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 >

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}

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}

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:

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:

= 2 3^{2}

Apply radical rule:

3^{2} = 3

= 32

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

511 = 511

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

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}}

Multiply by the conjugate

\frac{2}{2}

Apply radical rule:

a b = a \cdot b

52 = 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

322 = 6

322

Apply radical rule:

a a = a

22 = 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Rewrite

in standard complex form:

Factor

10 : 25

Factor

10 = 2 \cdot 5

= 2 \cdot 5

Apply radical rule:

= 25

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

Cancel

Apply radical rule:

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

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}}}

Subtract the numbers:

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

Apply radical rule:

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

Multiply fractions:

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

Multiply by the conjugate

\frac{2}{2}

Apply radical rule:

a b = a \cdot b

52 = 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

322 = 6

322

Apply radical rule:

a a = a

22 = 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Simplify

- - \frac{5 11}{18}

Simplify

- \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:

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:

= 2 3^{2}

Apply radical rule:

3^{2} = 3

= 32

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

511 = 511

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

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}}

Multiply by the conjugate

\frac{2}{2}

Apply radical rule:

a b = a \cdot b

52 = 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

322 = 6

322

Apply radical rule:

a a = a

22 = 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Rewrite

in standard complex form:

Factor

10 : 25

Factor

10 = 2 \cdot 5

= 2 \cdot 5

Apply radical rule:

= 25

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

Cancel

Apply radical rule:

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

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}}}

Subtract the numbers:

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

Apply radical rule:

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

Multiply fractions:

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

Multiply by the conjugate

\frac{2}{2}

Apply radical rule:

a b = a \cdot b

52 = 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

322 = 6

322

Apply radical rule:

a a = a

22 = 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Substitute back

u = sin (a)

No Solution

No Solution

No Solution

No Solution

Combine all the solutions

No Solution for a \in R

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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}