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

The general solution for 3-4sin^3(x)=sin^3(x) is x=1.00364…+2pin,x=pi-1.00364…+2pin

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Popular Trigonometry >

3-4sin^3(x)=sin^3(x)

Solution

3 - 4 sin^{3} (x) = sin^{3} (x)

Solution

x = 1.00364 \dots + 2 πn, x = π - 1.00364 \dots + 2 πn

Degrees

x = 57.50439 \dots^{\circ} + 36 0^{\circ} n, x = 122.49560 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 - 4 sin^{3} (x) = sin^{3} (x)

Solve by substitution

3 - 4 sin^{3} (x) = sin^{3} (x)

Let:

sin (x) = u

3 - 4 u^{3} = u^{3}

3 - 4 u^{3} = u^{3} : u = 3 \frac{3}{5}, u = - \frac{5 ^{\frac{2}{3}} 3 3}{10} + i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}, u = - \frac{5 ^{\frac{2}{3}} 3 3}{10} - i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}

3 - 4 u^{3} = u^{3}

Move

3

to the right side

3 - 4 u^{3} = u^{3}

Subtract

3

from both sides

3 - 4 u^{3} - 3 = u^{3} - 3

Simplify

- 4 u^{3} = u^{3} - 3

- 4 u^{3} = u^{3} - 3

Move

u^{3}

to the left side

- 4 u^{3} = u^{3} - 3

Subtract

u^{3}

from both sides

- 4 u^{3} - u^{3} = u^{3} - 3 - u^{3}

Simplify

- 5 u^{3} = - 3

- 5 u^{3} = - 3

Divide both sides by

- 5

- 5 u^{3} = - 3

Divide both sides by

- 5

\frac{- 5 u ^{3}}{- 5} = \frac{- 3}{- 5}

Simplify

u^{3} = \frac{3}{5}

u^{3} = \frac{3}{5}

For

x^{3} = f (a)

the solutions are

x = 3 f (a), 3 f (a) \frac{- 1 - 3 i}{2}, 3 f (a) \frac{- 1 + 3 i}{2}

u = 3 \frac{3}{5}, u = 3 \frac{3}{5} \frac{- 1 + 3 i}{2}, u = 3 \frac{3}{5} \frac{- 1 - 3 i}{2}

Simplify

3 \frac{3}{5} \frac{- 1 + 3 i}{2} : - \frac{5 ^{\frac{2}{3}} 3 3}{10} + i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}

3 \frac{3}{5} \frac{- 1 + 3 i}{2}

Multiply fractions:

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

= \frac{( - 1 + 3 i ) 3 \frac{3}{5}}{2}

3 \frac{3}{5} = \frac{3 3}{3 5}

3 \frac{3}{5}

Apply radical rule:

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

assuming

a \geq 0, b \geq 0

= \frac{3 3}{3 5}

= \frac{\frac{3 3}{3 5} ( - 1 + 3 i )}{2}

Multiply

(- 1 + 3 i) \frac{3 3}{3 5} : \frac{- 3 3 + 3 ^{\frac{5}{6}} i}{3 5}

(- 1 + 3 i) \frac{3 3}{3 5}

Multiply fractions:

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

= \frac{3 3 ( - 1 + 3 i )}{3 5}

Expand

33 (- 1 + 3 i) : - 33 + 3^{\frac{5}{6}} i

33 (- 1 + 3 i)

Apply the distributive law:

a (b + c) = ab + a c

a = 33, b = - 1, c = 3 i

= 33 (- 1) + 333 i

Apply minus-plus rules

+ (- a) = - a

= - 1 \cdot 33 + 333 i

Simplify

- 1 \cdot 33 + 333 i : - 33 + 3^{\frac{5}{6}} i

- 1 \cdot 33 + 333 i

1 \cdot 33 = 33

1 \cdot 33

Multiply:

1 \cdot 33 = 33

= 33

333 i = 3^{\frac{5}{6}} i

333 i

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

333 = 3^{\frac{1}{3}} \cdot 3^{\frac{1}{2}} = 3^{\frac{1}{3} + \frac{1}{2}}

= 3^{\frac{1}{3} + \frac{1}{2}} i

3^{\frac{1}{3} + \frac{1}{2}} = 3^{\frac{5}{6}}

3^{\frac{1}{3} + \frac{1}{2}}

Join

\frac{1}{3} + \frac{1}{2} : \frac{5}{6}

\frac{1}{3} + \frac{1}{2}

Least Common Multiplier of

3, 2 : 6

3, 2

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

3

2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{1}{3} :

multiply the denominator and numerator by

2

\frac{1}{3} = \frac{1 \cdot 2}{3 \cdot 2} = \frac{2}{6}

For

\frac{1}{2} :

multiply the denominator and numerator by

3

\frac{1}{2} = \frac{1 \cdot 3}{2 \cdot 3} = \frac{3}{6}

= \frac{2}{6} + \frac{3}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{2 + 3}{6}

Add the numbers:

2 + 3 = 5

= \frac{5}{6}

= 3^{\frac{5}{6}}

= 3^{\frac{5}{6}} i

= - 33 + 3^{\frac{5}{6}} i

= - 33 + 3^{\frac{5}{6}} i

= \frac{- 3 3 + 3 ^{\frac{5}{6}} i}{3 5}

= \frac{\frac{- 3 3 + 3 ^{\frac{5}{6}} i}{3 5}}{2}

Apply the fraction rule:

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

= \frac{- 3 3 + 3 ^{\frac{5}{6}} i}{3 5 \cdot 2}

Rationalize

\frac{- 3 3 + 3 ^{\frac{5}{6}} i}{2 3 5} : \frac{5 ^{\frac{2}{3}} ( - 3 3 + 3 ^{\frac{5}{6}} i )}{10}

\frac{- 3 3 + 3 ^{\frac{5}{6}} i}{2 3 5}

Multiply by the conjugate

\frac{5 ^{\frac{2}{3}}}{5 ^{\frac{2}{3}}}

= \frac{( - 3 3 + 3 ^{\frac{5}{6}} i ) \cdot 5 ^{\frac{2}{3}}}{3 5 \cdot 2 \cdot 5 ^{\frac{2}{3}}}

35 \cdot 2 \cdot 5^{\frac{2}{3}} = 10

35 \cdot 2 \cdot 5^{\frac{2}{3}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

5^{\frac{2}{3}} 35 = 5^{\frac{2}{3}} \cdot 5^{\frac{1}{3}} = 5^{\frac{2}{3} + \frac{1}{3}}

= 5^{\frac{2}{3} + \frac{1}{3}} \cdot 2

5^{\frac{2}{3} + \frac{1}{3}} = 5

5^{\frac{2}{3} + \frac{1}{3}}

Combine the fractions

\frac{2}{3} + \frac{1}{3} : 1

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 5^{1}

Apply rule

a^{1} = a

= 5

= 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 10

= \frac{5 ^{\frac{2}{3}} ( - 3 3 + 3 ^{\frac{5}{6}} i )}{10}

= \frac{5 ^{\frac{2}{3}} ( - 3 3 + 3 ^{\frac{5}{6}} i )}{10}

Rewrite

\frac{5 ^{\frac{2}{3}} ( - 3 3 + 3 ^{\frac{5}{6}} i )}{10}

in standard complex form:

- \frac{3 3 \cdot 5 ^{\frac{2}{3}}}{10} + \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10} i

\frac{5 ^{\frac{2}{3}} ( - 3 3 + 3 ^{\frac{5}{6}} i )}{10}

Factor

10 : 2 \cdot 5

Factor

10 = 2 \cdot 5

= \frac{5 ^{\frac{2}{3}} ( - 3 3 + 3 ^{\frac{5}{6}} i )}{2 \cdot 5}

Cancel

\frac{5 ^{\frac{2}{3}} ( - 3 3 + 3 ^{\frac{5}{6}} i )}{2 \cdot 5} : \frac{- 3 3 + 3 ^{\frac{5}{6}} i}{2 \cdot 5 ^{\frac{1}{3}}}

\frac{5 ^{\frac{2}{3}} ( - 3 3 + 3 ^{\frac{5}{6}} i )}{2 \cdot 5}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{5 ^{\frac{2}{3}}}{5} = \frac{1}{5 ^{1 - \frac{2}{3}}}

= \frac{- 3 3 + 3 ^{\frac{5}{6}} i}{2 \cdot 5 ^{- \frac{2}{3} + 1}}

Subtract the numbers:

1 - \frac{2}{3} = \frac{1}{3}

= \frac{- 3 3 + 3 ^{\frac{5}{6}} i}{2 \cdot 5 ^{\frac{1}{3}}}

= \frac{- 3 3 + 3 ^{\frac{5}{6}} i}{2 \cdot 5 ^{\frac{1}{3}}}

5^{\frac{1}{3}} = 35

Apply radical rule:

a^{\frac{1}{n}} = n a

5^{\frac{1}{3}} = 35

= \frac{- 3 3 + 3 ^{\frac{5}{6}} i}{2 3 5}

Apply the fraction rule:

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

\frac{- 3 3 + 3 ^{\frac{5}{6}} i}{2 3 5} = - \frac{3 3}{2 3 5} + \frac{3 ^{\frac{5}{6}} i}{2 3 5}

= - \frac{3 3}{2 3 5} + \frac{3 ^{\frac{5}{6}} i}{2 3 5}

\frac{3 ^{\frac{5}{6}}}{2 3 5} = \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}

\frac{3 ^{\frac{5}{6}}}{2 3 5}

Multiply by the conjugate

\frac{5 ^{\frac{2}{3}}}{5 ^{\frac{2}{3}}}

= \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{2 3 5 \cdot 5 ^{\frac{2}{3}}}

235 \cdot 5^{\frac{2}{3}} = 10

235 \cdot 5^{\frac{2}{3}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

5^{\frac{2}{3}} 35 = 5^{\frac{2}{3}} \cdot 5^{\frac{1}{3}} = 5^{\frac{2}{3} + \frac{1}{3}}

= 2 \cdot 5^{\frac{2}{3} + \frac{1}{3}}

5^{\frac{2}{3} + \frac{1}{3}} = 5

5^{\frac{2}{3} + \frac{1}{3}}

Combine the fractions

\frac{2}{3} + \frac{1}{3} : 1

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 5^{1}

Apply rule

a^{1} = a

= 5

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

= \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}

= - \frac{3 3}{2 3 5} + \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10} i

- \frac{3 3}{2 3 5} = - \frac{3 3 \cdot 5 ^{\frac{2}{3}}}{10}

- \frac{3 3}{2 3 5}

Multiply by the conjugate

\frac{5 ^{\frac{2}{3}}}{5 ^{\frac{2}{3}}}

= - \frac{3 3 \cdot 5 ^{\frac{2}{3}}}{2 3 5 \cdot 5 ^{\frac{2}{3}}}

235 \cdot 5^{\frac{2}{3}} = 10

235 \cdot 5^{\frac{2}{3}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

5^{\frac{2}{3}} 35 = 5^{\frac{2}{3}} \cdot 5^{\frac{1}{3}} = 5^{\frac{2}{3} + \frac{1}{3}}

= 2 \cdot 5^{\frac{2}{3} + \frac{1}{3}}

5^{\frac{2}{3} + \frac{1}{3}} = 5

5^{\frac{2}{3} + \frac{1}{3}}

Combine the fractions

\frac{2}{3} + \frac{1}{3} : 1

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 5^{1}

Apply rule

a^{1} = a

= 5

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

= - \frac{3 3 \cdot 5 ^{\frac{2}{3}}}{10}

= - \frac{3 3 \cdot 5 ^{\frac{2}{3}}}{10} + \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10} i

= - \frac{3 3 \cdot 5 ^{\frac{2}{3}}}{10} + \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10} i

Simplify

3 \frac{3}{5} \frac{- 1 - 3 i}{2} : - \frac{5 ^{\frac{2}{3}} 3 3}{10} - i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}

3 \frac{3}{5} \frac{- 1 - 3 i}{2}

Multiply fractions:

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

= \frac{( - 1 - 3 i ) 3 \frac{3}{5}}{2}

3 \frac{3}{5} = \frac{3 3}{3 5}

3 \frac{3}{5}

Apply radical rule:

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

assuming

a \geq 0, b \geq 0

= \frac{3 3}{3 5}

= \frac{\frac{3 3}{3 5} ( - 1 - 3 i )}{2}

Multiply

(- 1 - 3 i) \frac{3 3}{3 5} : \frac{- 3 3 - 3 ^{\frac{5}{6}} i}{3 5}

(- 1 - 3 i) \frac{3 3}{3 5}

Multiply fractions:

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

= \frac{3 3 ( - 1 - 3 i )}{3 5}

Expand

33 (- 1 - 3 i) : - 33 - 3^{\frac{5}{6}} i

33 (- 1 - 3 i)

Apply the distributive law:

a (b - c) = ab - a c

a = 33, b = - 1, c = 3 i

= 33 (- 1) - 333 i

Apply minus-plus rules

+ (- a) = - a

= - 1 \cdot 33 - 333 i

Simplify

- 1 \cdot 33 - 333 i : - 33 - 3^{\frac{5}{6}} i

- 1 \cdot 33 - 333 i

1 \cdot 33 = 33

1 \cdot 33

Multiply:

1 \cdot 33 = 33

= 33

333 i = 3^{\frac{5}{6}} i

333 i

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

333 = 3^{\frac{1}{3}} \cdot 3^{\frac{1}{2}} = 3^{\frac{1}{3} + \frac{1}{2}}

= 3^{\frac{1}{3} + \frac{1}{2}} i

3^{\frac{1}{3} + \frac{1}{2}} = 3^{\frac{5}{6}}

3^{\frac{1}{3} + \frac{1}{2}}

Join

\frac{1}{3} + \frac{1}{2} : \frac{5}{6}

\frac{1}{3} + \frac{1}{2}

Least Common Multiplier of

3, 2 : 6

3, 2

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

3

2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{1}{3} :

multiply the denominator and numerator by

2

\frac{1}{3} = \frac{1 \cdot 2}{3 \cdot 2} = \frac{2}{6}

For

\frac{1}{2} :

multiply the denominator and numerator by

3

\frac{1}{2} = \frac{1 \cdot 3}{2 \cdot 3} = \frac{3}{6}

= \frac{2}{6} + \frac{3}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{2 + 3}{6}

Add the numbers:

2 + 3 = 5

= \frac{5}{6}

= 3^{\frac{5}{6}}

= 3^{\frac{5}{6}} i

= - 33 - 3^{\frac{5}{6}} i

= - 33 - 3^{\frac{5}{6}} i

= \frac{- 3 3 - 3 ^{\frac{5}{6}} i}{3 5}

= \frac{\frac{- 3 3 - 3 ^{\frac{5}{6}} i}{3 5}}{2}

Apply the fraction rule:

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

= \frac{- 3 3 - 3 ^{\frac{5}{6}} i}{3 5 \cdot 2}

Rationalize

\frac{- 3 3 - 3 ^{\frac{5}{6}} i}{2 3 5} : \frac{5 ^{\frac{2}{3}} ( - 3 3 - 3 ^{\frac{5}{6}} i )}{10}

\frac{- 3 3 - 3 ^{\frac{5}{6}} i}{2 3 5}

Multiply by the conjugate

\frac{5 ^{\frac{2}{3}}}{5 ^{\frac{2}{3}}}

= \frac{( - 3 3 - 3 ^{\frac{5}{6}} i ) \cdot 5 ^{\frac{2}{3}}}{3 5 \cdot 2 \cdot 5 ^{\frac{2}{3}}}

35 \cdot 2 \cdot 5^{\frac{2}{3}} = 10

35 \cdot 2 \cdot 5^{\frac{2}{3}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

5^{\frac{2}{3}} 35 = 5^{\frac{2}{3}} \cdot 5^{\frac{1}{3}} = 5^{\frac{2}{3} + \frac{1}{3}}

= 5^{\frac{2}{3} + \frac{1}{3}} \cdot 2

5^{\frac{2}{3} + \frac{1}{3}} = 5

5^{\frac{2}{3} + \frac{1}{3}}

Combine the fractions

\frac{2}{3} + \frac{1}{3} : 1

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 5^{1}

Apply rule

a^{1} = a

= 5

= 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 10

= \frac{5 ^{\frac{2}{3}} ( - 3 3 - 3 ^{\frac{5}{6}} i )}{10}

= \frac{5 ^{\frac{2}{3}} ( - 3 3 - 3 ^{\frac{5}{6}} i )}{10}

Rewrite

\frac{5 ^{\frac{2}{3}} ( - 3 3 - 3 ^{\frac{5}{6}} i )}{10}

in standard complex form:

- \frac{3 3 \cdot 5 ^{\frac{2}{3}}}{10} - \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10} i

\frac{5 ^{\frac{2}{3}} ( - 3 3 - 3 ^{\frac{5}{6}} i )}{10}

Factor

10 : 2 \cdot 5

Factor

10 = 2 \cdot 5

= \frac{5 ^{\frac{2}{3}} ( - 3 3 - 3 ^{\frac{5}{6}} i )}{2 \cdot 5}

Cancel

\frac{5 ^{\frac{2}{3}} ( - 3 3 - 3 ^{\frac{5}{6}} i )}{2 \cdot 5} : \frac{- 3 3 - 3 ^{\frac{5}{6}} i}{2 \cdot 5 ^{\frac{1}{3}}}

\frac{5 ^{\frac{2}{3}} ( - 3 3 - 3 ^{\frac{5}{6}} i )}{2 \cdot 5}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{5 ^{\frac{2}{3}}}{5} = \frac{1}{5 ^{1 - \frac{2}{3}}}

= \frac{- 3 3 - 3 ^{\frac{5}{6}} i}{2 \cdot 5 ^{- \frac{2}{3} + 1}}

Subtract the numbers:

1 - \frac{2}{3} = \frac{1}{3}

= \frac{- 3 3 - 3 ^{\frac{5}{6}} i}{2 \cdot 5 ^{\frac{1}{3}}}

= \frac{- 3 3 - 3 ^{\frac{5}{6}} i}{2 \cdot 5 ^{\frac{1}{3}}}

5^{\frac{1}{3}} = 35

Apply radical rule:

a^{\frac{1}{n}} = n a

5^{\frac{1}{3}} = 35

= \frac{- 3 3 - 3 ^{\frac{5}{6}} i}{2 3 5}

Apply the fraction rule:

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

\frac{- 3 3 - 3 ^{\frac{5}{6}} i}{2 3 5} = - \frac{3 3}{2 3 5} - \frac{3 ^{\frac{5}{6}} i}{2 3 5}

= - \frac{3 3}{2 3 5} - \frac{3 ^{\frac{5}{6}} i}{2 3 5}

- \frac{3 ^{\frac{5}{6}}}{2 3 5} = - \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}

- \frac{3 ^{\frac{5}{6}}}{2 3 5}

Multiply by the conjugate

\frac{5 ^{\frac{2}{3}}}{5 ^{\frac{2}{3}}}

= - \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{2 3 5 \cdot 5 ^{\frac{2}{3}}}

235 \cdot 5^{\frac{2}{3}} = 10

235 \cdot 5^{\frac{2}{3}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

5^{\frac{2}{3}} 35 = 5^{\frac{2}{3}} \cdot 5^{\frac{1}{3}} = 5^{\frac{2}{3} + \frac{1}{3}}

= 2 \cdot 5^{\frac{2}{3} + \frac{1}{3}}

5^{\frac{2}{3} + \frac{1}{3}} = 5

5^{\frac{2}{3} + \frac{1}{3}}

Combine the fractions

\frac{2}{3} + \frac{1}{3} : 1

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 5^{1}

Apply rule

a^{1} = a

= 5

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

= - \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}

= - \frac{3 3}{2 3 5} - \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10} i

- \frac{3 3}{2 3 5} = - \frac{3 3 \cdot 5 ^{\frac{2}{3}}}{10}

- \frac{3 3}{2 3 5}

Multiply by the conjugate

\frac{5 ^{\frac{2}{3}}}{5 ^{\frac{2}{3}}}

= - \frac{3 3 \cdot 5 ^{\frac{2}{3}}}{2 3 5 \cdot 5 ^{\frac{2}{3}}}

235 \cdot 5^{\frac{2}{3}} = 10

235 \cdot 5^{\frac{2}{3}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

5^{\frac{2}{3}} 35 = 5^{\frac{2}{3}} \cdot 5^{\frac{1}{3}} = 5^{\frac{2}{3} + \frac{1}{3}}

= 2 \cdot 5^{\frac{2}{3} + \frac{1}{3}}

5^{\frac{2}{3} + \frac{1}{3}} = 5

5^{\frac{2}{3} + \frac{1}{3}}

Combine the fractions

\frac{2}{3} + \frac{1}{3} : 1

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 5^{1}

Apply rule

a^{1} = a

= 5

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

= - \frac{3 3 \cdot 5 ^{\frac{2}{3}}}{10}

= - \frac{3 3 \cdot 5 ^{\frac{2}{3}}}{10} - \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10} i

= - \frac{3 3 \cdot 5 ^{\frac{2}{3}}}{10} - \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10} i

u = 3 \frac{3}{5}, u = - \frac{5 ^{\frac{2}{3}} 3 3}{10} + i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}, u = - \frac{5 ^{\frac{2}{3}} 3 3}{10} - i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}

Substitute back

u = sin (x)

sin (x) = 3 \frac{3}{5}, sin (x) = - \frac{5 ^{\frac{2}{3}} 3 3}{10} + i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}, sin (x) = - \frac{5 ^{\frac{2}{3}} 3 3}{10} - i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}

sin (x) = 3 \frac{3}{5}, sin (x) = - \frac{5 ^{\frac{2}{3}} 3 3}{10} + i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}, sin (x) = - \frac{5 ^{\frac{2}{3}} 3 3}{10} - i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}

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

sin (x) = 3 \frac{3}{5}

Apply trig inverse properties

sin (x) = 3 \frac{3}{5}

General solutions for

sin (x) = 3 \frac{3}{5}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

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

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

sin (x) = - \frac{5 ^{\frac{2}{3}} 3 3}{10} + i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10} :

No Solution

sin (x) = - \frac{5 ^{\frac{2}{3}} 3 3}{10} + i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}

No Solution

sin (x) = - \frac{5 ^{\frac{2}{3}} 3 3}{10} - i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10} :

No Solution

sin (x) = - \frac{5 ^{\frac{2}{3}} 3 3}{10} - i \frac{3 ^{\frac{5}{6}} \cdot 5 ^{\frac{2}{3}}}{10}

No Solution

Combine all the solutions

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

Show solutions in decimal form

x = 1.00364 \dots + 2 πn, x = π - 1.00364 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 3-4sin^3(x)=sin^3(x) ?
The general solution for 3-4sin^3(x)=sin^3(x) is x=1.00364…+2pin,x=pi-1.00364…+2pin