What is the general solution for 2-sqrt(-1-6sin(x))=sqrt(-4sin(x)) ?

The general solution for 2-sqrt(-1-6sin(x))=sqrt(-4sin(x)) is x=-0.30674…+2pin,x=pi+0.30674…+2pin

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Popular Trigonometry >

2-sqrt(-1-6sin(x))=sqrt(-4sin(x))

Solution

2 - - 1 - 6 sin (x) = - 4 sin (x)

Solution

x = - 0.30674 \dots + 2 πn, x = π + 0.30674 \dots + 2 πn

Degrees

x = - 17.57542 \dots^{\circ} + 36 0^{\circ} n, x = 197.57542 \dots^{\circ} + 36 0^{\circ} n

Solution steps

2 - - 1 - 6 sin (x) = - 4 sin (x)

Solve by substitution

2 - - 1 - 6 sin (x) = - 4 sin (x)

Let:

sin (x) = u

2 - - 1 - 6 u = - 4 u

2 - - 1 - 6 u = - 4 u : u = \frac{- 21 + 4 26}{2}

2 - - 1 - 6 u = - 4 u

Remove square roots

2 - - 1 - 6 u = - 4 u

Square both sides:

- 6 u + 3 - 4 - 1 - 6 u = - 4 u

2 - - 1 - 6 u = - 4 u

(2 - - 1 - 6 u)^{2} = (- 4 u)^{2}

Expand

(2 - - 1 - 6 u)^{2} : - 6 u + 3 - 4 - 1 - 6 u

(2 - - 1 - 6 u)^{2}

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 2, b = - 1 - 6 u

= 2^{2} - 2 \cdot 2 - 1 - 6 u + (- 1 - 6 u)^{2}

Simplify

2^{2} - 2 \cdot 2 - 1 - 6 u + (- 1 - 6 u)^{2} : 4 - 4 - 1 - 6 u + - 1 - 6 u

2^{2} - 2 \cdot 2 - 1 - 6 u + (- 1 - 6 u)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

2 \cdot 2 - 1 - 6 u = 4 - 1 - 6 u

2 \cdot 2 - 1 - 6 u

Multiply the numbers:

2 \cdot 2 = 4

= 4 - 1 - 6 u

(- 1 - 6 u)^{2} = - 1 - 6 u

(- 1 - 6 u)^{2}

Apply radical rule:

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

= ((- 1 - 6 u)^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (- 1 - 6 u)^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= - 1 - 6 u

= 4 - 4 - 1 - 6 u - 1 - 6 u

= 4 - 4 - 1 - 6 u - 1 - 6 u

Refine

= - 6 u + 3 - 4 - 1 - 6 u

Expand

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

(- 4 u)^{2}

Apply radical rule:

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

= ((- 4 u)^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (- 4 u)^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= - 4 u

- 6 u + 3 - 4 - 1 - 6 u = - 4 u

- 6 u + 3 - 4 - 1 - 6 u = - 4 u

Add

6 u

to both sides

- 6 u + 3 - 4 - 1 - 6 u + 6 u = - 4 u + 6 u

Simplify

- 4 - 1 - 6 u + 3 = 2 u

Subtract

3

from both sides

- 4 - 1 - 6 u + 3 - 3 = 2 u - 3

Simplify

- 4 - 1 - 6 u = 2 u - 3

Square both sides:

- 16 - 96 u = 4 u^{2} - 12 u + 9

- 6 u + 3 - 4 - 1 - 6 u = - 4 u

(- 4 - 1 - 6 u)^{2} = (2 u - 3)^{2}

Expand

(- 4 - 1 - 6 u)^{2} : - 16 - 96 u

(- 4 - 1 - 6 u)^{2}

Apply exponent rule:

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

n

is even

(- 4 - 1 - 6 u)^{2} = (4 - 1 - 6 u)^{2}

= (4 - 1 - 6 u)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 4^{2} (- 1 - 6 u)^{2}

(- 1 - 6 u)^{2} : - 1 - 6 u

Apply radical rule:

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

= ((- 1 - 6 u)^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (- 1 - 6 u)^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= - 1 - 6 u

= 4^{2} (- 1 - 6 u)

4^{2} = 16

= 16 (- 1 - 6 u)

Expand

16 (- 1 - 6 u) : - 16 - 96 u

16 (- 1 - 6 u)

Apply the distributive law:

a (b - c) = ab - a c

a = 16, b = - 1, c = 6 u

= 16 (- 1) - 16 \cdot 6 u

Apply minus-plus rules

+ (- a) = - a

= - 16 \cdot 1 - 16 \cdot 6 u

Simplify

- 16 \cdot 1 - 16 \cdot 6 u : - 16 - 96 u

- 16 \cdot 1 - 16 \cdot 6 u

Multiply the numbers:

16 \cdot 1 = 16

= - 16 - 16 \cdot 6 u

Multiply the numbers:

16 \cdot 6 = 96

= - 16 - 96 u

= - 16 - 96 u

= - 16 - 96 u

Expand

(2 u - 3)^{2} : 4 u^{2} - 12 u + 9

(2 u - 3)^{2}

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 2 u, b = 3

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

Simplify

(2 u)^{2} - 2 \cdot 2 u \cdot 3 + 3^{2} : 4 u^{2} - 12 u + 9

(2 u)^{2} - 2 \cdot 2 u \cdot 3 + 3^{2}

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

(2 u)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} u^{2}

2^{2} = 4

= 4 u^{2}

2 \cdot 2 u \cdot 3 = 12 u

2 \cdot 2 u \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12 u

3^{2} = 9

3^{2}

3^{2} = 9

= 9

= 4 u^{2} - 12 u + 9

= 4 u^{2} - 12 u + 9

- 16 - 96 u = 4 u^{2} - 12 u + 9

- 16 - 96 u = 4 u^{2} - 12 u + 9

- 16 - 96 u = 4 u^{2} - 12 u + 9

Solve

- 16 - 96 u = 4 u^{2} - 12 u + 9 : u = \frac{- 21 + 4 26}{2}, u = - \frac{21 + 4 26}{2}

- 16 - 96 u = 4 u^{2} - 12 u + 9

Switch sides

4 u^{2} - 12 u + 9 = - 16 - 96 u

Move

96 u

to the left side

4 u^{2} - 12 u + 9 = - 16 - 96 u

Add

96 u

to both sides

4 u^{2} - 12 u + 9 + 96 u = - 16 - 96 u + 96 u

Simplify

4 u^{2} + 84 u + 9 = - 16

4 u^{2} + 84 u + 9 = - 16

Move

16

to the left side

4 u^{2} + 84 u + 9 = - 16

Add

16

to both sides

4 u^{2} + 84 u + 9 + 16 = - 16 + 16

Simplify

4 u^{2} + 84 u + 25 = 0

4 u^{2} + 84 u + 25 = 0

Solve with the quadratic formula

4 u^{2} + 84 u + 25 = 0

Quadratic Equation Formula:

For

a = 4, b = 84, c = 25

u_{1, 2} = \frac{- 84 \pm 8 4 ^{2} - 4 \cdot 4 \cdot 25}{2 \cdot 4}

u_{1, 2} = \frac{- 84 \pm 8 4 ^{2} - 4 \cdot 4 \cdot 25}{2 \cdot 4}

8 4^{2} - 4 \cdot 4 \cdot 25 = 1626

8 4^{2} - 4 \cdot 4 \cdot 25

Multiply the numbers:

4 \cdot 4 \cdot 25 = 400

= 8 4^{2} - 400

8 4^{2} = 7056

= 7056 - 400

Subtract the numbers:

7056 - 400 = 6656

= 6656

Prime factorization of

6656 : 2^{9} \cdot 13

6656

6656

divides by

26656 = 3328 \cdot 2

= 2 \cdot 3328

3328

divides by

23328 = 1664 \cdot 2

= 2 \cdot 2 \cdot 1664

1664

divides by

21664 = 832 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 832

832

divides by

2832 = 416 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 416

416

divides by

2416 = 208 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 208

208

divides by

2208 = 104 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 104

104

divides by

2104 = 52 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 52

52

divides by

252 = 26 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 26

26

divides by

226 = 13 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 13

2, 13

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 13

= 2^{9} \cdot 13

= 2^{9} \cdot 13

Apply exponent rule:

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

= 2^{8} \cdot 2 \cdot 13

Apply radical rule:

= 2^{8} 2 \cdot 13

Apply radical rule:

2^{8} = 2^{\frac{8}{2}} = 2^{4}

= 2^{4} 2 \cdot 13

Refine

= 1626

u_{1, 2} = \frac{- 84 \pm 16 26}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- 84 + 16 26}{2 \cdot 4}, u_{2} = \frac{- 84 - 16 26}{2 \cdot 4}

u = \frac{- 84 + 16 26}{2 \cdot 4} : \frac{- 21 + 4 26}{2}

\frac{- 84 + 16 26}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 84 + 16 26}{8}

Factor

- 84 + 1626 : 4 (- 21 + 426)

- 84 + 1626

Rewrite as

= - 4 \cdot 21 + 4 \cdot 426

Factor out common term

4

= 4 (- 21 + 426)

= \frac{4 ( - 21 + 4 26 )}{8}

Cancel the common factor:

4

= \frac{- 21 + 4 26}{2}

u = \frac{- 84 - 16 26}{2 \cdot 4} : - \frac{21 + 4 26}{2}

\frac{- 84 - 16 26}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 84 - 16 26}{8}

Factor

- 84 - 1626 : - 4 (21 + 426)

- 84 - 1626

Rewrite as

= - 4 \cdot 21 - 4 \cdot 426

Factor out common term

4

= - 4 (21 + 426)

= - \frac{4 ( 21 + 4 26 )}{8}

Cancel the common factor:

4

= - \frac{21 + 4 26}{2}

The solutions to the quadratic equation are:

u = \frac{- 21 + 4 26}{2}, u = - \frac{21 + 4 26}{2}

u = \frac{- 21 + 4 26}{2}, u = - \frac{21 + 4 26}{2}

Verify Solutions:

u = \frac{- 21 + 4 26}{2}

True

, u = - \frac{21 + 4 26}{2}

False

Check the solutions by plugging them into

2 - - 1 - 6 u = - 4 u

Remove the ones that don't agree with the equation.

Plug in

u = \frac{- 21 + 4 26}{2} :

True

2 - - 1 - 6 (\frac{- 21 + 4 26}{2}) = - 4 (\frac{- 21 + 4 26}{2})

2 - - 1 - 6 (\frac{- 21 + 4 26}{2}) = 26 - 4

2 - - 1 - 6 (\frac{- 21 + 4 26}{2})

Remove parentheses:

(a) = a

= 2 - - 1 - 6 \cdot \frac{- 21 + 4 26}{2}

- 1 - 6 \cdot \frac{- 21 + 4 26}{2} = 6 - 26

- 1 - 6 \cdot \frac{- 21 + 4 26}{2}

6 \cdot \frac{- 21 + 4 26}{2} = 3 (426 - 21)

6 \cdot \frac{- 21 + 4 26}{2}

Multiply fractions:

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

= \frac{( - 21 + 4 26 ) \cdot 6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 (426 - 21)

= - 1 - 3 (426 - 21)

Expand

- 1 - 3 (426 - 21) : 62 - 1226

- 1 - 3 (426 - 21)

Expand

- 3 (426 - 21) : - 1226 + 63

- 3 (426 - 21)

Apply the distributive law:

a (b - c) = ab - a c

a = - 3, b = 426, c = 21

= - 3 \cdot 426 - (- 3) \cdot 21

Apply minus-plus rules

- (- a) = a

= - 3 \cdot 426 + 3 \cdot 21

Simplify

- 3 \cdot 426 + 3 \cdot 21 : - 1226 + 63

- 3 \cdot 426 + 3 \cdot 21

Multiply the numbers:

3 \cdot 4 = 12

= - 1226 + 3 \cdot 21

Multiply the numbers:

3 \cdot 21 = 63

= - 1226 + 63

= - 1226 + 63

= - 1 - 1226 + 63

Add/Subtract the numbers:

- 1 + 63 = 62

= 62 - 1226

= 62 - 1226

= 26 - 1226 + 36

= (26)^{2} - 1226 + (36)^{2}

36 = 6

36

Factor the number:

36 = 6^{2}

= 6^{2}

Apply radical rule:

6^{2} = 6

= 6

= (26)^{2} - 1226 + 6^{2}

226 \cdot 6 = 1226

226 \cdot 6

Multiply the numbers:

2 \cdot 6 = 12

= 1226

= (26)^{2} - 226 \cdot 6 + 6^{2}

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

(26)^{2} - 226 \cdot 6 + 6^{2} = (26 - 6)^{2}

= (26 - 6)^{2}

Apply exponent rule:

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

n

is even

(26 - 6)^{2} = (6 - 26)^{2}

= (6 - 26)^{2}

Apply radical rule:

(6 - 26)^{2} = 6 - 26

= 6 - 26

= 2 - (6 - 26)

- (6 - 26) : - 6 + 26

- (6 - 26)

Distribute parentheses

= - (6) - (- 26)

Apply minus-plus rules

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

= - 6 + 26

= 2 - 6 + 26

Subtract the numbers:

2 - 6 = - 4

= 26 - 4

- 4 (\frac{- 21 + 4 26}{2}) = 2 - 426 + 21

- 4 (\frac{- 21 + 4 26}{2})

Remove parentheses:

(a) = a

= - 4 \cdot \frac{- 21 + 4 26}{2}

Multiply

- 4 \cdot \frac{- 21 + 4 26}{2} : - 2 (426 - 21)

- 4 \cdot \frac{- 21 + 4 26}{2}

Multiply fractions:

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

= - \frac{( - 21 + 4 26 ) \cdot 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= - 2 (426 - 21)

= - 2 (426 - 21)

Apply radical rule:

assuming

a \geq 0, b \geq 0

= 2 - (426 - 21)

Expand

- (426 - 21) : - 426 + 21

- (426 - 21)

Distribute parentheses

= - (426) - (- 21)

Apply minus-plus rules

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

= - 426 + 21

= 2 21 - 426

26 - 4 = 2 - 426 + 21

True

Plug in

u = - \frac{21 + 4 26}{2} :

False

2 - - 1 - 6 (- \frac{21 + 4 26}{2}) = - 4 (- \frac{21 + 4 26}{2})

2 - - 1 - 6 (- \frac{21 + 4 26}{2}) = - 4 - 26

2 - - 1 - 6 (- \frac{21 + 4 26}{2})

Apply rule

- (- a) = a

= 2 - - 1 + 6 \cdot \frac{21 + 4 26}{2}

- 1 + 6 \cdot \frac{21 + 4 26}{2} = 26 + 6

- 1 + 6 \cdot \frac{21 + 4 26}{2}

6 \cdot \frac{21 + 4 26}{2} = 3 (21 + 426)

6 \cdot \frac{21 + 4 26}{2}

Multiply fractions:

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

= \frac{( 21 + 4 26 ) \cdot 6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 (21 + 426)

= - 1 + 3 (21 + 426)

Expand

- 1 + 3 (21 + 426) : 62 + 1226

- 1 + 3 (21 + 426)

Expand

3 (21 + 426) : 63 + 1226

3 (21 + 426)

Apply the distributive law:

a (b + c) = ab + a c

a = 3, b = 21, c = 426

= 3 \cdot 21 + 3 \cdot 426

Simplify

3 \cdot 21 + 3 \cdot 426 : 63 + 1226

3 \cdot 21 + 3 \cdot 426

Multiply the numbers:

3 \cdot 21 = 63

= 63 + 3 \cdot 426

Multiply the numbers:

3 \cdot 4 = 12

= 63 + 1226

= 63 + 1226

= - 1 + 63 + 1226

Add/Subtract the numbers:

- 1 + 63 = 62

= 62 + 1226

= 62 + 1226

= 26 + 1226 + 36

= (26)^{2} + 1226 + (36)^{2}

36 = 6

36

Factor the number:

36 = 6^{2}

= 6^{2}

Apply radical rule:

6^{2} = 6

= 6

= (26)^{2} + 1226 + 6^{2}

226 \cdot 6 = 1226

226 \cdot 6

Multiply the numbers:

2 \cdot 6 = 12

= 1226

= (26)^{2} + 226 \cdot 6 + 6^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

(26)^{2} + 226 \cdot 6 + 6^{2} = (26 + 6)^{2}

= (26 + 6)^{2}

Apply radical rule:

(26 + 6)^{2} = 26 + 6

= 26 + 6

= 2 - (6 + 26)

- (26 + 6) : - 26 - 6

- (26 + 6)

Distribute parentheses

= - (26) - (6)

Apply minus-plus rules

+ (- a) = - a

= - 26 - 6

= 2 - 26 - 6

Subtract the numbers:

2 - 6 = - 4

= - 4 - 26

- 4 (- \frac{21 + 4 26}{2}) = 2 21 + 426

- 4 (- \frac{21 + 4 26}{2})

Apply rule

- (- a) = a

= 4 \cdot \frac{21 + 4 26}{2}

Multiply

4 \cdot \frac{21 + 4 26}{2} : 2 (21 + 426)

4 \cdot \frac{21 + 4 26}{2}

Multiply fractions:

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

= \frac{( 21 + 4 26 ) \cdot 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 (21 + 426)

= 2 (21 + 426)

Apply radical rule:

assuming

a \geq 0, b \geq 0

= 2 21 + 426

- 4 - 26 = 2 21 + 426

False

The solution is

u = \frac{- 21 + 4 26}{2}

Substitute back

u = sin (x)

sin (x) = \frac{- 21 + 4 26}{2}

sin (x) = \frac{- 21 + 4 26}{2}

sin (x) = \frac{- 21 + 4 26}{2} : x = arcsin (\frac{- 21 + 4 26}{2}) + 2 πn, x = π + arcsin (- \frac{- 21 + 4 26}{2}) + 2 πn

sin (x) = \frac{- 21 + 4 26}{2}

Apply trig inverse properties

sin (x) = \frac{- 21 + 4 26}{2}

General solutions for

sin (x) = \frac{- 21 + 4 26}{2}

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

x = arcsin (\frac{- 21 + 4 26}{2}) + 2 πn, x = π + arcsin (- \frac{- 21 + 4 26}{2}) + 2 πn

x = arcsin (\frac{- 21 + 4 26}{2}) + 2 πn, x = π + arcsin (- \frac{- 21 + 4 26}{2}) + 2 πn

Combine all the solutions

x = arcsin (\frac{- 21 + 4 26}{2}) + 2 πn, x = π + arcsin (- \frac{- 21 + 4 26}{2}) + 2 πn

Show solutions in decimal form

x = - 0.30674 \dots + 2 πn, x = π + 0.30674 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 2-sqrt(-1-6sin(x))=sqrt(-4sin(x)) ?
The general solution for 2-sqrt(-1-6sin(x))=sqrt(-4sin(x)) is x=-0.30674…+2pin,x=pi+0.30674…+2pin