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Popular >

12sin^2(x)+8sin(x)+1=0

Solution

12 sin^{2} (x) + 8 sin (x) + 1 = 0

Solution

x = - 0.16744 \dots + 2 πn, x = π + 0.16744 \dots + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = - 9.59406 \dots^{\circ} + 36 0^{\circ} n, x = 189.59406 \dots^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

12 sin^{2} (x) + 8 sin (x) + 1 = 0

Solve by substitution

12 sin^{2} (x) + 8 sin (x) + 1 = 0

Let:

sin (x) = u

12 u^{2} + 8 u + 1 = 0

12 u^{2} + 8 u + 1 = 0 : u = - \frac{1}{6}, u = - \frac{1}{2}

12 u^{2} + 8 u + 1 = 0

Solve with the quadratic formula

12 u^{2} + 8 u + 1 = 0

Quadratic Equation Formula:

For

a = 12, b = 8, c = 1

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 12 \cdot 1}{2 \cdot 12}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 12 \cdot 1}{2 \cdot 12}

8^{2} - 4 \cdot 12 \cdot 1 = 4

8^{2} - 4 \cdot 12 \cdot 1

Multiply the numbers:

4 \cdot 12 \cdot 1 = 48

= 8^{2} - 48

8^{2} = 64

= 64 - 48

Subtract the numbers:

64 - 48 = 16

= 16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

u_{1, 2} = \frac{- 8 \pm 4}{2 \cdot 12}

Separate the solutions

u_{1} = \frac{- 8 + 4}{2 \cdot 12}, u_{2} = \frac{- 8 - 4}{2 \cdot 12}

u = \frac{- 8 + 4}{2 \cdot 12} : - \frac{1}{6}

\frac{- 8 + 4}{2 \cdot 12}

Add/Subtract the numbers:

- 8 + 4 = - 4

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

Multiply the numbers:

2 \cdot 12 = 24

= \frac{- 4}{24}

Apply the fraction rule:

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

= - \frac{4}{24}

Cancel the common factor:

4

= - \frac{1}{6}

u = \frac{- 8 - 4}{2 \cdot 12} : - \frac{1}{2}

\frac{- 8 - 4}{2 \cdot 12}

Subtract the numbers:

- 8 - 4 = - 12

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

Multiply the numbers:

2 \cdot 12 = 24

= \frac{- 12}{24}

Apply the fraction rule:

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

= - \frac{12}{24}

Cancel the common factor:

12

= - \frac{1}{2}

The solutions to the quadratic equation are:

u = - \frac{1}{6}, u = - \frac{1}{2}

Substitute back

u = sin (x)

sin (x) = - \frac{1}{6}, sin (x) = - \frac{1}{2}

sin (x) = - \frac{1}{6}, sin (x) = - \frac{1}{2}

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

sin (x) = - \frac{1}{6}

Apply trig inverse properties

sin (x) = - \frac{1}{6}

General solutions for

sin (x) = - \frac{1}{6}

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

x = arcsin (- \frac{1}{6}) + 2 πn, x = π + arcsin (\frac{1}{6}) + 2 πn

x = arcsin (- \frac{1}{6}) + 2 πn, x = π + arcsin (\frac{1}{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

Combine all the solutions

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

Show solutions in decimal form

x = - 0.16744 \dots + 2 πn, x = π + 0.16744 \dots + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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