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Popular Trigonometry >

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

Solution

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

Solution

x = πn, x = \frac{π + 2 πn}{2}, x = \frac{π + 8 πn}{8}, x = \frac{7 π + 8 πn}{8}

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n, x = 22. 5^{\circ} + 18 0^{\circ} n, x = 157. 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

sin (4 x)

from both sides

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

Let:

u = 2 x

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

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

Factor

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

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

Factor out common term

sin (u)

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

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

Solving each part separately

sin (u) = 0 or 2 - 2 cos (u) = 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 - 2 cos (u) = 0 : u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

2 - 2 cos (u) = 0

Move

2

to the right side

2 - 2 cos (u) = 0

Subtract

2

from both sides

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

Simplify

- 2 cos (u) = - 2

- 2 cos (u) = - 2

Divide both sides by

- 2

- 2 cos (u) = - 2

Divide both sides by

- 2

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

Simplify

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

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

General solutions for

cos (u) = \frac{2}{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{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

u = \frac{π}{4} + 2 πn, u = \frac{7 π}{4} + 2 πn

Combine all the solutions

u = 2 πn, u = π + 2 πn, u = \frac{π}{4} + 2 πn, u = \frac{7 π}{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 = \frac{π}{4} + 2 πn : x = \frac{π + 8 πn}{8}

2 x = \frac{π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{π}{4} + 2 πn

Divide both sides by

2

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

Simplify

\frac{2 x}{2} = \frac{\frac{π}{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{\frac{π}{4}}{2} + \frac{2 πn}{2} : \frac{π + 8 πn}{8}

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

Apply rule

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

= \frac{\frac{π}{4} + 2 πn}{2}

Join

\frac{π}{4} + 2 πn : \frac{π + 8 πn}{4}

\frac{π}{4} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 4}{4}

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

Since the denominators are equal, combine the fractions:

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

= \frac{π + 2 πn \cdot 4}{4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{π + 8 πn}{4}

= \frac{\frac{π + 8 πn}{4}}{2}

Apply the fraction rule:

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

= \frac{π + 8 πn}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{π + 8 πn}{8}

x = \frac{π + 8 πn}{8}

x = \frac{π + 8 πn}{8}

x = \frac{π + 8 πn}{8}

2 x = \frac{7 π}{4} + 2 πn : x = \frac{7 π + 8 πn}{8}

2 x = \frac{7 π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{7 π}{4} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{7 π}{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{\frac{7 π}{4}}{2} + \frac{2 πn}{2} : \frac{7 π + 8 πn}{8}

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

Apply rule

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

= \frac{\frac{7 π}{4} + 2 πn}{2}

Join

\frac{7 π}{4} + 2 πn : \frac{7 π + 8 πn}{4}

\frac{7 π}{4} + 2 πn

Convert element to fraction:

2 πn = \frac{2 πn 4}{4}

= \frac{7 π}{4} + \frac{2 πn \cdot 4}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{7 π + 2 πn \cdot 4}{4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{7 π + 8 πn}{4}

= \frac{\frac{7 π + 8 πn}{4}}{2}

Apply the fraction rule:

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

= \frac{7 π + 8 πn}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{7 π + 8 πn}{8}

x = \frac{7 π + 8 πn}{8}

x = \frac{7 π + 8 πn}{8}

x = \frac{7 π + 8 πn}{8}

x = πn, x = \frac{π + 2 πn}{2}, x = \frac{π + 8 πn}{8}, x = \frac{7 π + 8 πn}{8}

Graph

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