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Popular >

-9(sin(x))^2+2sin(x)+2=0

Solution

- 9 (sin (x))^{2} + 2 sin (x) + 2 = 0

Solution

x = - 0.38246 \dots + 2 πn, x = π + 0.38246 \dots + 2 πn, x = 0.63780 \dots + 2 πn, x = π - 0.63780 \dots + 2 πn

Degrees

x = - 21.91378 \dots^{\circ} + 36 0^{\circ} n, x = 201.91378 \dots^{\circ} + 36 0^{\circ} n, x = 36.54352 \dots^{\circ} + 36 0^{\circ} n, x = 143.45647 \dots^{\circ} + 36 0^{\circ} n

Solution steps

- 9 (sin (x))^{2} + 2 sin (x) + 2 = 0

Solve by substitution

- 9 (sin (x))^{2} + 2 sin (x) + 2 = 0

Let:

sin (x) = u

- 9 u^{2} + 2 u + 2 = 0

- 9 u^{2} + 2 u + 2 = 0 : u = - \frac{- 1 + 19}{9}, u = \frac{1 + 19}{9}

- 9 u^{2} + 2 u + 2 = 0

Solve with the quadratic formula

- 9 u^{2} + 2 u + 2 = 0

Quadratic Equation Formula:

For

a = - 9, b = 2, c = 2

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 ( - 9 ) \cdot 2}{2 ( - 9 )}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 ( - 9 ) \cdot 2}{2 ( - 9 )}

2^{2} - 4 (- 9) \cdot 2 = 219

2^{2} - 4 (- 9) \cdot 2

Apply rule

- (- a) = a

= 2^{2} + 4 \cdot 9 \cdot 2

Multiply the numbers:

4 \cdot 9 \cdot 2 = 72

= 2^{2} + 72

2^{2} = 4

= 4 + 72

Add the numbers:

4 + 72 = 76

= 76

Prime factorization of

76 : 2^{2} \cdot 19

76

76

divides by

276 = 38 \cdot 2

= 2 \cdot 38

38

divides by

238 = 19 \cdot 2

= 2 \cdot 2 \cdot 19

2, 19

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 19

= 2^{2} \cdot 19

= 2^{2} \cdot 19

Apply radical rule:

= 19 2^{2}

Apply radical rule:

2^{2} = 2

= 219

u_{1, 2} = \frac{- 2 \pm 2 19}{2 ( - 9 )}

Separate the solutions

u_{1} = \frac{- 2 + 2 19}{2 ( - 9 )}, u_{2} = \frac{- 2 - 2 19}{2 ( - 9 )}

u = \frac{- 2 + 2 19}{2 ( - 9 )} : - \frac{- 1 + 19}{9}

\frac{- 2 + 2 19}{2 ( - 9 )}

Remove parentheses:

(- a) = - a

= \frac{- 2 + 2 19}{- 2 \cdot 9}

Multiply the numbers:

2 \cdot 9 = 18

= \frac{- 2 + 2 19}{- 18}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- 2 + 2 19}{18}

Cancel

\frac{- 2 + 2 19}{18} : \frac{19 - 1}{9}

\frac{- 2 + 2 19}{18}

Factor

- 2 + 219 : 2 (- 1 + 19)

- 2 + 219

Rewrite as

= - 2 \cdot 1 + 219

Factor out common term

2

= 2 (- 1 + 19)

= \frac{2 ( - 1 + 19 )}{18}

Cancel the common factor:

2

= \frac{- 1 + 19}{9}

= - \frac{19 - 1}{9}

= - \frac{- 1 + 19}{9}

u = \frac{- 2 - 2 19}{2 ( - 9 )} : \frac{1 + 19}{9}

\frac{- 2 - 2 19}{2 ( - 9 )}

Remove parentheses:

(- a) = - a

= \frac{- 2 - 2 19}{- 2 \cdot 9}

Multiply the numbers:

2 \cdot 9 = 18

= \frac{- 2 - 2 19}{- 18}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

- 2 - 219 = - (2 + 219)

= \frac{2 + 2 19}{18}

Factor

2 + 219 : 2 (1 + 19)

2 + 219

Rewrite as

= 2 \cdot 1 + 219

Factor out common term

2

= 2 (1 + 19)

= \frac{2 ( 1 + 19 )}{18}

Cancel the common factor:

2

= \frac{1 + 19}{9}

The solutions to the quadratic equation are:

u = - \frac{- 1 + 19}{9}, u = \frac{1 + 19}{9}

Substitute back

u = sin (x)

sin (x) = - \frac{- 1 + 19}{9}, sin (x) = \frac{1 + 19}{9}

sin (x) = - \frac{- 1 + 19}{9}, sin (x) = \frac{1 + 19}{9}

sin (x) = - \frac{- 1 + 19}{9} : x = arcsin (- \frac{- 1 + 19}{9}) + 2 πn, x = π + arcsin (\frac{- 1 + 19}{9}) + 2 πn

sin (x) = - \frac{- 1 + 19}{9}

Apply trig inverse properties

sin (x) = - \frac{- 1 + 19}{9}

General solutions for

sin (x) = - \frac{- 1 + 19}{9}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{- 1 + 19}{9}) + 2 πn, x = π + arcsin (\frac{- 1 + 19}{9}) + 2 πn

x = arcsin (- \frac{- 1 + 19}{9}) + 2 πn, x = π + arcsin (\frac{- 1 + 19}{9}) + 2 πn

sin (x) = \frac{1 + 19}{9} : x = arcsin (\frac{1 + 19}{9}) + 2 πn, x = π - arcsin (\frac{1 + 19}{9}) + 2 πn

sin (x) = \frac{1 + 19}{9}

Apply trig inverse properties

sin (x) = \frac{1 + 19}{9}

General solutions for

sin (x) = \frac{1 + 19}{9}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{1 + 19}{9}) + 2 πn, x = π - arcsin (\frac{1 + 19}{9}) + 2 πn

x = arcsin (\frac{1 + 19}{9}) + 2 πn, x = π - arcsin (\frac{1 + 19}{9}) + 2 πn

Combine all the solutions

x = arcsin (- \frac{- 1 + 19}{9}) + 2 πn, x = π + arcsin (\frac{- 1 + 19}{9}) + 2 πn, x = arcsin (\frac{1 + 19}{9}) + 2 πn, x = π - arcsin (\frac{1 + 19}{9}) + 2 πn

Show solutions in decimal form

x = - 0.38246 \dots + 2 πn, x = π + 0.38246 \dots + 2 πn, x = 0.63780 \dots + 2 πn, x = π - 0.63780 \dots + 2 πn

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