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Popular Trigonometry >

2sin(4x)=sin(2x)

Solution

2 sin (4 x) = sin (2 x)

Solution

x = πn, x = \frac{π + 2 πn}{2}, x = \frac{1.31811 \dots + 2 πn}{2}, x = \frac{2 π - 1.31811 \dots + 2 πn}{2}

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n, x = 37.76124 \dots^{\circ} + 18 0^{\circ} n, x = 142.23875 \dots^{\circ} + 18 0^{\circ} n

Solution steps

2 sin (4 x) = sin (2 x)

Subtract

sin (2 x)

from both sides

2 sin (4 x) - sin (2 x) = 0

Let:

u = 2 x

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

Rewrite using trig identities

- sin (u) + 2 sin (2 u)

Use the Double Angle identity:

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

= - sin (u) + 2 \cdot 2 sin (u) cos (u)

Simplify

= - sin (u) + 4 sin (u) cos (u)

- sin (u) + 4 cos (u) sin (u) = 0

Factor

- sin (u) + 4 cos (u) sin (u) : sin (u) (4 cos (u) - 1)

- sin (u) + 4 cos (u) sin (u)

Factor out common term

sin (u)

= sin (u) (- 1 + 4 cos (u))

sin (u) (4 cos (u) - 1) = 0

Solving each part separately

sin (u) = 0 or 4 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

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

4 cos (u) - 1 = 0

Move

1

to the right side

4 cos (u) - 1 = 0

Add

1

to both sides

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

Simplify

4 cos (u) = 1

4 cos (u) = 1

Divide both sides by

4

4 cos (u) = 1

Divide both sides by

4

\frac{4 cos ( u )}{4} = \frac{1}{4}

Simplify

cos (u) = \frac{1}{4}

cos (u) = \frac{1}{4}

Apply trig inverse properties

cos (u) = \frac{1}{4}

General solutions for

cos (u) = \frac{1}{4}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

u = arccos (\frac{1}{4}) + 2 πn, u = 2 π - arccos (\frac{1}{4}) + 2 πn

u = arccos (\frac{1}{4}) + 2 πn, u = 2 π - arccos (\frac{1}{4}) + 2 πn

Combine all the solutions

u = 2 πn, u = π + 2 πn, u = arccos (\frac{1}{4}) + 2 πn, u = 2 π - arccos (\frac{1}{4}) + 2 πn

Substitute back

u = 2 x

2 x = 2 πn : x = πn

2 x = 2 πn

Divide both sides by

2

2 x = 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2}

Simplify

x = πn

x = πn

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

2 x = π + 2 πn

Divide both sides by

2

2 x = π + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

\frac{π}{2} + \frac{2 πn}{2}

Apply rule

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

= \frac{π + 2 πn}{2}

x = \frac{π + 2 πn}{2}

x = \frac{π + 2 πn}{2}

x = \frac{π + 2 πn}{2}

2 x = arccos (\frac{1}{4}) + 2 πn : x = \frac{arccos ( \frac{1}{4} ) + 2 πn}{2}

2 x = arccos (\frac{1}{4}) + 2 πn

Divide both sides by

2

2 x = arccos (\frac{1}{4}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arccos ( \frac{1}{4} )}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{arccos ( \frac{1}{4} )}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{arccos ( \frac{1}{4} )}{2} + \frac{2 πn}{2} : \frac{arccos ( \frac{1}{4} ) + 2 πn}{2}

\frac{arccos ( \frac{1}{4} )}{2} + \frac{2 πn}{2}

Apply rule

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

= \frac{arccos ( \frac{1}{4} ) + 2 πn}{2}

x = \frac{arccos ( \frac{1}{4} ) + 2 πn}{2}

x = \frac{arccos ( \frac{1}{4} ) + 2 πn}{2}

x = \frac{arccos ( \frac{1}{4} ) + 2 πn}{2}

2 x = 2 π - arccos (\frac{1}{4}) + 2 πn : x = \frac{2 π - arccos ( \frac{1}{4} ) + 2 πn}{2}

2 x = 2 π - arccos (\frac{1}{4}) + 2 πn

Divide both sides by

2

2 x = 2 π - arccos (\frac{1}{4}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( \frac{1}{4} )}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( \frac{1}{4} )}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 π}{2} - \frac{arccos ( \frac{1}{4} )}{2} + \frac{2 πn}{2} : \frac{2 π - arccos ( \frac{1}{4} ) + 2 πn}{2}

\frac{2 π}{2} - \frac{arccos ( \frac{1}{4} )}{2} + \frac{2 πn}{2}

Apply rule

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

= \frac{2 π - arccos ( \frac{1}{4} ) + 2 πn}{2}

x = \frac{2 π - arccos ( \frac{1}{4} ) + 2 πn}{2}

x = \frac{2 π - arccos ( \frac{1}{4} ) + 2 πn}{2}

x = \frac{2 π - arccos ( \frac{1}{4} ) + 2 πn}{2}

x = πn, x = \frac{π + 2 πn}{2}, x = \frac{arccos ( \frac{1}{4} ) + 2 πn}{2}, x = \frac{2 π - arccos ( \frac{1}{4} ) + 2 πn}{2}

Show solutions in decimal form

x = πn, x = \frac{π + 2 πn}{2}, x = \frac{1.31811 \dots + 2 πn}{2}, x = \frac{2 π - 1.31811 \dots + 2 πn}{2}

Graph

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