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Popular >

sin(x/3)cos(6x)-cos(x/3)sin(6x)=0

Solution

sin (\frac{x}{3}) cos (6 x) - cos (\frac{x}{3}) sin (6 x) = 0

Solution

x = - \frac{6 πn}{17}, x = - \frac{3 π}{17} - \frac{6 πn}{17}

Degrees

x = 0^{\circ} - 63.52941 \dots^{\circ} n, x = - 31.76470 \dots^{\circ} - 63.52941 \dots^{\circ} n

Solution steps

sin (\frac{x}{3}) cos (6 x) - cos (\frac{x}{3}) sin (6 x) = 0

Rewrite using trig identities

sin (\frac{x}{3}) cos (6 x) - cos (\frac{x}{3}) sin (6 x)

Use the Angle Difference identity:

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

= sin (\frac{x}{3} - 6 x)

sin (\frac{x}{3} - 6 x) = 0

General solutions for

sin (\frac{x}{3} - 6 x) = 0

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}{3} - 6 x = 0 + 2 πn, \frac{x}{3} - 6 x = π + 2 πn

\frac{x}{3} - 6 x = 0 + 2 πn, \frac{x}{3} - 6 x = π + 2 πn

Solve

\frac{x}{3} - 6 x = 0 + 2 πn : x = - \frac{6 πn}{17}

\frac{x}{3} - 6 x = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{3} - 6 x = 2 πn

Multiply both sides by

3

\frac{x}{3} - 6 x = 2 πn

Multiply both sides by

3

\frac{x}{3} \cdot 3 - 6 x \cdot 3 = 2 πn \cdot 3

Simplify

\frac{x}{3} \cdot 3 - 6 x \cdot 3 = 2 πn \cdot 3

Simplify

\frac{x}{3} \cdot 3 : x

\frac{x}{3} \cdot 3

Multiply fractions:

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

= \frac{x \cdot 3}{3}

Cancel the common factor:

3

= x

Simplify

- 6 x \cdot 3 : - 18 x

- 6 x \cdot 3

Multiply the numbers:

6 \cdot 3 = 18

= - 18 x

Simplify

2 πn \cdot 3 : 6 πn

2 πn \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6 πn

x - 18 x = 6 πn

- 17 x = 6 πn

- 17 x = 6 πn

- 17 x = 6 πn

Divide both sides by

- 17

- 17 x = 6 πn

Divide both sides by

- 17

\frac{- 17 x}{- 17} = \frac{6 πn}{- 17}

Simplify

x = - \frac{6 πn}{17}

x = - \frac{6 πn}{17}

Solve

\frac{x}{3} - 6 x = π + 2 πn : x = - \frac{3 π}{17} - \frac{6 πn}{17}

\frac{x}{3} - 6 x = π + 2 πn

Multiply both sides by

3

\frac{x}{3} - 6 x = π + 2 πn

Multiply both sides by

3

\frac{x}{3} \cdot 3 - 6 x \cdot 3 = π 3 + 2 πn \cdot 3

Simplify

\frac{x}{3} \cdot 3 - 6 x \cdot 3 = π 3 + 2 πn \cdot 3

Simplify

\frac{x}{3} \cdot 3 : x

\frac{x}{3} \cdot 3

Multiply fractions:

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

= \frac{x \cdot 3}{3}

Cancel the common factor:

3

= x

Simplify

- 6 x \cdot 3 : - 18 x

- 6 x \cdot 3

Multiply the numbers:

6 \cdot 3 = 18

= - 18 x

Simplify

π 3 : 3 π

π 3

Apply the commutative law:

π 3 = 3 π

3 π

Simplify

2 πn \cdot 3 : 6 πn

2 πn \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6 πn

x - 18 x = 3 π + 6 πn

- 17 x = 3 π + 6 πn

- 17 x = 3 π + 6 πn

- 17 x = 3 π + 6 πn

Divide both sides by

- 17

- 17 x = 3 π + 6 πn

Divide both sides by

- 17

\frac{- 17 x}{- 17} = \frac{3 π}{- 17} + \frac{6 πn}{- 17}

Simplify

\frac{- 17 x}{- 17} = \frac{3 π}{- 17} + \frac{6 πn}{- 17}

Simplify

\frac{- 17 x}{- 17} : x

\frac{- 17 x}{- 17}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{17 x}{17}

Divide the numbers:

\frac{17}{17} = 1

= x

Simplify

\frac{3 π}{- 17} + \frac{6 πn}{- 17} : - \frac{3 π}{17} - \frac{6 πn}{17}

\frac{3 π}{- 17} + \frac{6 πn}{- 17}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{3 π}{17} + \frac{6 πn}{- 17}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{3 π}{17} - \frac{6 πn}{17}

x = - \frac{3 π}{17} - \frac{6 πn}{17}

x = - \frac{3 π}{17} - \frac{6 πn}{17}

x = - \frac{3 π}{17} - \frac{6 πn}{17}

x = - \frac{6 πn}{17}, x = - \frac{3 π}{17} - \frac{6 πn}{17}

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