x - \frac{π}{3}) = - \frac{3}{2}

General solutions for

sin (x - \frac{π}{3}) = - \frac{3}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x - \frac{π}{3} = \frac{4 π}{3} + 2 πn, x - \frac{π}{3} = \frac{5 π}{3} + 2 πn

x - \frac{π}{3} = \frac{4 π}{3} + 2 πn, x - \frac{π}{3} = \frac{5 π}{3} + 2 πn

Solve

x - \frac{π}{3} = \frac{4 π}{3} + 2 πn : x = 2 πn + \frac{5 π}{3}

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

x - \frac{π}{3} + \frac{π}{3} : x

x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= x

Simplify

\frac{4 π}{3} + 2 πn + \frac{π}{3} : 2 πn + \frac{5 π}{3}

\frac{4 π}{3} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{3} + \frac{4 π}{3}

Combine the fractions

\frac{π}{3} + \frac{4 π}{3} : \frac{5 π}{3}

Apply rule

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

= \frac{π + 4 π}{3}

Add similar elements:

π + 4 π = 5 π

= \frac{5 π}{3}

= 2 πn + \frac{5 π}{3}

x = 2 πn + \frac{5 π}{3}

x = 2 πn + \frac{5 π}{3}

x = 2 πn + \frac{5 π}{3}

Solve

x - \frac{π}{3} = \frac{5 π}{3} + 2 πn : x = 2 πn + 2 π

x - \frac{π}{3} = \frac{5 π}{3} + 2 πn

Move

\frac{π}{3}

to the right side

x - \frac{π}{3} = \frac{5 π}{3} + 2 πn

Add

\frac{π}{3}

to both sides

x - \frac{π}{3} + \frac{π}{3} = \frac{5 π}{3} + 2 πn + \frac{π}{3}

Simplify

x - \frac{π}{3} + \frac{π}{3} = \frac{5 π}{3} + 2 πn + \frac{π}{3}

Simplify

x - \frac{π}{3} + \frac{π}{3} : x

x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= x

Simplify

\frac{5 π}{3} + 2 πn + \frac{π}{3} : 2 πn + 2 π

\frac{5 π}{3} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{3} + \frac{5 π}{3}

Combine the fractions

\frac{π}{3} + \frac{5 π}{3} : 2 π

Apply rule

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

= \frac{π + 5 π}{3}

Add similar elements:

π + 5 π = 6 π

= \frac{6 π}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 π

= 2 πn + 2 π

x = 2 πn + 2 π

x = 2 πn + 2 π

x = 2 πn + 2 π

x = 2 πn + \frac{5 π}{3}, x = 2 πn + 2 π

Combine all the solutions

x = \frac{2 π}{3} + 2 πn, x = 2 πn + π, x = 2 πn + \frac{5 π}{3}, x = 2 πn + 2 π

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 4sin^2(x-pi/3)=3 ?
The general solution for 4sin^2(x-pi/3)=3 is x=(2pi)/3+2pin,x=2pin+pi,x=2pin+(5pi)/3 ,x=2pin+2pi