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Popular >

sin(8x)=sin(6x)

Solution

sin (8 x) = sin (6 x)

Solution

x = \frac{π}{14} + \frac{2 πn}{7}, x = \frac{3 π}{14} + \frac{2 πn}{7}, x = 2 πn, x = π + 2 πn

Degrees

x = 12.85714 \dots^{\circ} + 51.42857 \dots^{\circ} n, x = 38.57142 \dots^{\circ} + 51.42857 \dots^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (8 x) = sin (6 x)

Subtract

sin (6 x)

from both sides

sin (8 x) - sin (6 x) = 0

Rewrite using trig identities

- sin (6 x) + sin (8 x)

Use the Sum to Product identity:

sin (s) - sin (t) = 2 sin (\frac{s - t}{2}) cos (\frac{s + t}{2})

= 2 sin (\frac{8 x - 6 x}{2}) cos (\frac{8 x + 6 x}{2})

Simplify

2 sin (\frac{8 x - 6 x}{2}) cos (\frac{8 x + 6 x}{2}) : 2 sin (x) cos (7 x)

2 sin (\frac{8 x - 6 x}{2}) cos (\frac{8 x + 6 x}{2})

\frac{8 x - 6 x}{2} = x

\frac{8 x - 6 x}{2}

Add similar elements:

8 x - 6 x = 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

= 2 sin (x) cos (\frac{8 x + 6 x}{2})

\frac{8 x + 6 x}{2} = 7 x

\frac{8 x + 6 x}{2}

Add similar elements:

8 x + 6 x = 14 x

= \frac{14 x}{2}

Divide the numbers:

\frac{14}{2} = 7

= 7 x

= 2 sin (x) cos (7 x)

= 2 sin (x) cos (7 x)

2 cos (7 x) sin (x) = 0

Solving each part separately

cos (7 x) = 0 or sin (x) = 0

cos (7 x) = 0 : x = \frac{π}{14} + \frac{2 πn}{7}, x = \frac{3 π}{14} + \frac{2 πn}{7}

cos (7 x) = 0

General solutions for

cos (7 x) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

7 x = \frac{π}{2} + 2 πn, 7 x = \frac{3 π}{2} + 2 πn

7 x = \frac{π}{2} + 2 πn, 7 x = \frac{3 π}{2} + 2 πn

Solve

7 x = \frac{π}{2} + 2 πn : x = \frac{π}{14} + \frac{2 πn}{7}

7 x = \frac{π}{2} + 2 πn

Divide both sides by

7

7 x = \frac{π}{2} + 2 πn

Divide both sides by

7

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

Simplify

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

Simplify

\frac{7 x}{7} : x

\frac{7 x}{7}

Divide the numbers:

\frac{7}{7} = 1

= x

Simplify

\frac{\frac{π}{2}}{7} + \frac{2 πn}{7} : \frac{π}{14} + \frac{2 πn}{7}

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

\frac{\frac{π}{2}}{7} = \frac{π}{14}

\frac{\frac{π}{2}}{7}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{π}{14}

= \frac{π}{14} + \frac{2 πn}{7}

x = \frac{π}{14} + \frac{2 πn}{7}

x = \frac{π}{14} + \frac{2 πn}{7}

x = \frac{π}{14} + \frac{2 πn}{7}

Solve

7 x = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{14} + \frac{2 πn}{7}

7 x = \frac{3 π}{2} + 2 πn

Divide both sides by

7

7 x = \frac{3 π}{2} + 2 πn

Divide both sides by

7

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

Simplify

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

Simplify

\frac{7 x}{7} : x

\frac{7 x}{7}

Divide the numbers:

\frac{7}{7} = 1

= x

Simplify

\frac{\frac{3 π}{2}}{7} + \frac{2 πn}{7} : \frac{3 π}{14} + \frac{2 πn}{7}

\frac{\frac{3 π}{2}}{7} + \frac{2 πn}{7}

\frac{\frac{3 π}{2}}{7} = \frac{3 π}{14}

\frac{\frac{3 π}{2}}{7}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{3 π}{14}

= \frac{3 π}{14} + \frac{2 πn}{7}

x = \frac{3 π}{14} + \frac{2 πn}{7}

x = \frac{3 π}{14} + \frac{2 πn}{7}

x = \frac{3 π}{14} + \frac{2 πn}{7}

x = \frac{π}{14} + \frac{2 πn}{7}, x = \frac{3 π}{14} + \frac{2 πn}{7}

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

Combine all the solutions

x = \frac{π}{14} + \frac{2 πn}{7}, x = \frac{3 π}{14} + \frac{2 πn}{7}, x = 2 πn, x = π + 2 πn

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