What is the general solution for cos^3(x)=66 ?

The general solution for cos^3(x)=66 is No Solution for x\in\mathbb{R}

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Popular Trigonometry >

cos^3(x)=66

Solution

cos^{3} (x) = 66

Solution

No Solution for x \in R

Solution steps

cos^{3} (x) = 66

Solve by substitution

cos^{3} (x) = 66

Let:

cos (x) = u

u^{3} = 66

u^{3} = 66

For

x^{3} = f (a)

the solutions are

Simplify

Multiply fractions:

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

Factor

66 = 2 \cdot 3 \cdot 11

Apply radical rule:

Cancel

Apply radical rule:

Apply exponent rule:

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

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

Subtract the numbers:

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

Simplify

Apply radical rule:

Multiply the numbers:

3 \cdot 11 = 33

Expand

Apply the distributive law:

a (b + c) = ab + a c

Apply minus-plus rules

+ (- a) = - a

Simplify

Multiply:

Factor integer

33 = 3 \cdot 11

Apply radical rule:

Apply exponent rule:

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

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}}

Rationalize

Multiply by the conjugate

Apply exponent rule:

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

= 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

Rewrite

in standard complex form:

Apply radical rule:

Apply exponent rule:

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

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

Subtract the numbers:

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

Apply the fraction rule:

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

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

Combine same powers :

2^{2} = 4

Multiply by the conjugate

Apply radical rule:

Multiply the numbers:

11 \cdot 2 = 22

Apply exponent rule:

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

= 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

Simplify

Multiply fractions:

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

Factor

66 = 2 \cdot 3 \cdot 11

Apply radical rule:

Cancel

Apply radical rule:

Apply exponent rule:

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

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

Subtract the numbers:

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

Simplify

Apply radical rule:

Multiply the numbers:

3 \cdot 11 = 33

Expand

Apply the distributive law:

a (b - c) = ab - a c

Apply minus-plus rules

+ (- a) = - a

Simplify

Multiply:

Factor integer

33 = 3 \cdot 11

Apply radical rule:

Apply exponent rule:

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

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}}

Rationalize

Multiply by the conjugate

Apply exponent rule:

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

= 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

Rewrite

in standard complex form:

Apply radical rule:

Apply exponent rule:

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

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

Subtract the numbers:

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

Apply the fraction rule:

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

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

Combine same powers :

2^{2} = 4

Multiply by the conjugate

Apply radical rule:

Multiply the numbers:

11 \cdot 2 = 22

Apply exponent rule:

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

= 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

Substitute back

u = cos (x)

No Solution

- 1 \leq cos (x) \leq 1

No Solution

No Solution

No Solution

No Solution

No Solution

Combine all the solutions

No Solution for x \in R

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos^3(x)=66 ?
The general solution for cos^3(x)=66 is No Solution for x\in\mathbb{R}