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

The general solution for sqrt(2)sin(x/2)-1=0 is x= pi/2+4pin,x=(3pi)/2+4pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Solution

2 sin (\frac{x}{2}) - 1 = 0

Solution

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

Degrees

x = 9 0^{\circ} + 72 0^{\circ} n, x = 27 0^{\circ} + 72 0^{\circ} n

Solution steps

2 sin (\frac{x}{2}) - 1 = 0

Move

1

to the right side

2 sin (\frac{x}{2}) - 1 = 0

Add

1

to both sides

2 sin (\frac{x}{2}) - 1 + 1 = 0 + 1

Simplify

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

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

Simplify

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

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

Cancel the common factor:

2

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

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

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

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

General solutions for

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

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

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

Solve

\frac{x}{2} = \frac{π}{4} + 2 πn : x = \frac{π}{2} + 4 πn

\frac{x}{2} = \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{4} + 2 \cdot 2 πn : \frac{π}{2} + 4 πn

2 \cdot \frac{π}{4} + 2 \cdot 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn

Solve

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

\frac{x}{2} = \frac{3 π}{4} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{3 π}{4} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn : \frac{3 π}{2} + 4 πn

2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{4} = \frac{3 π}{2}

2 \cdot \frac{3 π}{4}

Multiply fractions:

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

= \frac{3 π 2}{4}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{3 π}{2} + 4 πn

x = \frac{3 π}{2} + 4 πn

x = \frac{3 π}{2} + 4 πn

x = \frac{3 π}{2} + 4 πn

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

Popular Examples

cos^2(x)-sin^2(x)+sin(x)=0

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

2tan^2(t)=-3sec(t)

2 tan^{2} (t) = - 3 sec (t)

tan^2(x)+3=0

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

4cos(θ)+1=0

4 cos (θ) + 1 = 0

cos(a)= 3/(sqrt(-6))

cos (a) = \frac{3}{- 6}

Frequently Asked Questions (FAQ)

What is the general solution for sqrt(2)sin(x/2)-1=0 ?
The general solution for sqrt(2)sin(x/2)-1=0 is x= pi/2+4pin,x=(3pi)/2+4pin