What is the general solution for 1/2-sin(x)=1 ?

The general solution for 1/2-sin(x)=1 is x=(7pi)/6+2pin,x=(11pi)/6+2pin

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1/2-sin(x)=1

Solution

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

Solution

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

Degrees

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Move

\frac{1}{2}

to the right side

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

Subtract

\frac{1}{2}

from both sides

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

Simplify

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

Simplify

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

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

Add similar elements:

\frac{1}{2} - \frac{1}{2} = 0

= - sin (x)

Simplify

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

1 - \frac{1}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 2 - 1 = 1

1 \cdot 2 - 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= \frac{1}{2}

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

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

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

Divide both sides by

- 1

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

Divide both sides by

- 1

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

Simplify

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

Simplify

\frac{- sin ( x )}{- 1} : sin (x)

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sin (x)

Simplify

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

\frac{\frac{1}{2}}{- 1}

Apply the fraction rule:

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

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

Apply the fraction rule:

\frac{a}{1} = a

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

= - \frac{1}{2}

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

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

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

Popular Examples

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sin (t) + 2 = 3

tan(3θ)=-1

tan (3 θ) = - 1

1+2sin(2x)=0

1 + 2 sin (2 x) = 0

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2 sech^{2} (x) + tanh (x) = 0

8cos(2θ)=-24cos(θ)-16

8 cos (2 θ) = - 24 cos (θ) - 16

Frequently Asked Questions (FAQ)

What is the general solution for 1/2-sin(x)=1 ?
The general solution for 1/2-sin(x)=1 is x=(7pi)/6+2pin,x=(11pi)/6+2pin