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

The general solution for sqrt(2)sin(x)=1 is x= pi/4+2pin,x=(3pi)/4+2pin

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sqrt(2)sin(x)=1

2 sin (x) = 1

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

Solution steps

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}

Cancel the common factor:

2

= sin (x)

Simplify

\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}

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{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

4 sin^{2} (x) cot (x) - cot (x) = 0

2 sin (θ) - 1 = csc (θ)

sin^{2} (x) + 3 sin (x) - 4 = 0

sin (3 x) - sin (x) = cos (2 x)

2 tan^{2} (x) + 3 sec (x) = 0

What is the general solution for sqrt(2)sin(x)=1 ?
The general solution for sqrt(2)sin(x)=1 is x= pi/4+2pin,x=(3pi)/4+2pin