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Popular >

-6sin^2(x)+sin(x)=-3

Solution

- 6 sin^{2} (x) + sin (x) = - 3

Solution

x = - 0.67983 \dots + 2 πn, x = π + 0.67983 \dots + 2 πn, x = 0.91955 \dots + 2 πn, x = π - 0.91955 \dots + 2 πn

Degrees

x = - 38.95184 \dots^{\circ} + 36 0^{\circ} n, x = 218.95184 \dots^{\circ} + 36 0^{\circ} n, x = 52.68678 \dots^{\circ} + 36 0^{\circ} n, x = 127.31321 \dots^{\circ} + 36 0^{\circ} n

Solution steps

- 6 sin^{2} (x) + sin (x) = - 3

Solve by substitution

- 6 sin^{2} (x) + sin (x) = - 3

Let:

sin (x) = u

- 6 u^{2} + u = - 3

- 6 u^{2} + u = - 3 : u = - \frac{- 1 + 73}{12}, u = \frac{1 + 73}{12}

- 6 u^{2} + u = - 3

Move

3

to the left side

- 6 u^{2} + u = - 3

Add

3

to both sides

- 6 u^{2} + u + 3 = - 3 + 3

Simplify

- 6 u^{2} + u + 3 = 0

- 6 u^{2} + u + 3 = 0

Solve with the quadratic formula

- 6 u^{2} + u + 3 = 0

Quadratic Equation Formula:

For

a = - 6, b = 1, c = 3

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 6 ) \cdot 3}{2 ( - 6 )}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 6 ) \cdot 3}{2 ( - 6 )}

1^{2} - 4 (- 6) \cdot 3 = 73

1^{2} - 4 (- 6) \cdot 3

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 6) \cdot 3

Apply rule

- (- a) = a

= 1 + 4 \cdot 6 \cdot 3

Multiply the numbers:

4 \cdot 6 \cdot 3 = 72

= 1 + 72

Add the numbers:

1 + 72 = 73

= 73

u_{1, 2} = \frac{- 1 \pm 73}{2 ( - 6 )}

Separate the solutions

u_{1} = \frac{- 1 + 73}{2 ( - 6 )}, u_{2} = \frac{- 1 - 73}{2 ( - 6 )}

u = \frac{- 1 + 73}{2 ( - 6 )} : - \frac{- 1 + 73}{12}

\frac{- 1 + 73}{2 ( - 6 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 + 73}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 1 + 73}{- 12}

Apply the fraction rule:

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

= - \frac{- 1 + 73}{12}

u = \frac{- 1 - 73}{2 ( - 6 )} : \frac{1 + 73}{12}

\frac{- 1 - 73}{2 ( - 6 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 - 73}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 1 - 73}{- 12}

Apply the fraction rule:

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

- 1 - 73 = - (1 + 73)

= \frac{1 + 73}{12}

The solutions to the quadratic equation are:

u = - \frac{- 1 + 73}{12}, u = \frac{1 + 73}{12}

Substitute back

u = sin (x)

sin (x) = - \frac{- 1 + 73}{12}, sin (x) = \frac{1 + 73}{12}

sin (x) = - \frac{- 1 + 73}{12}, sin (x) = \frac{1 + 73}{12}

sin (x) = - \frac{- 1 + 73}{12} : x = arcsin (- \frac{- 1 + 73}{12}) + 2 πn, x = π + arcsin (\frac{- 1 + 73}{12}) + 2 πn

sin (x) = - \frac{- 1 + 73}{12}

Apply trig inverse properties

sin (x) = - \frac{- 1 + 73}{12}

General solutions for

sin (x) = - \frac{- 1 + 73}{12}

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

x = arcsin (- \frac{- 1 + 73}{12}) + 2 πn, x = π + arcsin (\frac{- 1 + 73}{12}) + 2 πn

x = arcsin (- \frac{- 1 + 73}{12}) + 2 πn, x = π + arcsin (\frac{- 1 + 73}{12}) + 2 πn

sin (x) = \frac{1 + 73}{12} : x = arcsin (\frac{1 + 73}{12}) + 2 πn, x = π - arcsin (\frac{1 + 73}{12}) + 2 πn

sin (x) = \frac{1 + 73}{12}

Apply trig inverse properties

sin (x) = \frac{1 + 73}{12}

General solutions for

sin (x) = \frac{1 + 73}{12}

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

x = arcsin (\frac{1 + 73}{12}) + 2 πn, x = π - arcsin (\frac{1 + 73}{12}) + 2 πn

x = arcsin (\frac{1 + 73}{12}) + 2 πn, x = π - arcsin (\frac{1 + 73}{12}) + 2 πn

Combine all the solutions

x = arcsin (- \frac{- 1 + 73}{12}) + 2 πn, x = π + arcsin (\frac{- 1 + 73}{12}) + 2 πn, x = arcsin (\frac{1 + 73}{12}) + 2 πn, x = π - arcsin (\frac{1 + 73}{12}) + 2 πn

Show solutions in decimal form

x = - 0.67983 \dots + 2 πn, x = π + 0.67983 \dots + 2 πn, x = 0.91955 \dots + 2 πn, x = π - 0.91955 \dots + 2 πn

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