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Popular >

sin(9x)=sin(x)

Solution

sin (9 x) = sin (x)

Solution

x = \frac{π}{10} + \frac{2 πn}{5}, x = \frac{3 π}{10} + \frac{2 πn}{5}, x = \frac{πn}{2}, x = \frac{π}{4} + \frac{πn}{2}

Degrees

x = 1 8^{\circ} + 7 2^{\circ} n, x = 5 4^{\circ} + 7 2^{\circ} n, x = 0^{\circ} + 9 0^{\circ} n, x = 4 5^{\circ} + 9 0^{\circ} n

Solution steps

sin (9 x) = sin (x)

Subtract

sin (x)

from both sides

sin (9 x) - sin (x) = 0

Rewrite using trig identities

sin (9 x) - sin (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{9 x - x}{2}) cos (\frac{9 x + x}{2})

Simplify

2 sin (\frac{9 x - x}{2}) cos (\frac{9 x + x}{2}) : 2 sin (4 x) cos (5 x)

2 sin (\frac{9 x - x}{2}) cos (\frac{9 x + x}{2})

\frac{9 x - x}{2} = 4 x

\frac{9 x - x}{2}

Add similar elements:

9 x - x = 8 x

= \frac{8 x}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4 x

= 2 sin (4 x) cos (\frac{9 x + x}{2})

\frac{9 x + x}{2} = 5 x

\frac{9 x + x}{2}

Add similar elements:

9 x + x = 10 x

= \frac{10 x}{2}

Divide the numbers:

\frac{10}{2} = 5

= 5 x

= 2 sin (4 x) cos (5 x)

= 2 sin (4 x) cos (5 x)

2 cos (5 x) sin (4 x) = 0

Solving each part separately

cos (5 x) = 0 or sin (4 x) = 0

cos (5 x) = 0 : x = \frac{π}{10} + \frac{2 πn}{5}, x = \frac{3 π}{10} + \frac{2 πn}{5}

cos (5 x) = 0

General solutions for

cos (5 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}

5 x = \frac{π}{2} + 2 πn, 5 x = \frac{3 π}{2} + 2 πn

5 x = \frac{π}{2} + 2 πn, 5 x = \frac{3 π}{2} + 2 πn

Solve

5 x = \frac{π}{2} + 2 πn : x = \frac{π}{10} + \frac{2 πn}{5}

5 x = \frac{π}{2} + 2 πn

Divide both sides by

5

5 x = \frac{π}{2} + 2 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{\frac{π}{2}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 x}{5} = \frac{\frac{π}{2}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{\frac{π}{2}}{5} + \frac{2 πn}{5} : \frac{π}{10} + \frac{2 πn}{5}

\frac{\frac{π}{2}}{5} + \frac{2 πn}{5}

\frac{\frac{π}{2}}{5} = \frac{π}{10}

\frac{\frac{π}{2}}{5}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{π}{10}

= \frac{π}{10} + \frac{2 πn}{5}

x = \frac{π}{10} + \frac{2 πn}{5}

x = \frac{π}{10} + \frac{2 πn}{5}

x = \frac{π}{10} + \frac{2 πn}{5}

Solve

5 x = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{10} + \frac{2 πn}{5}

5 x = \frac{3 π}{2} + 2 πn

Divide both sides by

5

5 x = \frac{3 π}{2} + 2 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 x}{5} = \frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5} : \frac{3 π}{10} + \frac{2 πn}{5}

\frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5}

\frac{\frac{3 π}{2}}{5} = \frac{3 π}{10}

\frac{\frac{3 π}{2}}{5}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{3 π}{10}

= \frac{3 π}{10} + \frac{2 πn}{5}

x = \frac{3 π}{10} + \frac{2 πn}{5}

x = \frac{3 π}{10} + \frac{2 πn}{5}

x = \frac{3 π}{10} + \frac{2 πn}{5}

x = \frac{π}{10} + \frac{2 πn}{5}, x = \frac{3 π}{10} + \frac{2 πn}{5}

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

sin (4 x) = 0

General solutions for

sin (4 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}

4 x = 0 + 2 πn, 4 x = π + 2 πn

4 x = 0 + 2 πn, 4 x = π + 2 πn

Solve

4 x = 0 + 2 πn : x = \frac{πn}{2}

4 x = 0 + 2 πn

0 + 2 πn = 2 πn

4 x = 2 πn

Divide both sides by

4

4 x = 2 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{2 πn}{4}

Simplify

x = \frac{πn}{2}

x = \frac{πn}{2}

Solve

4 x = π + 2 πn : x = \frac{π}{4} + \frac{πn}{2}

4 x = π + 2 πn

Divide both sides by

4

4 x = π + 2 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{π}{4} + \frac{2 πn}{4}

Simplify

x = \frac{π}{4} + \frac{πn}{2}

x = \frac{π}{4} + \frac{πn}{2}

x = \frac{πn}{2}, x = \frac{π}{4} + \frac{πn}{2}

Combine all the solutions

x = \frac{π}{10} + \frac{2 πn}{5}, x = \frac{3 π}{10} + \frac{2 πn}{5}, x = \frac{πn}{2}, x = \frac{π}{4} + \frac{πn}{2}

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