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Popular >

-sin(4x)=sin(8x)

Solution

- sin (4 x) = sin (8 x)

Solution

x = \frac{πn}{2}, x = \frac{π + 2 πn}{4}, x = \frac{π ( 3 n + 1 )}{6}, x = \frac{π ( 3 n + 2 )}{6}

Degrees

x = 0^{\circ} + 9 0^{\circ} n, x = 4 5^{\circ} + 9 0^{\circ} n, x = 3 0^{\circ} + 9 0^{\circ} n, x = 6 0^{\circ} + 9 0^{\circ} n

Solution steps

- sin (4 x) = sin (8 x)

Subtract

sin (8 x)

from both sides

- sin (4 x) - sin (8 x) = 0

Let:

u = 4 x

- sin (u) - sin (2 u) = 0

Rewrite using trig identities

- sin (2 u) - sin (u)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= - 2 sin (u) cos (u) - sin (u)

- sin (u) - 2 cos (u) sin (u) = 0

Factor

- sin (u) - 2 cos (u) sin (u) : - sin (u) (2 cos (u) + 1)

- sin (u) - 2 cos (u) sin (u)

Factor out common term

- sin (u)

= - sin (u) (1 + 2 cos (u))

- sin (u) (2 cos (u) + 1) = 0

Solving each part separately

sin (u) = 0 or 2 cos (u) + 1 = 0

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = 0

General solutions for

sin (u) = 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}

u = 0 + 2 πn, u = π + 2 πn

u = 0 + 2 πn, u = π + 2 πn

Solve

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn, u = π + 2 πn

2 cos (u) + 1 = 0 : u = \frac{2 π}{3} + 2 πn, u = \frac{4 π}{3} + 2 πn

2 cos (u) + 1 = 0

Move

1

to the right side

2 cos (u) + 1 = 0

Subtract

1

from both sides

2 cos (u) + 1 - 1 = 0 - 1

Simplify

2 cos (u) = - 1

2 cos (u) = - 1

Divide both sides by

2

2 cos (u) = - 1

Divide both sides by

2

\frac{2 cos ( u )}{2} = \frac{- 1}{2}

Simplify

cos (u) = - \frac{1}{2}

cos (u) = - \frac{1}{2}

General solutions for

cos (u) = - \frac{1}{2}

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}

u = \frac{2 π}{3} + 2 πn, u = \frac{4 π}{3} + 2 πn

u = \frac{2 π}{3} + 2 πn, u = \frac{4 π}{3} + 2 πn

Combine all the solutions

u = 2 πn, u = π + 2 πn, u = \frac{2 π}{3} + 2 πn, u = \frac{4 π}{3} + 2 πn

Substitute back

u = 4 x

4 x = 2 πn : x = \frac{πn}{2}

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}

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

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

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

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

\frac{π}{4} + \frac{2 πn}{4}

Apply rule

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

= \frac{π + 2 πn}{4}

x = \frac{π + 2 πn}{4}

x = \frac{π + 2 πn}{4}

x = \frac{π + 2 πn}{4}

4 x = \frac{2 π}{3} + 2 πn : x = \frac{π ( 3 n + 1 )}{6}

4 x = \frac{2 π}{3} + 2 πn

Divide both sides by

4

4 x = \frac{2 π}{3} + 2 πn

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{\frac{2 π}{3}}{4} + \frac{2 πn}{4} : \frac{π ( 3 n + 1 )}{6}

\frac{\frac{2 π}{3}}{4} + \frac{2 πn}{4}

Apply rule

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

= \frac{\frac{2 π}{3} + 2 πn}{4}

Join

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

\frac{2 π}{3} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 3}{3}

= \frac{2 π}{3} + \frac{2 πn \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{2 π + 2 πn \cdot 3}{3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{2 π + 6 πn}{3}

= \frac{\frac{2 π + 6 πn}{3}}{4}

Apply the fraction rule:

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

= \frac{2 π + 6 πn}{3 \cdot 4}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{2 π + 6 πn}{12}

Factor

2 π + 6 πn : 2 π (1 + 3 n)

2 π + 6 πn

Rewrite as

= 1 \cdot 2 π + 3 \cdot 2 πn

Factor out common term

2 π

= 2 π (1 + 3 n)

= \frac{2 π ( 1 + 3 n )}{12}

Cancel the common factor:

2

= \frac{π ( 3 n + 1 )}{6}

x = \frac{π ( 3 n + 1 )}{6}

x = \frac{π ( 3 n + 1 )}{6}

x = \frac{π ( 3 n + 1 )}{6}

4 x = \frac{4 π}{3} + 2 πn : x = \frac{π ( 3 n + 2 )}{6}

4 x = \frac{4 π}{3} + 2 πn

Divide both sides by

4

4 x = \frac{4 π}{3} + 2 πn

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{\frac{4 π}{3}}{4} + \frac{2 πn}{4} : \frac{π ( 3 n + 2 )}{6}

\frac{\frac{4 π}{3}}{4} + \frac{2 πn}{4}

Apply rule

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

= \frac{\frac{4 π}{3} + 2 πn}{4}

Join

\frac{4 π}{3} + 2 πn : \frac{4 π + 6 πn}{3}

\frac{4 π}{3} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 3}{3}

= \frac{4 π}{3} + \frac{2 πn \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{4 π + 2 πn \cdot 3}{3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{4 π + 6 πn}{3}

= \frac{\frac{4 π + 6 πn}{3}}{4}

Apply the fraction rule:

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

= \frac{4 π + 6 πn}{3 \cdot 4}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{4 π + 6 πn}{12}

Factor

4 π + 6 πn : 2 π (2 + 3 n)

4 π + 6 πn

Rewrite as

= 2 \cdot 2 π + 3 \cdot 2 πn

Factor out common term

2 π

= 2 π (2 + 3 n)

= \frac{2 π ( 2 + 3 n )}{12}

Cancel the common factor:

2

= \frac{π ( 3 n + 2 )}{6}

x = \frac{π ( 3 n + 2 )}{6}

x = \frac{π ( 3 n + 2 )}{6}

x = \frac{π ( 3 n + 2 )}{6}

x = \frac{πn}{2}, x = \frac{π + 2 πn}{4}, x = \frac{π ( 3 n + 1 )}{6}, x = \frac{π ( 3 n + 2 )}{6}

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