What is the general solution for cos(4x)cos(x)+sin(4x)sin(x)= 1/2 ?

The general solution for cos(4x)cos(x)+sin(4x)sin(x)= 1/2 is x= pi/9+(2pin)/3 ,x=(5pi)/9+(2pin)/3

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cos(4x)cos(x)+sin(4x)sin(x)= 1/2

Solution

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

Solution

x = \frac{π}{9} + \frac{2 πn}{3}, x = \frac{5 π}{9} + \frac{2 πn}{3}

Degrees

x = 2 0^{\circ} + 12 0^{\circ} n, x = 10 0^{\circ} + 12 0^{\circ} n

Solution steps

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

Rewrite using trig identities

cos (4 x) cos (x) + sin (4 x) sin (x)

Use the Angle Difference identity:

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

= cos (4 x - x)

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

General solutions for

cos (4 x - x) = \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}

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

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

Solve

4 x - x = \frac{π}{3} + 2 πn : x = \frac{π}{9} + \frac{2 πn}{3}

4 x - x = \frac{π}{3} + 2 πn

Add similar elements:

4 x - x = 3 x

3 x = \frac{π}{3} + 2 πn

Divide both sides by

3

3 x = \frac{π}{3} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{π}{3}}{3} + \frac{2 πn}{3} : \frac{π}{9} + \frac{2 πn}{3}

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

\frac{\frac{π}{3}}{3} = \frac{π}{9}

\frac{\frac{π}{3}}{3}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{π}{9}

= \frac{π}{9} + \frac{2 πn}{3}

x = \frac{π}{9} + \frac{2 πn}{3}

x = \frac{π}{9} + \frac{2 πn}{3}

x = \frac{π}{9} + \frac{2 πn}{3}

Solve

4 x - x = \frac{5 π}{3} + 2 πn : x = \frac{5 π}{9} + \frac{2 πn}{3}

4 x - x = \frac{5 π}{3} + 2 πn

Add similar elements:

4 x - x = 3 x

3 x = \frac{5 π}{3} + 2 πn

Divide both sides by

3

3 x = \frac{5 π}{3} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{5 π}{3}}{3} + \frac{2 πn}{3} : \frac{5 π}{9} + \frac{2 πn}{3}

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

\frac{\frac{5 π}{3}}{3} = \frac{5 π}{9}

\frac{\frac{5 π}{3}}{3}

Apply the fraction rule:

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

= \frac{5 π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{5 π}{9}

= \frac{5 π}{9} + \frac{2 πn}{3}

x = \frac{5 π}{9} + \frac{2 πn}{3}

x = \frac{5 π}{9} + \frac{2 πn}{3}

x = \frac{5 π}{9} + \frac{2 πn}{3}

x = \frac{π}{9} + \frac{2 πn}{3}, x = \frac{5 π}{9} + \frac{2 πn}{3}

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tan^2(θ)+7tan(θ)+5=0

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Frequently Asked Questions (FAQ)

What is the general solution for cos(4x)cos(x)+sin(4x)sin(x)= 1/2 ?
The general solution for cos(4x)cos(x)+sin(4x)sin(x)= 1/2 is x= pi/9+(2pin)/3 ,x=(5pi)/9+(2pin)/3