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2-1/2 sin(3x+330)=(8-sqrt(3))/4

Solution

2 - \frac{1}{2} sin (3 x + 330) = \frac{8 - 3}{4}

Solution

x = - 110 + \frac{2 πn}{3} + \frac{π}{9}, x = \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} - 110

Degrees

x = - 6282.53574 \dots^{\circ} + 12 0^{\circ} n, x = - 6262.53574 \dots^{\circ} + 12 0^{\circ} n

Solution steps

2 - \frac{1}{2} sin (3 x + 330) = \frac{8 - 3}{4}

Move

2

to the right side

2 - \frac{1}{2} sin (3 x + 330) = \frac{8 - 3}{4}

Subtract

2

from both sides

2 - \frac{1}{2} sin (3 x + 330) - 2 = \frac{8 - 3}{4} - 2

Simplify

- \frac{1}{2} sin (3 x + 330) = \frac{8 - 3}{4} - 2

- \frac{1}{2} sin (3 x + 330) = \frac{8 - 3}{4} - 2

Multiply both sides by

- 2

- \frac{1}{2} sin (3 x + 330) = \frac{8 - 3}{4} - 2

Multiply both sides by

- 2

(- \frac{1}{2} sin (3 x + 330)) (- 2) = \frac{8 - 3}{4} (- 2) - 2 (- 2)

Simplify

(- \frac{1}{2} sin (3 x + 330)) (- 2) = \frac{8 - 3}{4} (- 2) - 2 (- 2)

Simplify

(- \frac{1}{2} sin (3 x + 330)) (- 2) : sin (3 x + 330)

(- \frac{1}{2} sin (3 x + 330)) (- 2)

Remove parentheses:

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

= \frac{1}{2} sin (3 x + 330) \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2} sin (3 x + 330)

Cancel the common factor:

2

= sin (3 x + 330) \cdot 1

Multiply:

sin (3 x + 330) \cdot 1 = sin (3 x + 330)

= sin (3 x + 330)

Simplify

\frac{8 - 3}{4} (- 2) - 2 (- 2) : - \frac{8 - 3}{2} + 4

\frac{8 - 3}{4} (- 2) - 2 (- 2)

Remove parentheses:

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

= - \frac{8 - 3}{4} \cdot 2 + 2 \cdot 2

\frac{8 - 3}{4} \cdot 2 = \frac{8 - 3}{2}

\frac{8 - 3}{4} \cdot 2

Multiply fractions:

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

= \frac{( 8 - 3 ) \cdot 2}{4}

Cancel the common factor:

2

= \frac{8 - 3}{2}

2 \cdot 2 = 4

2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

= - \frac{8 - 3}{2} + 4

sin (3 x + 330) = - \frac{8 - 3}{2} + 4

sin (3 x + 330) = - \frac{8 - 3}{2} + 4

sin (3 x + 330) = - \frac{8 - 3}{2} + 4

Apply trig inverse properties

sin (3 x + 330) = - \frac{8 - 3}{2} + 4

General solutions for

sin (3 x + 330) = - \frac{8 - 3}{2} + 4

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

3 x + 330 = arcsin (- \frac{8 - 3}{2} + 4) + 2 πn, 3 x + 330 = π - arcsin (- \frac{8 - 3}{2} + 4) + 2 πn

3 x + 330 = arcsin (- \frac{8 - 3}{2} + 4) + 2 πn, 3 x + 330 = π - arcsin (- \frac{8 - 3}{2} + 4) + 2 πn

Solve

3 x + 330 = arcsin (- \frac{8 - 3}{2} + 4) + 2 πn : x = - 110 + \frac{2 πn}{3} + \frac{π}{9}

3 x + 330 = arcsin (- \frac{8 - 3}{2} + 4) + 2 πn

Simplify

arcsin (- \frac{8 - 3}{2} + 4) + 2 πn : \frac{π}{3} + 2 πn

arcsin (- \frac{8 - 3}{2} + 4) + 2 πn

Use the following trivial identity:

arcsin (- \frac{8 - 3}{2} + 4) = \frac{π}{3}

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{3} + 2 πn

3 x + 330 = \frac{π}{3} + 2 πn

Move

330

to the right side

3 x + 330 = \frac{π}{3} + 2 πn

Subtract

330

from both sides

3 x + 330 - 330 = \frac{π}{3} + 2 πn - 330

Simplify

3 x = \frac{π}{3} + 2 πn - 330

3 x = \frac{π}{3} + 2 πn - 330

Divide both sides by

3

3 x = \frac{π}{3} + 2 πn - 330

Divide both sides by

3

\frac{3 x}{3} = \frac{\frac{π}{3}}{3} + \frac{2 πn}{3} - \frac{330}{3}

Simplify

\frac{3 x}{3} = \frac{\frac{π}{3}}{3} + \frac{2 πn}{3} - \frac{330}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{π}{3}}{3} + \frac{2 πn}{3} - \frac{330}{3} : - 110 + \frac{2 πn}{3} + \frac{π}{9}

\frac{\frac{π}{3}}{3} + \frac{2 πn}{3} - \frac{330}{3}

Group like terms

= - \frac{330}{3} + \frac{2 πn}{3} + \frac{\frac{π}{3}}{3}

\frac{330}{3} = 110

\frac{330}{3}

Divide the numbers:

\frac{330}{3} = 110

= 110

\frac{\frac{π}{3}}{3} = \frac{π}{9}

\frac{\frac{π}{3}}{3}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{π}{9}

= - 110 + \frac{2 πn}{3} + \frac{π}{9}

x = - 110 + \frac{2 πn}{3} + \frac{π}{9}

x = - 110 + \frac{2 πn}{3} + \frac{π}{9}

x = - 110 + \frac{2 πn}{3} + \frac{π}{9}

Solve

3 x + 330 = π - arcsin (- \frac{8 - 3}{2} + 4) + 2 πn : x = \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} - 110

3 x + 330 = π - arcsin (- \frac{8 - 3}{2} + 4) + 2 πn

Simplify

π - arcsin (- \frac{8 - 3}{2} + 4) + 2 πn : π - \frac{π}{3} + 2 πn

π - arcsin (- \frac{8 - 3}{2} + 4) + 2 πn

Use the following trivial identity:

arcsin (- \frac{8 - 3}{2} + 4) = \frac{π}{3}

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= π - \frac{π}{3} + 2 πn

3 x + 330 = π - \frac{π}{3} + 2 πn

Move

330

to the right side

3 x + 330 = π - \frac{π}{3} + 2 πn

Subtract

330

from both sides

3 x + 330 - 330 = π - \frac{π}{3} + 2 πn - 330

Simplify

3 x = π - \frac{π}{3} + 2 πn - 330

3 x = π - \frac{π}{3} + 2 πn - 330

Divide both sides by

3

3 x = π - \frac{π}{3} + 2 πn - 330

Divide both sides by

3

\frac{3 x}{3} = \frac{π}{3} - \frac{\frac{π}{3}}{3} + \frac{2 πn}{3} - \frac{330}{3}

Simplify

\frac{3 x}{3} = \frac{π}{3} - \frac{\frac{π}{3}}{3} + \frac{2 πn}{3} - \frac{330}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{π}{3} - \frac{\frac{π}{3}}{3} + \frac{2 πn}{3} - \frac{330}{3} : \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} - 110

\frac{π}{3} - \frac{\frac{π}{3}}{3} + \frac{2 πn}{3} - \frac{330}{3}

Group like terms

= \frac{π}{3} - \frac{330}{3} + \frac{2 πn}{3} - \frac{\frac{π}{3}}{3}

\frac{330}{3} = 110

\frac{330}{3}

Divide the numbers:

\frac{330}{3} = 110

= 110

\frac{\frac{π}{3}}{3} = \frac{π}{9}

\frac{\frac{π}{3}}{3}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{π}{9}

= \frac{π}{3} - 110 + \frac{2 πn}{3} - \frac{π}{9}

Group like terms

= \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} - 110

x = \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} - 110

x = \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} - 110

x = \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} - 110

x = - 110 + \frac{2 πn}{3} + \frac{π}{9}, x = \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} - 110

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