What is the general solution for sin(6x)=-1/2 ?

The general solution for sin(6x)=-1/2 is x=(7pi}{36}+(pin)/3 ,x=(11pi)/(36)+\frac{pin)/3

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sin(6x)=-1/2

Solution

sin (6 x) = - \frac{1}{2}

Solution

x = \frac{7 π}{36} + \frac{πn}{3}, x = \frac{11 π}{36} + \frac{πn}{3}

Degrees

x = 3 5^{\circ} + 6 0^{\circ} n, x = 5 5^{\circ} + 6 0^{\circ} n

Solution steps

sin (6 x) = - \frac{1}{2}

General solutions for

sin (6 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}

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

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

Solve

6 x = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{36} + \frac{πn}{3}

6 x = \frac{7 π}{6} + 2 πn

Divide both sides by

6

6 x = \frac{7 π}{6} + 2 πn

Divide both sides by

6

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

Simplify

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

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

\frac{\frac{7 π}{6}}{6} + \frac{2 πn}{6} : \frac{7 π}{36} + \frac{πn}{3}

\frac{\frac{7 π}{6}}{6} + \frac{2 πn}{6}

\frac{\frac{7 π}{6}}{6} = \frac{7 π}{36}

\frac{\frac{7 π}{6}}{6}

Apply the fraction rule:

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

= \frac{7 π}{6 \cdot 6}

Multiply the numbers:

6 \cdot 6 = 36

= \frac{7 π}{36}

\frac{2 πn}{6} = \frac{πn}{3}

\frac{2 πn}{6}

Cancel the common factor:

2

= \frac{πn}{3}

= \frac{7 π}{36} + \frac{πn}{3}

x = \frac{7 π}{36} + \frac{πn}{3}

x = \frac{7 π}{36} + \frac{πn}{3}

x = \frac{7 π}{36} + \frac{πn}{3}

Solve

6 x = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{36} + \frac{πn}{3}

6 x = \frac{11 π}{6} + 2 πn

Divide both sides by

6

6 x = \frac{11 π}{6} + 2 πn

Divide both sides by

6

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

Simplify

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

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

\frac{\frac{11 π}{6}}{6} + \frac{2 πn}{6} : \frac{11 π}{36} + \frac{πn}{3}

\frac{\frac{11 π}{6}}{6} + \frac{2 πn}{6}

\frac{\frac{11 π}{6}}{6} = \frac{11 π}{36}

\frac{\frac{11 π}{6}}{6}

Apply the fraction rule:

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

= \frac{11 π}{6 \cdot 6}

Multiply the numbers:

6 \cdot 6 = 36

= \frac{11 π}{36}

\frac{2 πn}{6} = \frac{πn}{3}

\frac{2 πn}{6}

Cancel the common factor:

2

= \frac{πn}{3}

= \frac{11 π}{36} + \frac{πn}{3}

x = \frac{11 π}{36} + \frac{πn}{3}

x = \frac{11 π}{36} + \frac{πn}{3}

x = \frac{11 π}{36} + \frac{πn}{3}

x = \frac{7 π}{36} + \frac{πn}{3}, x = \frac{11 π}{36} + \frac{πn}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(6x)=-1/2 ?
The general solution for sin(6x)=-1/2 is x=(7pi}{36}+(pin)/3 ,x=(11pi)/(36)+\frac{pin)/3