What is the general solution for 0=-sqrt(2)sin(x)-1 ?

The general solution for 0=-sqrt(2)sin(x)-1 is x=(5pi)/4+2pin,x=(7pi)/4+2pin

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0=-sqrt(2)sin(x)-1

0 = - 2 sin (x) - 1

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

0 = - 2 sin (x) - 1

Switch sides

- 2 sin (x) - 1 = 0

Move

1

to the right side

- 2 sin (x) - 1 = 0

Add

1

to both sides

- 2 sin (x) - 1 + 1 = 0 + 1

Simplify

- 2 sin (x) = 1

- 2 sin (x) = 1

Divide both sides by

- 2

- 2 sin (x) = 1

Divide both sides by

- 2

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

Simplify

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

Simplify

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

2

= sin (x)

Simplify

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

\frac{1}{- 2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

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

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

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

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

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

General solutions for

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

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x, z = sin (2 x + 3 y)

2 sin (3 θ) = 2

2 tan (x) csc (x) + 2 csc (x) + tan (x) = - 1

4 cos (x) + 1 = 2 cos (x)

cos (2 x) + cos (4 x) + cos (6 x) = 0

What is the general solution for 0=-sqrt(2)sin(x)-1 ?
The general solution for 0=-sqrt(2)sin(x)-1 is x=(5pi)/4+2pin,x=(7pi)/4+2pin