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

The general solution for sin(6x)=1 is x= pi/(12)+(pin)/3

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sin(6x)=1

sin (6 x) = 1

x = \frac{π}{12} + \frac{πn}{3}

Degrees

x = 1 5^{\circ} + 6 0^{\circ} n

Solution steps

sin (6 x) = 1

General solutions for

sin (6 x) = 1

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{π}{2} + 2 πn

6 x = \frac{π}{2} + 2 πn

Solve

6 x = \frac{π}{2} + 2 πn : x = \frac{π}{12} + \frac{πn}{3}

6 x = \frac{π}{2} + 2 πn

Divide both sides by

6

6 x = \frac{π}{2} + 2 πn

Divide both sides by

6

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

Simplify

\frac{6 x}{6} = \frac{\frac{π}{2}}{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{π}{2}}{6} + \frac{2 πn}{6} : \frac{π}{12} + \frac{πn}{3}

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

\frac{\frac{π}{2}}{6} = \frac{π}{12}

\frac{\frac{π}{2}}{6}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{π}{12}

\frac{2 πn}{6} = \frac{πn}{3}

\frac{2 πn}{6}

Cancel the common factor:

2

= \frac{πn}{3}

= \frac{π}{12} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}

3 sin (θ) = 2 cos (θ)

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

tan (θ) = \frac{12}{6}

tan (θ) = \frac{12}{7}

2 sin (x) + \frac{3}{4} = \frac{3}{2}, (0, 2 π)

What is the general solution for sin(6x)=1 ?
The general solution for sin(6x)=1 is x= pi/(12)+(pin)/3