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8sin^4(x)-3sin^2(x)+1/4 =0

Solution

8 sin^{4} (x) - 3 sin^{2} (x) + \frac{1}{4} = 0

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = 0.36136 \dots + 2 πn, x = π - 0.36136 \dots + 2 πn, x = - 0.36136 \dots + 2 πn, x = π + 0.36136 \dots + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 20.70481 \dots^{\circ} + 36 0^{\circ} n, x = 159.29518 \dots^{\circ} + 36 0^{\circ} n, x = - 20.70481 \dots^{\circ} + 36 0^{\circ} n, x = 200.70481 \dots^{\circ} + 36 0^{\circ} n

Solution steps

8 sin^{4} (x) - 3 sin^{2} (x) + \frac{1}{4} = 0

Solve by substitution

8 sin^{4} (x) - 3 sin^{2} (x) + \frac{1}{4} = 0

Let:

sin (x) = u

8 u^{4} - 3 u^{2} + \frac{1}{4} = 0

8 u^{4} - 3 u^{2} + \frac{1}{4} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}, u = \frac{2}{4}, u = - \frac{2}{4}

8 u^{4} - 3 u^{2} + \frac{1}{4} = 0

Multiply both sides by

4

8 u^{4} - 3 u^{2} + \frac{1}{4} = 0

Multiply both sides by

4

8 u^{4} \cdot 4 - 3 u^{2} \cdot 4 + \frac{1}{4} \cdot 4 = 0 \cdot 4

Simplify

32 u^{4} - 12 u^{2} + 1 = 0

32 u^{4} - 12 u^{2} + 1 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

32 v^{2} - 12 v + 1 = 0

Solve

32 v^{2} - 12 v + 1 = 0 : v = \frac{1}{4}, v = \frac{1}{8}

32 v^{2} - 12 v + 1 = 0

Solve with the quadratic formula

32 v^{2} - 12 v + 1 = 0

Quadratic Equation Formula:

For

a = 32, b = - 12, c = 1

v_{1, 2} = \frac{- ( - 12 ) \pm ( - 12 ) ^{2} - 4 \cdot 32 \cdot 1}{2 \cdot 32}

v_{1, 2} = \frac{- ( - 12 ) \pm ( - 12 ) ^{2} - 4 \cdot 32 \cdot 1}{2 \cdot 32}

(- 12)^{2} - 4 \cdot 32 \cdot 1 = 4

(- 12)^{2} - 4 \cdot 32 \cdot 1

Apply exponent rule:

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

n

is even

(- 12)^{2} = 1 2^{2}

= 1 2^{2} - 4 \cdot 32 \cdot 1

Multiply the numbers:

4 \cdot 32 \cdot 1 = 128

= 1 2^{2} - 128

1 2^{2} = 144

= 144 - 128

Subtract the numbers:

144 - 128 = 16

= 16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

v_{1, 2} = \frac{- ( - 12 ) \pm 4}{2 \cdot 32}

Separate the solutions

v_{1} = \frac{- ( - 12 ) + 4}{2 \cdot 32}, v_{2} = \frac{- ( - 12 ) - 4}{2 \cdot 32}

v = \frac{- ( - 12 ) + 4}{2 \cdot 32} : \frac{1}{4}

\frac{- ( - 12 ) + 4}{2 \cdot 32}

Apply rule

- (- a) = a

= \frac{12 + 4}{2 \cdot 32}

Add the numbers:

12 + 4 = 16

= \frac{16}{2 \cdot 32}

Multiply the numbers:

2 \cdot 32 = 64

= \frac{16}{64}

Cancel the common factor:

16

= \frac{1}{4}

v = \frac{- ( - 12 ) - 4}{2 \cdot 32} : \frac{1}{8}

\frac{- ( - 12 ) - 4}{2 \cdot 32}

Apply rule

- (- a) = a

= \frac{12 - 4}{2 \cdot 32}

Subtract the numbers:

12 - 4 = 8

= \frac{8}{2 \cdot 32}

Multiply the numbers:

2 \cdot 32 = 64

= \frac{8}{64}

Cancel the common factor:

8

= \frac{1}{8}

The solutions to the quadratic equation are:

v = \frac{1}{4}, v = \frac{1}{8}

v = \frac{1}{4}, v = \frac{1}{8}

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = \frac{1}{4} : u = \frac{1}{2}, u = - \frac{1}{2}

u^{2} = \frac{1}{4}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{4}, u = - \frac{1}{4}

\frac{1}{4} = \frac{1}{2}

\frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

Apply rule

1 = 1

= \frac{1}{2}

- \frac{1}{4} = - \frac{1}{2}

- \frac{1}{4}

Simplify

\frac{1}{4} : \frac{1}{2}

\frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

= - \frac{1}{2}

Apply rule

1 = 1

= - \frac{1}{2}

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

Solve

u^{2} = \frac{1}{8} : u = \frac{2}{4}, u = - \frac{2}{4}

u^{2} = \frac{1}{8}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{8}, u = - \frac{1}{8}

\frac{1}{8} = \frac{2}{4}

\frac{1}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{1}{2 2}

Apply rule

1 = 1

= \frac{1}{2 2}

Rationalize

\frac{1}{2 2} : \frac{2}{4}

\frac{1}{2 2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2}{4}

= \frac{2}{4}

- \frac{1}{8} = - \frac{2}{4}

- \frac{1}{8}

Simplify

\frac{1}{8} : \frac{1}{2 2}

\frac{1}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{1}{2 2}

= - \frac{1}{2 2}

Apply rule

1 = 1

= - \frac{1}{2 2}

Rationalize

- \frac{1}{2 2} : - \frac{2}{4}

- \frac{1}{2 2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{2}{4}

= - \frac{2}{4}

u = \frac{2}{4}, u = - \frac{2}{4}

The solutions are

u = \frac{1}{2}, u = - \frac{1}{2}, u = \frac{2}{4}, u = - \frac{2}{4}

Substitute back

u = sin (x)

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}, sin (x) = \frac{2}{4}, sin (x) = - \frac{2}{4}

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}, sin (x) = \frac{2}{4}, sin (x) = - \frac{2}{4}

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sin (x) = - \frac{1}{2}

General solutions for

sin (x) = - \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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

sin (x) = \frac{2}{4}

Apply trig inverse properties

sin (x) = \frac{2}{4}

General solutions for

sin (x) = \frac{2}{4}

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

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

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

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

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = arcsin (\frac{2}{4}) + 2 πn, x = π - arcsin (\frac{2}{4}) + 2 πn, x = arcsin (- \frac{2}{4}) + 2 πn, x = π + arcsin (\frac{2}{4}) + 2 πn

Show solutions in decimal form

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = 0.36136 \dots + 2 πn, x = π - 0.36136 \dots + 2 πn, x = - 0.36136 \dots + 2 πn, x = π + 0.36136 \dots + 2 πn

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