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

The general solution for cos^3(x)=-1/2 is x=2.48766…+2pin,x=-2.48766…+2pin

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cos^3(x)=-1/2

Solution

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

Solution

x = 2.48766 \dots + 2 πn, x = - 2.48766 \dots + 2 πn

Degrees

x = 142.53268 \dots^{\circ} + 36 0^{\circ} n, x = - 142.53268 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

cos (x) = u

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

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

For

x^{3} = f (a)

the solutions are

Apply radical rule:

n

is odd

Simplify

Apply radical rule:

n

is odd

Multiply fractions:

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

Apply rule

Multiply

Multiply fractions:

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

1 \cdot (- 1 + 3 i) = - 1 + 3 i

1 \cdot (- 1 + 3 i)

Multiply:

1 \cdot (- 1 + 3 i) = (- 1 + 3 i)

= (- 1 + 3 i)

Remove parentheses:

(- a) = - a

= - 1 + 3 i

Apply the fraction rule:

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

Rationalize

Multiply by the conjugate

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

Apply exponent rule:

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

= 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}} ( - 1 + 3 i )}{4}

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

Rewrite

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

in standard complex form:

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

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

Cancel

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

Factor

4 : 2^{2}

Factor

4 = 2^{2}

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

Cancel

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

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

Subtract the numbers:

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

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

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

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

Apply the fraction rule:

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

Remove parentheses:

(a) = a, - (- a) = a

Multiply by the conjugate

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

Apply exponent rule:

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

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

Multiply by the conjugate

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

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

Apply exponent rule:

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

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

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

Simplify

Apply radical rule:

n

is odd

Multiply fractions:

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

Apply rule

Multiply

Multiply fractions:

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

1 \cdot (- 1 - 3 i) = - 1 - 3 i

1 \cdot (- 1 - 3 i)

Multiply:

1 \cdot (- 1 - 3 i) = (- 1 - 3 i)

= (- 1 - 3 i)

Remove parentheses:

(- a) = - a

= - 1 - 3 i

Apply the fraction rule:

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

Rationalize

Multiply by the conjugate

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

Apply exponent rule:

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

= 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}} ( - 1 - 3 i )}{4}

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

Rewrite

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

in standard complex form:

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

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

Cancel

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

Factor

4 : 2^{2}

Factor

4 = 2^{2}

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

Cancel

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

\frac{2 ^{\frac{2}{3}} ( - 1 - 3 i )}{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 - 3 i}{2 ^{2 - \frac{2}{3}}}

Subtract the numbers:

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

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

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

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

Apply the fraction rule:

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

Apply rule

- (- a) = a

Multiply by the conjugate

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

Apply exponent rule:

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

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

Multiply by the conjugate

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

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

Apply exponent rule:

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

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

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

Substitute back

u = cos (x)

Apply trig inverse properties

General solutions for

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

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

No Solution

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

No Solution

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

No Solution

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

No Solution

Combine all the solutions

Show solutions in decimal form

x = 2.48766 \dots + 2 πn, x = - 2.48766 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos^3(x)=-1/2 ?
The general solution for cos^3(x)=-1/2 is x=2.48766…+2pin,x=-2.48766…+2pin