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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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}

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

Simplify

Multiply fractions:

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

Multiply

Multiply fractions:

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

Expand

Apply the distributive law:

a (b + c) = ab + a c

Apply minus-plus rules

+ (- a) = - a

Simplify

Multiply:

Apply exponent rule:

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

= 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

Apply the fraction rule:

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

Rationalize

Multiply by the conjugate

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

Apply exponent rule:

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

= 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

Rewrite

in standard complex form:

Factor

10 : 2 \cdot 5

Factor

10 = 2 \cdot 5

Cancel

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}}}

Subtract the numbers:

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

Apply radical rule:

Apply the fraction rule:

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

Multiply by the conjugate

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

Apply exponent rule:

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

= 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}

Multiply by the conjugate

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

Apply exponent rule:

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

= 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

Simplify

Multiply fractions:

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

Multiply

Multiply fractions:

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

Expand

Apply the distributive law:

a (b - c) = ab - a c

Apply minus-plus rules

+ (- a) = - a

Simplify

Multiply:

Apply exponent rule:

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

= 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

Apply the fraction rule:

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

Rationalize

Multiply by the conjugate

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

Apply exponent rule:

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

= 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

Rewrite

in standard complex form:

Factor

10 : 2 \cdot 5

Factor

10 = 2 \cdot 5

Cancel

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}}}

Subtract the numbers:

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

Apply radical rule:

Apply the fraction rule:

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

Multiply by the conjugate

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

Apply exponent rule:

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

= 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}

Multiply by the conjugate

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

Apply exponent rule:

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

= 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

Substitute back

u = sin (x)

Apply trig inverse properties

General solutions for

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

No Solution

No Solution

No Solution

No Solution

Combine all the solutions

Show solutions in decimal form

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

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sin(2x)=5cos(x)

sin (2 x) = 5 cos (x)

sin(a)=0.4848

sin (a) = 0.4848

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sin^3(x)+cos^3(x)=(1-1)/(2sin^2(x))

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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