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

The general solution for sin(1/2 x)=-1/2 is x=(7pi)/3+4pin,x=(11pi)/3+4pin

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

Solution

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

Solution

x = \frac{7 π}{3} + 4 πn, x = \frac{11 π}{3} + 4 πn

Degrees

x = 42 0^{\circ} + 72 0^{\circ} n, x = 66 0^{\circ} + 72 0^{\circ} n

Solution steps

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

General solutions for

sin (\frac{1}{2} 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}

\frac{1}{2} x = \frac{7 π}{6} + 2 πn, \frac{1}{2} x = \frac{11 π}{6} + 2 πn

\frac{1}{2} x = \frac{7 π}{6} + 2 πn, \frac{1}{2} x = \frac{11 π}{6} + 2 πn

Solve

\frac{1}{2} x = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{3} + 4 πn

\frac{1}{2} x = \frac{7 π}{6} + 2 πn

Multiply both sides by

2

\frac{1}{2} x = \frac{7 π}{6} + 2 πn

Multiply both sides by

2

2 \cdot \frac{1}{2} x = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x : x

2 \cdot \frac{1}{2} x

Multiply fractions:

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

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

Cancel the common factor:

2

= x \cdot 1

Multiply:

x \cdot 1 = x

= x

Simplify

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn : \frac{7 π}{3} + 4 πn

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{7 π}{6} = \frac{7 π}{3}

2 \cdot \frac{7 π}{6}

Multiply fractions:

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

= \frac{7 π 2}{6}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14 π}{6}

Cancel the common factor:

2

= \frac{7 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn

Solve

\frac{1}{2} x = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{3} + 4 πn

\frac{1}{2} x = \frac{11 π}{6} + 2 πn

Multiply both sides by

2

\frac{1}{2} x = \frac{11 π}{6} + 2 πn

Multiply both sides by

2

2 \cdot \frac{1}{2} x = 2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x = 2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x : x

2 \cdot \frac{1}{2} x

Multiply fractions:

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

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

Cancel the common factor:

2

= x \cdot 1

Multiply:

x \cdot 1 = x

= x

Simplify

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn : \frac{11 π}{3} + 4 πn

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{11 π}{6} = \frac{11 π}{3}

2 \cdot \frac{11 π}{6}

Multiply fractions:

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

= \frac{11 π 2}{6}

Multiply the numbers:

11 \cdot 2 = 22

= \frac{22 π}{6}

Cancel the common factor:

2

= \frac{11 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

x = \frac{11 π}{3} + 4 πn

x = \frac{7 π}{3} + 4 πn, x = \frac{11 π}{3} + 4 πn

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3sin(60-(3x)/4)=5sin((3x)/4-30)

3 sin (6 0^{\circ} - \frac{3 x}{4}) = 5 sin (\frac{3 x}{4} - 3 0^{\circ})

tan(u)=(sqrt(3))/3

tan (u) = \frac{3}{3}

(sin(θ))/(60)=(sin(80))/(280.87)

\frac{sin ( θ )}{60} = \frac{sin ( 8 0 ^{\circ} )}{280.87}

5.63=11cos((2pi)/T 0.137)

5.63 = 11 cos (\frac{2 π}{T} 0.137)

Frequently Asked Questions (FAQ)

What is the general solution for sin(1/2 x)=-1/2 ?
The general solution for sin(1/2 x)=-1/2 is x=(7pi)/3+4pin,x=(11pi)/3+4pin