What is the general solution for sin(2x)sin(4x)=cos(2x)cos(4x) ?

The general solution for sin(2x)sin(4x)=cos(2x)cos(4x) is x= pi/(12)+(pin)/3 ,x= pi/4+(pin)/3

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

Solution

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

Solution

x = \frac{π}{12} + \frac{πn}{3}, x = \frac{π}{4} + \frac{πn}{3}

Degrees

x = 1 5^{\circ} + 6 0^{\circ} n, x = 4 5^{\circ} + 6 0^{\circ} n

Solution steps

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

Subtract

cos (2 x) cos (4 x)

from both sides

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

Rewrite using trig identities

sin (2 x) sin (4 x) - cos (2 x) cos (4 x)

Use the Angle Sum identity:

cos (s) cos (t) - sin (s) sin (t) = cos (s + t)

- cos (s) cos (t) + sin (s) sin (t) = - cos (s + t)

= - cos (2 x + 4 x)

- cos (2 x + 4 x) = 0

Divide both sides by

- 1

- cos (2 x + 4 x) = 0

Divide both sides by

- 1

\frac{- cos ( 2 x + 4 x )}{- 1} = \frac{0}{- 1}

Simplify

cos (2 x + 4 x) = 0

cos (2 x + 4 x) = 0

General solutions for

cos (2 x + 4 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}

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

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

Solve

2 x + 4 x = \frac{π}{2} + 2 πn : x = \frac{π}{12} + \frac{πn}{3}

2 x + 4 x = \frac{π}{2} + 2 πn

Add similar elements:

2 x + 4 x = 6 x

6 x = \frac{π}{2} + 2 πn

Divide both sides by

6

6 x = \frac{π}{2} + 2 πn

Divide both sides by

6

\frac{6 x}{6} = \frac{\frac{π}{2}}{6} + \frac{2 πn}{6}

Simplify

\frac{6 x}{6} = \frac{\frac{π}{2}}{6} + \frac{2 πn}{6}

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

\frac{\frac{π}{2}}{6} + \frac{2 πn}{6} : \frac{π}{12} + \frac{πn}{3}

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

\frac{\frac{π}{2}}{6} = \frac{π}{12}

\frac{\frac{π}{2}}{6}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{π}{12}

\frac{2 πn}{6} = \frac{πn}{3}

\frac{2 πn}{6}

Cancel the common factor:

2

= \frac{πn}{3}

= \frac{π}{12} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}

Solve

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

2 x + 4 x = \frac{3 π}{2} + 2 πn

Add similar elements:

2 x + 4 x = 6 x

6 x = \frac{3 π}{2} + 2 πn

Divide both sides by

6

6 x = \frac{3 π}{2} + 2 πn

Divide both sides by

6

\frac{6 x}{6} = \frac{\frac{3 π}{2}}{6} + \frac{2 πn}{6}

Simplify

\frac{6 x}{6} = \frac{\frac{3 π}{2}}{6} + \frac{2 πn}{6}

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

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

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

\frac{\frac{3 π}{2}}{6} = \frac{π}{4}

\frac{\frac{3 π}{2}}{6}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{3 π}{12}

Cancel the common factor:

3

= \frac{π}{4}

\frac{2 πn}{6} = \frac{πn}{3}

\frac{2 πn}{6}

Cancel the common factor:

2

= \frac{πn}{3}

= \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}, x = \frac{π}{4} + \frac{πn}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x)sin(4x)=cos(2x)cos(4x) ?
The general solution for sin(2x)sin(4x)=cos(2x)cos(4x) is x= pi/(12)+(pin)/3 ,x= pi/4+(pin)/3