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Popular Trigonometry >

tan^3(3x)-2sin^3(3x)=0

Solution

tan^{3} (3 x) - 2 sin^{3} (3 x) = 0

Solution

x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{0.65392 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{0.65392 \dots}{3} + \frac{2 πn}{3}

Degrees

x = 0^{\circ} + 12 0^{\circ} n, x = 6 0^{\circ} + 12 0^{\circ} n, x = 12.48910 \dots^{\circ} + 12 0^{\circ} n, x = 107.51089 \dots^{\circ} + 12 0^{\circ} n

Solution steps

tan^{3} (3 x) - 2 sin^{3} (3 x) = 0

Factor

tan^{3} (3 x) - 2 sin^{3} (3 x) : (tan (3 x) - 32 sin (3 x)) (tan^{2} (3 x) + 32 tan (3 x) sin (3 x) + 2^{\frac{2}{3}} sin^{2} (3 x))

tan^{3} (3 x) - 2 sin^{3} (3 x)

Rewrite

tan^{3} (3 x) - 2 sin^{3} (3 x)

tan^{3} (3 x) - (32 sin (3 x))^{3}

tan^{3} (3 x) - 2 sin^{3} (3 x)

Apply radical rule:

a = (a)^{2}

2 = (32)^{3}

= tan^{3} (3 x) - (32)^{3} sin^{3} (3 x)

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(32)^{3} sin^{3} (3 x) = (32 sin (3 x))^{3}

= tan^{3} (3 x) - (32 sin (3 x))^{3}

= tan^{3} (3 x) - (32 sin (3 x))^{3}

Apply Difference of Cubes Formula:

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

tan^{3} (3 x) - (32 sin (3 x))^{3} = (tan (3 x) - 32 sin (3 x)) (tan^{2} (3 x) + 32 tan (3 x) sin (3 x) + (32)^{2} sin^{2} (3 x))

= (tan (3 x) - 32 sin (3 x)) (tan^{2} (3 x) + 32 tan (3 x) sin (3 x) + (32)^{2} sin^{2} (3 x))

Refine

= (tan (3 x) - 32 sin (3 x)) (tan^{2} (3 x) + 32 tan (3 x) sin (3 x) + 2^{\frac{2}{3}} sin^{2} (3 x))

(tan (3 x) - 32 sin (3 x)) (tan^{2} (3 x) + 32 tan (3 x) sin (3 x) + 2^{\frac{2}{3}} sin^{2} (3 x)) = 0

Solving each part separately

tan (3 x) - 32 sin (3 x) = 0 or tan^{2} (3 x) + 32 tan (3 x) sin (3 x) + 2^{\frac{2}{3}} sin^{2} (3 x) = 0

tan (3 x) - 32 sin (3 x) = 0 : x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

tan (3 x) - 32 sin (3 x) = 0

Express with sin, cos

tan (3 x) - sin (3 x) 32

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{sin ( 3 x )}{cos ( 3 x )} - sin (3 x) 32

Simplify

\frac{sin ( 3 x )}{cos ( 3 x )} - sin (3 x) 32 : \frac{sin ( 3 x ) - 3 2 sin ( 3 x ) cos ( 3 x )}{cos ( 3 x )}

\frac{sin ( 3 x )}{cos ( 3 x )} - sin (3 x) 32

Convert element to fraction:

32 sin (3 x) = \frac{sin ( 3 x ) 3 2 cos ( 3 x )}{cos ( 3 x )}

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

sin (3 x) - cos (3 x) sin (3 x) 32 = 0

Factor

sin (3 x) - cos (3 x) sin (3 x) 32 : - sin (3 x) (32 cos (3 x) - 1)

sin (3 x) - cos (3 x) sin (3 x) 32

Factor out common term

- sin (3 x)

= - sin (3 x) (- 1 + 32 cos (3 x))

- sin (3 x) (32 cos (3 x) - 1) = 0

Solving each part separately

sin (3 x) = 0 or 32 cos (3 x) - 1 = 0

sin (3 x) = 0 : x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

sin (3 x) = 0

General solutions for

sin (3 x) = 0

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}

3 x = 0 + 2 πn, 3 x = π + 2 πn

3 x = 0 + 2 πn, 3 x = π + 2 πn

Solve

3 x = 0 + 2 πn : x = \frac{2 πn}{3}

3 x = 0 + 2 πn

0 + 2 πn = 2 πn

3 x = 2 πn

Divide both sides by

3

3 x = 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3}

Simplify

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

Solve

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

3 x = π + 2 πn

Divide both sides by

3

3 x = π + 2 πn

Divide both sides by

3

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

Simplify

x = \frac{π}{3} + \frac{2 πn}{3}

x = \frac{π}{3} + \frac{2 πn}{3}

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

32 cos (3 x) - 1 = 0 : x = \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

32 cos (3 x) - 1 = 0

Move

1

to the right side

32 cos (3 x) - 1 = 0

Add

1

to both sides

32 cos (3 x) - 1 + 1 = 0 + 1

Simplify

32 cos (3 x) = 1

32 cos (3 x) = 1

Divide both sides by

32

32 cos (3 x) = 1

Divide both sides by

32

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

Simplify

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

Simplify

\frac{3 2 cos ( 3 x )}{3 2} : cos (3 x)

\frac{3 2 cos ( 3 x )}{3 2}

Cancel the common factor:

32

= cos (3 x)

Simplify

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

\frac{1}{3 2}

Multiply by the conjugate

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

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

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

32 \cdot 2^{\frac{2}{3}} = 2

32 \cdot 2^{\frac{2}{3}}

Apply exponent rule:

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

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

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

Join

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

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

Since the denominators are equal, combine the fractions:

\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

= 2^{1}

Apply rule

a^{1} = a

= 2

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

cos (3 x) = \frac{2 ^{\frac{2}{3}}}{2}

cos (3 x) = \frac{2 ^{\frac{2}{3}}}{2}

cos (3 x) = \frac{2 ^{\frac{2}{3}}}{2}

Apply trig inverse properties

cos (3 x) = \frac{2 ^{\frac{2}{3}}}{2}

General solutions for

cos (3 x) = \frac{2 ^{\frac{2}{3}}}{2}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

3 x = arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn, 3 x = 2 π - arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

3 x = arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn, 3 x = 2 π - arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

Solve

3 x = arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn : x = \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

3 x = arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

Simplify

arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn : arccos (\frac{1}{2 ^{\frac{1}{3}}}) + 2 πn

arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

= arccos (\frac{1}{2 ^{\frac{1}{3}}}) + 2 πn

3 x = arccos (\frac{1}{2 ^{\frac{1}{3}}}) + 2 πn

Divide both sides by

3

3 x = arccos (\frac{1}{2 ^{\frac{1}{3}}}) + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{arccos ( \frac{1}{2 ^{\frac{1}{3}}} )}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} = \frac{arccos ( \frac{1}{2 ^{\frac{1}{3}}} )}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{arccos ( \frac{1}{2 ^{\frac{1}{3}}} )}{3} + \frac{2 πn}{3} : \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

\frac{arccos ( \frac{1}{2 ^{\frac{1}{3}}} )}{3} + \frac{2 πn}{3}

arccos (\frac{1}{2 ^{\frac{1}{3}}}) = arccos (\frac{2 ^{\frac{2}{3}}}{2})

arccos (\frac{1}{2 ^{\frac{1}{3}}})

= arccos (\frac{2 ^{\frac{2}{3}}}{2})

= \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

x = \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

x = \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

x = \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

Solve

3 x = 2 π - arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn : x = \frac{2 π}{3} - \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

3 x = 2 π - arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

Simplify

2 π - arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn : 2 π - arccos (\frac{1}{2 ^{\frac{1}{3}}}) + 2 πn

2 π - arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

= 2 π - arccos (\frac{1}{2 ^{\frac{1}{3}}}) + 2 πn

3 x = 2 π - arccos (\frac{1}{2 ^{\frac{1}{3}}}) + 2 πn

Divide both sides by

3

3 x = 2 π - arccos (\frac{1}{2 ^{\frac{1}{3}}}) + 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{2 π}{3} - \frac{arccos ( \frac{1}{2 ^{\frac{1}{3}}} )}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} = \frac{2 π}{3} - \frac{arccos ( \frac{1}{2 ^{\frac{1}{3}}} )}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 π}{3} - \frac{arccos ( \frac{1}{2 ^{\frac{1}{3}}} )}{3} + \frac{2 πn}{3} : \frac{2 π}{3} - \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

\frac{2 π}{3} - \frac{arccos ( \frac{1}{2 ^{\frac{1}{3}}} )}{3} + \frac{2 πn}{3}

arccos (\frac{1}{2 ^{\frac{1}{3}}}) = arccos (\frac{2 ^{\frac{2}{3}}}{2})

arccos (\frac{1}{2 ^{\frac{1}{3}}})

= arccos (\frac{2 ^{\frac{2}{3}}}{2})

= \frac{2 π}{3} - \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

x = \frac{2 π}{3} - \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

x = \frac{2 π}{3} - \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

x = \frac{2 π}{3} - \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

x = \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

Combine all the solutions

x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

tan^{2} (3 x) + 32 tan (3 x) sin (3 x) + 2^{\frac{2}{3}} sin^{2} (3 x) = 0 : x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

tan^{2} (3 x) + 32 tan (3 x) sin (3 x) + 2^{\frac{2}{3}} sin^{2} (3 x) = 0

Express with sin, cos

tan^{2} (3 x) + 2^{\frac{2}{3}} sin^{2} (3 x) + sin (3 x) 32 tan (3 x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= (\frac{sin ( 3 x )}{cos ( 3 x )})^{2} + 2^{\frac{2}{3}} sin^{2} (3 x) + sin (3 x) 32 \frac{sin ( 3 x )}{cos ( 3 x )}

Simplify

(\frac{sin ( 3 x )}{cos ( 3 x )})^{2} + 2^{\frac{2}{3}} sin^{2} (3 x) + sin (3 x) 32 \frac{sin ( 3 x )}{cos ( 3 x )} : \frac{sin ^{2} ( 3 x ) + 2 ^{\frac{2}{3}} sin ^{2} ( 3 x ) cos ^{2} ( 3 x ) + 3 2 sin ^{2} ( 3 x ) cos ( 3 x )}{cos ^{2} ( 3 x )}

(\frac{sin ( 3 x )}{cos ( 3 x )})^{2} + 2^{\frac{2}{3}} sin^{2} (3 x) + sin (3 x) 32 \frac{sin ( 3 x )}{cos ( 3 x )}

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

(\frac{sin ( 3 x )}{cos ( 3 x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

sin (3 x) 32 \frac{sin ( 3 x )}{cos ( 3 x )} = \frac{3 2 sin ^{2} ( 3 x )}{cos ( 3 x )}

sin (3 x) 32 \frac{sin ( 3 x )}{cos ( 3 x )}

Multiply fractions:

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

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

sin (3 x) sin (3 x) 32 = 32 sin^{2} (3 x)

sin (3 x) sin (3 x) 32

Apply exponent rule:

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

sin (3 x) sin (3 x) = sin^{1 + 1} (3 x)

= sin^{1 + 1} (3 x) 32

Add the numbers:

1 + 1 = 2

= sin^{2} (3 x) 32

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

= \frac{sin ^{2} ( 3 x )}{cos ^{2} ( 3 x )} + 2^{\frac{2}{3}} sin^{2} (3 x) + \frac{3 2 sin ^{2} ( 3 x )}{cos ( 3 x )}

Convert element to fraction:

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

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

Least Common Multiplier of

cos^{2} (3 x), 1, cos (3 x) : cos^{2} (3 x)

cos^{2} (3 x), 1, cos (3 x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear in at least one of the factored expressions

= cos^{2} (3 x)

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

cos^{2} (3 x)

For

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

multiply the denominator and numerator by

cos^{2} (3 x)

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

For

\frac{sin ^{2} ( 3 x ) 3 2}{cos ( 3 x )} :

multiply the denominator and numerator by

cos (3 x)

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

= \frac{sin ^{2} ( 3 x )}{cos ^{2} ( 3 x )} + \frac{2 ^{\frac{2}{3}} sin ^{2} ( 3 x ) cos ^{2} ( 3 x )}{cos ^{2} ( 3 x )} + \frac{sin ^{2} ( 3 x ) 3 2 cos ( 3 x )}{cos ^{2} ( 3 x )}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ^{2} ( 3 x ) + 2 ^{\frac{2}{3}} sin ^{2} ( 3 x ) cos ^{2} ( 3 x ) + sin ^{2} ( 3 x ) 3 2 cos ( 3 x )}{cos ^{2} ( 3 x )}

= \frac{sin ^{2} ( 3 x ) + 2 ^{\frac{2}{3}} sin ^{2} ( 3 x ) cos ^{2} ( 3 x ) + 3 2 sin ^{2} ( 3 x ) cos ( 3 x )}{cos ^{2} ( 3 x )}

\frac{sin ^{2} ( 3 x ) + 2 ^{\frac{2}{3}} cos ^{2} ( 3 x ) sin ^{2} ( 3 x ) + cos ( 3 x ) sin ^{2} ( 3 x ) 3 2}{cos ^{2} ( 3 x )} = 0

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

sin^{2} (3 x) + 2^{\frac{2}{3}} cos^{2} (3 x) sin^{2} (3 x) + cos (3 x) sin^{2} (3 x) 32 = 0

Factor

sin^{2} (3 x) + 2^{\frac{2}{3}} cos^{2} (3 x) sin^{2} (3 x) + cos (3 x) sin^{2} (3 x) 32 : sin^{2} (3 x) (2^{\frac{2}{3}} cos^{2} (3 x) + 32 cos (3 x) + 1)

sin^{2} (3 x) + 2^{\frac{2}{3}} cos^{2} (3 x) sin^{2} (3 x) + cos (3 x) sin^{2} (3 x) 32

Factor out common term

sin^{2} (3 x)

= sin^{2} (3 x) (1 + 2^{\frac{2}{3}} cos^{2} (3 x) + 32 cos (3 x))

sin^{2} (3 x) (2^{\frac{2}{3}} cos^{2} (3 x) + 32 cos (3 x) + 1) = 0

Solving each part separately

sin^{2} (3 x) = 0 or 2^{\frac{2}{3}} cos^{2} (3 x) + 32 cos (3 x) + 1 = 0

sin^{2} (3 x) = 0 : x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

sin^{2} (3 x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

sin (3 x) = 0

General solutions for

sin (3 x) = 0

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}

3 x = 0 + 2 πn, 3 x = π + 2 πn

3 x = 0 + 2 πn, 3 x = π + 2 πn

Solve

3 x = 0 + 2 πn : x = \frac{2 πn}{3}

3 x = 0 + 2 πn

0 + 2 πn = 2 πn

3 x = 2 πn

Divide both sides by

3

3 x = 2 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3}

Simplify

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

Solve

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

3 x = π + 2 πn

Divide both sides by

3

3 x = π + 2 πn

Divide both sides by

3

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

Simplify

x = \frac{π}{3} + \frac{2 πn}{3}

x = \frac{π}{3} + \frac{2 πn}{3}

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

2^{\frac{2}{3}} cos^{2} (3 x) + 32 cos (3 x) + 1 = 0 :

No Solution

2^{\frac{2}{3}} cos^{2} (3 x) + 32 cos (3 x) + 1 = 0

Solve by substitution

2^{\frac{2}{3}} cos^{2} (3 x) + 32 cos (3 x) + 1 = 0

Let:

cos (3 x) = u

2^{\frac{2}{3}} u^{2} + 32 u + 1 = 0

2^{\frac{2}{3}} u^{2} + 32 u + 1 = 0 : u = - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{3 2 3 2 ^{\frac{2}{3}}}{4}, u = - \frac{2 ^{\frac{2}{3}}}{4} - i \frac{3 2 3 2 ^{\frac{2}{3}}}{4}

2^{\frac{2}{3}} u^{2} + 32 u + 1 = 0

Solve with the quadratic formula

2^{\frac{2}{3}} u^{2} + 32 u + 1 = 0

Quadratic Equation Formula:

For

a = 2^{\frac{2}{3}}, b = 32, c = 1

u_{1, 2} = \frac{- 3 2 \pm ( 3 2 ) ^{2} - 4 \cdot 2 ^{\frac{2}{3}} \cdot 1}{2 \cdot 2 ^{\frac{2}{3}}}

u_{1, 2} = \frac{- 3 2 \pm ( 3 2 ) ^{2} - 4 \cdot 2 ^{\frac{2}{3}} \cdot 1}{2 \cdot 2 ^{\frac{2}{3}}}

Simplify

(32)^{2} - 4 \cdot 2^{\frac{2}{3}} \cdot 1 : 3 i 2^{\frac{2}{3}}

(32)^{2} - 4 \cdot 2^{\frac{2}{3}} \cdot 1

(32)^{2} = 2^{\frac{2}{3}}

(32)^{2}

Apply radical rule:

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

= (2^{\frac{1}{3}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{3} \cdot 2

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{3}

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

4 \cdot 2^{\frac{2}{3}} \cdot 1 = 4 \cdot 2^{\frac{2}{3}}

4 \cdot 2^{\frac{2}{3}} \cdot 1

Multiply the numbers:

4 \cdot 1 = 4

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

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

Add similar elements:

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

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

Apply radical rule:

- a = - 1 a

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

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

Apply imaginary number rule:

- 1 = i

= i 3 \cdot 2^{\frac{2}{3}}

Apply radical rule:

n ab = n a n b,

assuming

a \geq 0, b \geq 0

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

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

u_{1, 2} = \frac{- 3 2 \pm 3 i 2 ^{\frac{2}{3}}}{2 \cdot 2 ^{\frac{2}{3}}}

Separate the solutions

u_{1} = \frac{- 3 2 + 3 i 2 ^{\frac{2}{3}}}{2 \cdot 2 ^{\frac{2}{3}}}, u_{2} = \frac{- 3 2 - 3 i 2 ^{\frac{2}{3}}}{2 \cdot 2 ^{\frac{2}{3}}}

u = \frac{- 3 2 + 3 i 2 ^{\frac{2}{3}}}{2 \cdot 2 ^{\frac{2}{3}}} : - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{3 2 3 2 ^{\frac{2}{3}}}{4}

\frac{- 3 2 + 3 i 2 ^{\frac{2}{3}}}{2 \cdot 2 ^{\frac{2}{3}}}

Multiply by the conjugate

\frac{3 2}{3 2}

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

Simplify

(- 32 + 3 i 2^{\frac{2}{3}}) 32 : - 2^{\frac{2}{3}} + 323 i 2^{\frac{2}{3}}

(- 32 + 3 i 2^{\frac{2}{3}}) 32

= 32 (- 32 + 3 i 2^{\frac{2}{3}})

Apply the distributive law:

a (b + c) = ab + a c

a = 32, b = - 32, c = 3 i 2^{\frac{2}{3}}

= 32 (- 32) + 323 i 2^{\frac{2}{3}}

Apply minus-plus rules

+ (- a) = - a

= - 3232 + 323 i 2^{\frac{2}{3}}

3232 = 2^{\frac{2}{3}}

3232

Apply exponent rule:

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

3232 = 2^{\frac{1}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{1}{3} + \frac{1}{3}}

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

Add similar elements:

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

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

Multiply

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

2 \cdot \frac{1}{3}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{3}

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

= - 2^{\frac{2}{3}} + 323 i 2^{\frac{2}{3}}

2 \cdot 2^{\frac{2}{3}} 32 = 4

2 \cdot 2^{\frac{2}{3}} 32

Apply exponent rule:

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

2 \cdot 2^{\frac{2}{3}} 32 = 2 \cdot 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{1 + \frac{2}{3} + \frac{1}{3}}

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

Join

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

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

Convert element to fraction:

1 = \frac{1}{1}

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

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

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

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

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

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

Since the denominators are equal, combine the fractions:

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

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

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= \frac{- 2 ^{\frac{2}{3}} + 3 2 3 i 2 ^{\frac{2}{3}}}{4}

Rewrite

\frac{- 2 ^{\frac{2}{3}} + 3 2 3 i 2 ^{\frac{2}{3}}}{4}

in standard complex form:

- \frac{2 ^{\frac{2}{3}}}{4} + \frac{3 3 2 2 ^{\frac{2}{3}}}{4} i

\frac{- 2 ^{\frac{2}{3}} + 3 2 3 i 2 ^{\frac{2}{3}}}{4}

Apply the fraction rule:

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

\frac{- 2 ^{\frac{2}{3}} + 3 2 3 i 2 ^{\frac{2}{3}}}{4} = - \frac{2 ^{\frac{2}{3}}}{4} + \frac{3 2 3 i 2 ^{\frac{2}{3}}}{4}

= - \frac{2 ^{\frac{2}{3}}}{4} + \frac{3 2 3 i 2 ^{\frac{2}{3}}}{4}

\frac{2 ^{\frac{2}{3}}}{4} = \frac{1}{2 3 2}

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

Factor

4 : 2^{2}

Factor

4 = 2^{2}

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

Cancel

\frac{2 ^{\frac{2}{3}}}{2 ^{2}} : \frac{1}{2 ^{\frac{4}{3}}}

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

= \frac{1}{2 ^{\frac{4}{3}}}

= \frac{1}{2 ^{\frac{4}{3}}}

2^{\frac{4}{3}} = 232

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

2^{\frac{4}{3}} = 2^{1 + \frac{1}{3}}

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 232

= \frac{1}{2 3 2}

\frac{3 2 3 i 2 ^{\frac{2}{3}}}{4} = \frac{3 i 2 ^{\frac{2}{3}}}{2 \cdot 2 ^{\frac{2}{3}}}

\frac{3 2 3 i 2 ^{\frac{2}{3}}}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

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

Cancel

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

\frac{3 2 3 i 2 ^{\frac{2}{3}}}{2 ^{2}}

Apply radical rule:

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

32 = 2^{\frac{1}{3}}

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

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

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

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

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

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

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

= \frac{3 i 2 ^{\frac{2}{3}}}{2 \cdot 2 ^{\frac{2}{3}}}

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

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

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

Multiply by the conjugate

\frac{3 2}{3 2}

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

2 \cdot 2^{\frac{2}{3}} 32 = 4

2 \cdot 2^{\frac{2}{3}} 32

Apply exponent rule:

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

2 \cdot 2^{\frac{2}{3}} 32 = 2 \cdot 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{1 + \frac{2}{3} + \frac{1}{3}}

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

Join

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

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

Convert element to fraction:

1 = \frac{1}{1}

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

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

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

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

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

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

Since the denominators are equal, combine the fractions:

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

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

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

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

= - \frac{1}{2 3 2} + \frac{3 3 2 2 ^{\frac{2}{3}}}{4} i

- \frac{1}{2 3 2} = - \frac{2 ^{\frac{2}{3}}}{4}

- \frac{1}{2 3 2}

Multiply by the conjugate

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

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

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

232 \cdot 2^{\frac{2}{3}} = 4

232 \cdot 2^{\frac{2}{3}}

Apply exponent rule:

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

2 \cdot 2^{\frac{2}{3}} 32 = 2 \cdot 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{1 + \frac{2}{3} + \frac{1}{3}}

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

Join

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

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

Convert element to fraction:

1 = \frac{1}{1}

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

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

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

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

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

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

Since the denominators are equal, combine the fractions:

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

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

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

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

= - \frac{2 ^{\frac{2}{3}}}{4} + \frac{3 3 2 2 ^{\frac{2}{3}}}{4} i

= - \frac{2 ^{\frac{2}{3}}}{4} + \frac{3 3 2 2 ^{\frac{2}{3}}}{4} i

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

\frac{- 3 2 - 3 i 2 ^{\frac{2}{3}}}{2 \cdot 2 ^{\frac{2}{3}}}

Multiply by the conjugate

\frac{3 2}{3 2}

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

Simplify

(- 32 - 3 i 2^{\frac{2}{3}}) 32 : - 2^{\frac{2}{3}} - 323 i 2^{\frac{2}{3}}

(- 32 - 3 i 2^{\frac{2}{3}}) 32

= 32 (- 32 - 3 i 2^{\frac{2}{3}})

Apply the distributive law:

a (b - c) = ab - a c

a = 32, b = - 32, c = 3 i 2^{\frac{2}{3}}

= 32 (- 32) - 323 i 2^{\frac{2}{3}}

Apply minus-plus rules

+ (- a) = - a

= - 3232 - 323 i 2^{\frac{2}{3}}

3232 = 2^{\frac{2}{3}}

3232

Apply exponent rule:

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

3232 = 2^{\frac{1}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{1}{3} + \frac{1}{3}}

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

Add similar elements:

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

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

Multiply

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

2 \cdot \frac{1}{3}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{3}

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

= - 2^{\frac{2}{3}} - 323 i 2^{\frac{2}{3}}

2 \cdot 2^{\frac{2}{3}} 32 = 4

2 \cdot 2^{\frac{2}{3}} 32

Apply exponent rule:

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

2 \cdot 2^{\frac{2}{3}} 32 = 2 \cdot 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{1 + \frac{2}{3} + \frac{1}{3}}

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

Join

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

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

Convert element to fraction:

1 = \frac{1}{1}

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

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

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

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

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

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

Since the denominators are equal, combine the fractions:

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

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

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

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

Rewrite

\frac{- 2 ^{\frac{2}{3}} - 3 2 3 i 2 ^{\frac{2}{3}}}{4}

in standard complex form:

- \frac{2 ^{\frac{2}{3}}}{4} - \frac{3 3 2 2 ^{\frac{2}{3}}}{4} i

\frac{- 2 ^{\frac{2}{3}} - 3 2 3 i 2 ^{\frac{2}{3}}}{4}

Apply the fraction rule:

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

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

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

\frac{2 ^{\frac{2}{3}}}{4} = \frac{1}{2 3 2}

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

Factor

4 : 2^{2}

Factor

4 = 2^{2}

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

Cancel

\frac{2 ^{\frac{2}{3}}}{2 ^{2}} : \frac{1}{2 ^{\frac{4}{3}}}

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

= \frac{1}{2 ^{\frac{4}{3}}}

= \frac{1}{2 ^{\frac{4}{3}}}

2^{\frac{4}{3}} = 232

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

2^{\frac{4}{3}} = 2^{1 + \frac{1}{3}}

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 232

= \frac{1}{2 3 2}

\frac{3 2 3 i 2 ^{\frac{2}{3}}}{4} = \frac{3 i 2 ^{\frac{2}{3}}}{2 \cdot 2 ^{\frac{2}{3}}}

\frac{3 2 3 i 2 ^{\frac{2}{3}}}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

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

Cancel

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

\frac{3 2 3 i 2 ^{\frac{2}{3}}}{2 ^{2}}

Apply radical rule:

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

32 = 2^{\frac{1}{3}}

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

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

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

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

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

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

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

= \frac{3 i 2 ^{\frac{2}{3}}}{2 \cdot 2 ^{\frac{2}{3}}}

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

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

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

Multiply by the conjugate

\frac{3 2}{3 2}

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

2 \cdot 2^{\frac{2}{3}} 32 = 4

2 \cdot 2^{\frac{2}{3}} 32

Apply exponent rule:

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

2 \cdot 2^{\frac{2}{3}} 32 = 2 \cdot 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{1 + \frac{2}{3} + \frac{1}{3}}

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

Join

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

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

Convert element to fraction:

1 = \frac{1}{1}

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

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

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

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

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

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

Since the denominators are equal, combine the fractions:

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

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

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

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

= - \frac{1}{2 3 2} - \frac{3 3 2 2 ^{\frac{2}{3}}}{4} i

- \frac{1}{2 3 2} = - \frac{2 ^{\frac{2}{3}}}{4}

- \frac{1}{2 3 2}

Multiply by the conjugate

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

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

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

232 \cdot 2^{\frac{2}{3}} = 4

232 \cdot 2^{\frac{2}{3}}

Apply exponent rule:

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

2 \cdot 2^{\frac{2}{3}} 32 = 2 \cdot 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{1 + \frac{2}{3} + \frac{1}{3}}

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

Join

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

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

Convert element to fraction:

1 = \frac{1}{1}

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

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

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

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

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

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

Since the denominators are equal, combine the fractions:

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

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

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

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

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

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

The solutions to the quadratic equation are:

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

Substitute back

u = cos (3 x)

cos (3 x) = - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{3 2 3 2 ^{\frac{2}{3}}}{4}, cos (3 x) = - \frac{2 ^{\frac{2}{3}}}{4} - i \frac{3 2 3 2 ^{\frac{2}{3}}}{4}

cos (3 x) = - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{3 2 3 2 ^{\frac{2}{3}}}{4}, cos (3 x) = - \frac{2 ^{\frac{2}{3}}}{4} - i \frac{3 2 3 2 ^{\frac{2}{3}}}{4}

cos (3 x) = - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{3 2 3 2 ^{\frac{2}{3}}}{4} :

No Solution

cos (3 x) = - \frac{2 ^{\frac{2}{3}}}{4} + i \frac{3 2 3 2 ^{\frac{2}{3}}}{4}

No Solution

cos (3 x) = - \frac{2 ^{\frac{2}{3}}}{4} - i \frac{3 2 3 2 ^{\frac{2}{3}}}{4} :

No Solution

cos (3 x) = - \frac{2 ^{\frac{2}{3}}}{4} - i \frac{3 2 3 2 ^{\frac{2}{3}}}{4}

No Solution

Combine all the solutions

No Solution

Combine all the solutions

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

Combine all the solutions

x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{2 ^{\frac{2}{3}}}{2} )}{3} + \frac{2 πn}{3}

Show solutions in decimal form

x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{0.65392 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{0.65392 \dots}{3} + \frac{2 πn}{3}

Graph

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