What is the general solution for 4sin^2(x/3)=1 ?

The general solution for 4sin^2(x/3)=1 is x= pi/2+6pin,x=(5pi)/2+6pin,x=(7pi)/2+6pin,x=(11pi)/2+6pin

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4sin^2(x/3)=1

Solution

4 sin^{2} (\frac{x}{3}) = 1

Solution

x = \frac{π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn, x = \frac{7 π}{2} + 6 πn, x = \frac{11 π}{2} + 6 πn

Degrees

x = 9 0^{\circ} + 108 0^{\circ} n, x = 45 0^{\circ} + 108 0^{\circ} n, x = 63 0^{\circ} + 108 0^{\circ} n, x = 99 0^{\circ} + 108 0^{\circ} n

Solution steps

4 sin^{2} (\frac{x}{3}) = 1

Solve by substitution

4 sin^{2} (\frac{x}{3}) = 1

Let:

sin (\frac{x}{3}) = u

4 u^{2} = 1

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

4 u^{2} = 1

Divide both sides by

4

4 u^{2} = 1

Divide both sides by

4

\frac{4 u ^{2}}{4} = \frac{1}{4}

Simplify

u^{2} = \frac{1}{4}

u^{2} = \frac{1}{4}

For

x^{2} = f (a)

the solutions are

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

u = \frac{1}{4}, u = - \frac{1}{4}

\frac{1}{4} = \frac{1}{2}

\frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

Apply rule

1 = 1

= \frac{1}{2}

- \frac{1}{4} = - \frac{1}{2}

- \frac{1}{4}

Simplify

\frac{1}{4} : \frac{1}{2}

\frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

= - \frac{1}{2}

Apply rule

1 = 1

= - \frac{1}{2}

u = \frac{1}{2}, u = - \frac{1}{2}

Substitute back

u = sin (\frac{x}{3})

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

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

sin (\frac{x}{3}) = \frac{1}{2} : x = \frac{π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn

sin (\frac{x}{3}) = \frac{1}{2}

General solutions for

sin (\frac{x}{3}) = \frac{1}{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}

\frac{x}{3} = \frac{π}{6} + 2 πn, \frac{x}{3} = \frac{5 π}{6} + 2 πn

\frac{x}{3} = \frac{π}{6} + 2 πn, \frac{x}{3} = \frac{5 π}{6} + 2 πn

Solve

\frac{x}{3} = \frac{π}{6} + 2 πn : x = \frac{π}{2} + 6 πn

\frac{x}{3} = \frac{π}{6} + 2 πn

Multiply both sides by

3

\frac{x}{3} = \frac{π}{6} + 2 πn

Multiply both sides by

3

\frac{3 x}{3} = 3 \cdot \frac{π}{6} + 3 \cdot 2 πn

Simplify

\frac{3 x}{3} = 3 \cdot \frac{π}{6} + 3 \cdot 2 πn

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

3 \cdot \frac{π}{6} + 3 \cdot 2 πn : \frac{π}{2} + 6 πn

3 \cdot \frac{π}{6} + 3 \cdot 2 πn

3 \cdot \frac{π}{6} = \frac{π}{2}

3 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 3}{6}

Cancel the common factor:

3

= \frac{π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= \frac{π}{2} + 6 πn

x = \frac{π}{2} + 6 πn

x = \frac{π}{2} + 6 πn

x = \frac{π}{2} + 6 πn

Solve

\frac{x}{3} = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{2} + 6 πn

\frac{x}{3} = \frac{5 π}{6} + 2 πn

Multiply both sides by

3

\frac{x}{3} = \frac{5 π}{6} + 2 πn

Multiply both sides by

3

\frac{3 x}{3} = 3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

Simplify

\frac{3 x}{3} = 3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn : \frac{5 π}{2} + 6 πn

3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

3 \cdot \frac{5 π}{6} = \frac{5 π}{2}

3 \cdot \frac{5 π}{6}

Multiply fractions:

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

= \frac{5 π 3}{6}

Multiply the numbers:

5 \cdot 3 = 15

= \frac{15 π}{6}

Cancel the common factor:

3

= \frac{5 π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= \frac{5 π}{2} + 6 πn

x = \frac{5 π}{2} + 6 πn

x = \frac{5 π}{2} + 6 πn

x = \frac{5 π}{2} + 6 πn

x = \frac{π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn

sin (\frac{x}{3}) = - \frac{1}{2} : x = \frac{7 π}{2} + 6 πn, x = \frac{11 π}{2} + 6 πn

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

General solutions for

sin (\frac{x}{3}) = - \frac{1}{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}

\frac{x}{3} = \frac{7 π}{6} + 2 πn, \frac{x}{3} = \frac{11 π}{6} + 2 πn

\frac{x}{3} = \frac{7 π}{6} + 2 πn, \frac{x}{3} = \frac{11 π}{6} + 2 πn

Solve

\frac{x}{3} = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{2} + 6 πn

\frac{x}{3} = \frac{7 π}{6} + 2 πn

Multiply both sides by

3

\frac{x}{3} = \frac{7 π}{6} + 2 πn

Multiply both sides by

3

\frac{3 x}{3} = 3 \cdot \frac{7 π}{6} + 3 \cdot 2 πn

Simplify

\frac{3 x}{3} = 3 \cdot \frac{7 π}{6} + 3 \cdot 2 πn

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

3 \cdot \frac{7 π}{6} + 3 \cdot 2 πn : \frac{7 π}{2} + 6 πn

3 \cdot \frac{7 π}{6} + 3 \cdot 2 πn

3 \cdot \frac{7 π}{6} = \frac{7 π}{2}

3 \cdot \frac{7 π}{6}

Multiply fractions:

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

= \frac{7 π 3}{6}

Multiply the numbers:

7 \cdot 3 = 21

= \frac{21 π}{6}

Cancel the common factor:

3

= \frac{7 π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= \frac{7 π}{2} + 6 πn

x = \frac{7 π}{2} + 6 πn

x = \frac{7 π}{2} + 6 πn

x = \frac{7 π}{2} + 6 πn

Solve

\frac{x}{3} = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{2} + 6 πn

\frac{x}{3} = \frac{11 π}{6} + 2 πn

Multiply both sides by

3

\frac{x}{3} = \frac{11 π}{6} + 2 πn

Multiply both sides by

3

\frac{3 x}{3} = 3 \cdot \frac{11 π}{6} + 3 \cdot 2 πn

Simplify

\frac{3 x}{3} = 3 \cdot \frac{11 π}{6} + 3 \cdot 2 πn

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

3 \cdot \frac{11 π}{6} + 3 \cdot 2 πn : \frac{11 π}{2} + 6 πn

3 \cdot \frac{11 π}{6} + 3 \cdot 2 πn

3 \cdot \frac{11 π}{6} = \frac{11 π}{2}

3 \cdot \frac{11 π}{6}

Multiply fractions:

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

= \frac{11 π 3}{6}

Multiply the numbers:

11 \cdot 3 = 33

= \frac{33 π}{6}

Cancel the common factor:

3

= \frac{11 π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= \frac{11 π}{2} + 6 πn

x = \frac{11 π}{2} + 6 πn

x = \frac{11 π}{2} + 6 πn

x = \frac{11 π}{2} + 6 πn

x = \frac{7 π}{2} + 6 πn, x = \frac{11 π}{2} + 6 πn

Combine all the solutions

x = \frac{π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn, x = \frac{7 π}{2} + 6 πn, x = \frac{11 π}{2} + 6 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 4sin^2(x/3)=1 ?
The general solution for 4sin^2(x/3)=1 is x= pi/2+6pin,x=(5pi)/2+6pin,x=(7pi)/2+6pin,x=(11pi)/2+6pin