What is the general solution for 3-sin(x)(4+6sin(x))=0 ?

The general solution for 3-sin(x)(4+6sin(x))=0 is x=0.46497…+2pin,x=pi-0.46497…+2pin

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3-sin(x)(4+6sin(x))=0

Solution

3 - sin (x) (4 + 6 sin (x)) = 0

Solution

x = 0.46497 \dots + 2 πn, x = π - 0.46497 \dots + 2 πn

Degrees

x = 26.64124 \dots^{\circ} + 36 0^{\circ} n, x = 153.35875 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 - sin (x) (4 + 6 sin (x)) = 0

Solve by substitution

3 - sin (x) (4 + 6 sin (x)) = 0

Let:

sin (x) = u

3 - u (4 + 6 u) = 0

3 - u (4 + 6 u) = 0 : u = - \frac{2 + 22}{6}, u = \frac{22 - 2}{6}

3 - u (4 + 6 u) = 0

Expand

3 - u (4 + 6 u) : 3 - 4 u - 6 u^{2}

3 - u (4 + 6 u)

Expand

- u (4 + 6 u) : - 4 u - 6 u^{2}

- u (4 + 6 u)

Apply the distributive law:

a (b + c) = ab + a c

a = - u, b = 4, c = 6 u

= - u \cdot 4 + (- u) \cdot 6 u

Apply minus-plus rules

+ (- a) = - a

= - 4 u - 6 uu

6 uu = 6 u^{2}

6 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 6 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 6 u^{2}

= - 4 u - 6 u^{2}

= 3 - 4 u - 6 u^{2}

3 - 4 u - 6 u^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

- 6 u^{2} - 4 u + 3 = 0

Solve with the quadratic formula

- 6 u^{2} - 4 u + 3 = 0

Quadratic Equation Formula:

For

a = - 6, b = - 4, c = 3

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

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

(- 4)^{2} - 4 (- 6) \cdot 3 = 222

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

Apply rule

- (- a) = a

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 4)^{2} = 4^{2}

= 4^{2} + 4 \cdot 6 \cdot 3

Multiply the numbers:

4 \cdot 6 \cdot 3 = 72

= 4^{2} + 72

4^{2} = 16

= 16 + 72

Add the numbers:

16 + 72 = 88

= 88

Prime factorization of

88 : 2^{3} \cdot 11

88

88

divides by

288 = 44 \cdot 2

= 2 \cdot 44

44

divides by

244 = 22 \cdot 2

= 2 \cdot 2 \cdot 22

22

divides by

222 = 11 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 11

2, 11

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 11

= 2^{3} \cdot 11

= 2^{3} \cdot 11

Apply exponent rule:

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

= 2^{2} \cdot 2 \cdot 11

Apply radical rule:

= 2^{2} 2 \cdot 11

Apply radical rule:

2^{2} = 2

= 2 2 \cdot 11

Refine

= 222

u_{1, 2} = \frac{- ( - 4 ) \pm 2 22}{2 ( - 6 )}

Separate the solutions

u_{1} = \frac{- ( - 4 ) + 2 22}{2 ( - 6 )}, u_{2} = \frac{- ( - 4 ) - 2 22}{2 ( - 6 )}

u = \frac{- ( - 4 ) + 2 22}{2 ( - 6 )} : - \frac{2 + 22}{6}

\frac{- ( - 4 ) + 2 22}{2 ( - 6 )}

Remove parentheses:

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

= \frac{4 + 2 22}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{4 + 2 22}{- 12}

Apply the fraction rule:

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

= - \frac{4 + 2 22}{12}

Cancel

\frac{4 + 2 22}{12} : \frac{2 + 22}{6}

\frac{4 + 2 22}{12}

Factor

4 + 222 : 2 (2 + 22)

4 + 222

Rewrite as

= 2 \cdot 2 + 222

Factor out common term

2

= 2 (2 + 22)

= \frac{2 ( 2 + 22 )}{12}

Cancel the common factor:

2

= \frac{2 + 22}{6}

= - \frac{2 + 22}{6}

u = \frac{- ( - 4 ) - 2 22}{2 ( - 6 )} : \frac{22 - 2}{6}

\frac{- ( - 4 ) - 2 22}{2 ( - 6 )}

Remove parentheses:

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

= \frac{4 - 2 22}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{4 - 2 22}{- 12}

Apply the fraction rule:

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

4 - 222 = - (222 - 4)

= \frac{2 22 - 4}{12}

Factor

222 - 4 : 2 (22 - 2)

222 - 4

Rewrite as

= 222 - 2 \cdot 2

Factor out common term

2

= 2 (22 - 2)

= \frac{2 ( 22 - 2 )}{12}

Cancel the common factor:

2

= \frac{22 - 2}{6}

The solutions to the quadratic equation are:

u = - \frac{2 + 22}{6}, u = \frac{22 - 2}{6}

Substitute back

u = sin (x)

sin (x) = - \frac{2 + 22}{6}, sin (x) = \frac{22 - 2}{6}

sin (x) = - \frac{2 + 22}{6}, sin (x) = \frac{22 - 2}{6}

sin (x) = - \frac{2 + 22}{6} :

No Solution

sin (x) = - \frac{2 + 22}{6}

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = \frac{22 - 2}{6} : x = arcsin (\frac{22 - 2}{6}) + 2 πn, x = π - arcsin (\frac{22 - 2}{6}) + 2 πn

sin (x) = \frac{22 - 2}{6}

Apply trig inverse properties

sin (x) = \frac{22 - 2}{6}

General solutions for

sin (x) = \frac{22 - 2}{6}

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

x = arcsin (\frac{22 - 2}{6}) + 2 πn, x = π - arcsin (\frac{22 - 2}{6}) + 2 πn

x = arcsin (\frac{22 - 2}{6}) + 2 πn, x = π - arcsin (\frac{22 - 2}{6}) + 2 πn

Combine all the solutions

x = arcsin (\frac{22 - 2}{6}) + 2 πn, x = π - arcsin (\frac{22 - 2}{6}) + 2 πn

Show solutions in decimal form

x = 0.46497 \dots + 2 πn, x = π - 0.46497 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3-sin(x)(4+6sin(x))=0 ?
The general solution for 3-sin(x)(4+6sin(x))=0 is x=0.46497…+2pin,x=pi-0.46497…+2pin