What is the general solution for sin^2(a)=-((5sqrt(11)))/((18)) ?

The general solution for sin^2(a)=-((5sqrt(11)))/((18)) is No Solution for a\in\mathbb{R}

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Popular Trigonometry >

sin^2(a)=-((5sqrt(11)))/((18))

Solution

sin^{2} (a) = - \frac{( 5 11 )}{( 18 )}

Solution

No Solution for a \in R

Solution steps

sin^{2} (a) = - \frac{( 5 11 )}{( 18 )}

Solve by substitution

sin^{2} (a) = - \frac{5 11}{18}

Let:

sin (a) = u

u^{2} = - \frac{5 11}{18}

u^{2} = - \frac{5 11}{18} : u = i \frac{4 11 10}{6}, u = - i \frac{4 11 10}{6}

u^{2} = - \frac{5 11}{18}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - \frac{5 11}{18}, u = - - \frac{5 11}{18}

Simplify

- \frac{5 11}{18} : i \frac{4 11 10}{6}

- \frac{5 11}{18}

Apply radical rule:

- a = - 1 a

- \frac{5 11}{18} = - 1 \frac{5 11}{18}

= - 1 \frac{5 11}{18}

Apply imaginary number rule:

- 1 = i

= i \frac{5 11}{18}

Apply radical rule:

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

assuming

a \geq 0, b \geq 0

\frac{5 11}{18} = \frac{5 11}{18}

= i \frac{5 11}{18}

18 = 32

18

Prime factorization of

18 : 2 \cdot 3^{2}

18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 3

= 2 \cdot 3^{2}

= 3^{2} \cdot 2

Apply radical rule:

n ab = n a n b

= 2 3^{2}

Apply radical rule:

n a^{n} = a

3^{2} = 3

= 32

= i \frac{5 11}{3 2}

Apply radical rule:

n ab = n a n b,

assuming

a \geq 0, b \geq 0

511 = 511

= i \frac{5 11}{3 2}

11 : 411

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot \frac{1}{2}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

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

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

= 411

= i \frac{5 4 11}{3 2}

\frac{5 4 11}{3 2} = \frac{10 4 11}{6}

\frac{5 4 11}{3 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{5 4 11 2}{3 2 2}

54112 = 10411

54112

Apply radical rule:

a b = a \cdot b

52 = 5 \cdot 2

= 411 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 10411

322 = 6

322

Apply radical rule:

a a = a

22 = 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

= \frac{10 4 11}{6}

= i \frac{10 4 11}{6}

Rewrite

i \frac{10 4 11}{6}

in standard complex form:

\frac{10 4 11}{6} i

i \frac{10 4 11}{6}

\frac{10 4 11}{6} = \frac{5 4 11}{3 2}

\frac{10 4 11}{6}

Factor

10 : 25

Factor

10 = 2 \cdot 5

= 2 \cdot 5

Apply radical rule:

n ab = n a n b

= 25

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

= \frac{2 5 4 11}{2 \cdot 3}

Cancel

\frac{2 5 4 11}{2 \cdot 3} : \frac{5 4 11}{2 \cdot 3}

\frac{2 5 4 11}{2 \cdot 3}

Apply radical rule:

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

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

= \frac{2 ^{\frac{1}{2}} 5 4 11}{2 \cdot 3}

Apply exponent rule:

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

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

= \frac{5 4 11}{3 \cdot 2 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

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

= \frac{5 4 11}{3 \cdot 2 ^{\frac{1}{2}}}

Apply radical rule:

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

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

= \frac{5 4 11}{3 2}

= \frac{5 4 11}{2 \cdot 3}

= i \frac{5 4 11}{3 2}

Multiply fractions:

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

= \frac{5 4 11 i}{2 \cdot 3}

\frac{5 4 11}{3 2} = \frac{10 4 11}{6}

\frac{5 4 11}{3 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{5 4 11 2}{3 2 2}

54112 = 10411

54112

Apply radical rule:

a b = a \cdot b

52 = 5 \cdot 2

= 411 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 10411

322 = 6

322

Apply radical rule:

a a = a

22 = 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

= \frac{10 4 11}{6}

= \frac{10 4 11}{6} i

= \frac{10 4 11}{6} i

Simplify

- - \frac{5 11}{18} : - i \frac{4 11 10}{6}

- - \frac{5 11}{18}

Simplify

- \frac{5 11}{18} : i \frac{5 4 11}{3 2}

- \frac{5 11}{18}

Apply radical rule:

- a = - 1 a

- \frac{5 11}{18} = - 1 \frac{5 11}{18}

= - 1 \frac{5 11}{18}

Apply imaginary number rule:

- 1 = i

= i \frac{5 11}{18}

Apply radical rule:

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

assuming

a \geq 0, b \geq 0

\frac{5 11}{18} = \frac{5 11}{18}

= i \frac{5 11}{18}

18 = 32

18

Prime factorization of

18 : 2 \cdot 3^{2}

18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3 \cdot 3

= 2 \cdot 3^{2}

= 3^{2} \cdot 2

Apply radical rule:

n ab = n a n b

= 2 3^{2}

Apply radical rule:

n a^{n} = a

3^{2} = 3

= 32

= i \frac{5 11}{3 2}

Apply radical rule:

n ab = n a n b,

assuming

a \geq 0, b \geq 0

511 = 511

= i \frac{5 11}{3 2}

11 : 411

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot \frac{1}{2}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

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

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

= 411

= i \frac{5 4 11}{3 2}

= - i \frac{5 4 11}{3 2}

\frac{5 4 11}{3 2} = \frac{10 4 11}{6}

\frac{5 4 11}{3 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{5 4 11 2}{3 2 2}

54112 = 10411

54112

Apply radical rule:

a b = a \cdot b

52 = 5 \cdot 2

= 411 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 10411

322 = 6

322

Apply radical rule:

a a = a

22 = 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

= \frac{10 4 11}{6}

= - i \frac{10 4 11}{6}

Rewrite

- i \frac{10 4 11}{6}

in standard complex form:

- \frac{10 4 11}{6} i

- i \frac{10 4 11}{6}

\frac{10 4 11}{6} = \frac{5 4 11}{3 2}

\frac{10 4 11}{6}

Factor

10 : 25

Factor

10 = 2 \cdot 5

= 2 \cdot 5

Apply radical rule:

n ab = n a n b

= 25

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

= \frac{2 5 4 11}{2 \cdot 3}

Cancel

\frac{2 5 4 11}{2 \cdot 3} : \frac{5 4 11}{2 \cdot 3}

\frac{2 5 4 11}{2 \cdot 3}

Apply radical rule:

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

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

= \frac{2 ^{\frac{1}{2}} 5 4 11}{2 \cdot 3}

Apply exponent rule:

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

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

= \frac{5 4 11}{3 \cdot 2 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

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

= \frac{5 4 11}{3 \cdot 2 ^{\frac{1}{2}}}

Apply radical rule:

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

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

= \frac{5 4 11}{3 2}

= \frac{5 4 11}{2 \cdot 3}

= - i \frac{5 4 11}{3 2}

Multiply fractions:

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

= - \frac{5 4 11 i}{2 \cdot 3}

- \frac{5 4 11}{3 2} = - \frac{10 4 11}{6}

- \frac{5 4 11}{3 2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{5 4 11 2}{3 2 2}

54112 = 10411

54112

Apply radical rule:

a b = a \cdot b

52 = 5 \cdot 2

= 411 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 10411

322 = 6

322

Apply radical rule:

a a = a

22 = 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

= - \frac{10 4 11}{6}

= - \frac{10 4 11}{6} i

= - \frac{10 4 11}{6} i

u = i \frac{4 11 10}{6}, u = - i \frac{4 11 10}{6}

Substitute back

u = sin (a)

sin (a) = i \frac{4 11 10}{6}, sin (a) = - i \frac{4 11 10}{6}

sin (a) = i \frac{4 11 10}{6}, sin (a) = - i \frac{4 11 10}{6}

sin (a) = i \frac{4 11 10}{6} :

No Solution

sin (a) = i \frac{4 11 10}{6}

No Solution

sin (a) = - i \frac{4 11 10}{6} :

No Solution

sin (a) = - i \frac{4 11 10}{6}

No Solution

Combine all the solutions

No Solution for a \in R

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(a)=-((5sqrt(11)))/((18)) ?
The general solution for sin^2(a)=-((5sqrt(11)))/((18)) is No Solution for a\in\mathbb{R}