What is the general solution for sin(θ/2)=(sqrt(2))/2 ?

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

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sin(θ/2)=(sqrt(2))/2

Solution

sin (\frac{θ}{2}) = \frac{2}{2}

Solution

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

Degrees

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

Solution steps

sin (\frac{θ}{2}) = \frac{2}{2}

General solutions for

sin (\frac{θ}{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{θ}{2} = \frac{π}{4} + 2 πn, \frac{θ}{2} = \frac{3 π}{4} + 2 πn

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

Solve

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

\frac{θ}{2} = \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{θ}{2} = \frac{π}{4} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

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

θ = \frac{π}{2} + 4 πn

θ = \frac{π}{2} + 4 πn

θ = \frac{π}{2} + 4 πn

Solve

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

\frac{θ}{2} = \frac{3 π}{4} + 2 πn

Multiply both sides by

2

\frac{θ}{2} = \frac{3 π}{4} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

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

θ = \frac{3 π}{2} + 4 πn

θ = \frac{3 π}{2} + 4 πn

θ = \frac{3 π}{2} + 4 πn

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

Popular Examples

9tan(3x)=9

9 tan (3 x) = 9

sin(θ)= 40/41

sin (θ) = \frac{40}{41}

cos(2x)-sin^2(x)= 1/4

cos (2 x) - sin^{2} (x) = \frac{1}{4}

sec(θ)csc(θ)+sqrt(2)csc(θ)=0

sec (θ) csc (θ) + 2 csc (θ) = 0

2sin^2(θ)-sin(θ)-1=0,0<= θ<= 2pi

2 sin^{2} (θ) - sin (θ) - 1 = 0, 0 \leq θ \leq 2 π

Frequently Asked Questions (FAQ)

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