What is the general solution for sin(5x)-sin(x)+3cos(3x)=0 ?

The general solution for sin(5x)-sin(x)+3cos(3x)=0 is x= pi/6+(2pin)/3 ,x= pi/2+(2pin)/3

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Popular >

sin(5x)-sin(x)+3cos(3x)=0

Solution

sin (5 x) - sin (x) + 3 cos (3 x) = 0

Solution

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

Degrees

x = 3 0^{\circ} + 12 0^{\circ} n, x = 9 0^{\circ} + 12 0^{\circ} n

Solution steps

sin (5 x) - sin (x) + 3 cos (3 x) = 0

Rewrite using trig identities

sin (5 x) - sin (x) + 3 cos (3 x)

Use the Sum to Product identity:

sin (s) - sin (t) = 2 sin (\frac{s - t}{2}) cos (\frac{s + t}{2})

= 3 cos (3 x) + 2 sin (\frac{5 x - x}{2}) cos (\frac{5 x + x}{2})

2 sin (\frac{5 x - x}{2}) cos (\frac{5 x + x}{2}) = 2 sin (2 x) cos (3 x)

2 sin (\frac{5 x - x}{2}) cos (\frac{5 x + x}{2})

\frac{5 x - x}{2} = 2 x

\frac{5 x - x}{2}

Add similar elements:

5 x - x = 4 x

= \frac{4 x}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 x

= 2 sin (2 x) cos (\frac{5 x + x}{2})

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

\frac{5 x + x}{2}

Add similar elements:

5 x + x = 6 x

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

= 2 sin (2 x) cos (3 x)

= 3 cos (3 x) + 2 sin (2 x) cos (3 x)

3 cos (3 x) + 2 cos (3 x) sin (2 x) = 0

Factor

3 cos (3 x) + 2 cos (3 x) sin (2 x) : cos (3 x) (2 sin (2 x) + 3)

3 cos (3 x) + 2 cos (3 x) sin (2 x)

Factor out common term

cos (3 x)

= cos (3 x) (3 + 2 sin (2 x))

cos (3 x) (2 sin (2 x) + 3) = 0

Solving each part separately

cos (3 x) = 0 or 2 sin (2 x) + 3 = 0

cos (3 x) = 0 : x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

cos (3 x) = 0

General solutions for

cos (3 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}

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

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

Solve

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

3 x = \frac{π}{2} + 2 πn

Divide both sides by

3

3 x = \frac{π}{2} + 2 πn

Divide both sides by

3

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

Simplify

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

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

\frac{\frac{π}{2}}{3} = \frac{π}{6}

\frac{\frac{π}{2}}{3}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π}{6}

= \frac{π}{6} + \frac{2 πn}{3}

x = \frac{π}{6} + \frac{2 πn}{3}

x = \frac{π}{6} + \frac{2 πn}{3}

x = \frac{π}{6} + \frac{2 πn}{3}

Solve

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

3 x = \frac{3 π}{2} + 2 πn

Divide both sides by

3

3 x = \frac{3 π}{2} + 2 πn

Divide both sides by

3

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

Simplify

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

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

\frac{\frac{3 π}{2}}{3} = \frac{π}{2}

\frac{\frac{3 π}{2}}{3}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{3 π}{6}

Cancel the common factor:

3

= \frac{π}{2}

= \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

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

2 sin (2 x) + 3 = 0 :

No Solution

2 sin (2 x) + 3 = 0

Move

3

to the right side

2 sin (2 x) + 3 = 0

Subtract

3

from both sides

2 sin (2 x) + 3 - 3 = 0 - 3

Simplify

2 sin (2 x) = - 3

2 sin (2 x) = - 3

Divide both sides by

2

2 sin (2 x) = - 3

Divide both sides by

2

\frac{2 sin ( 2 x )}{2} = \frac{- 3}{2}

Simplify

sin (2 x) = - \frac{3}{2}

sin (2 x) = - \frac{3}{2}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(5x)-sin(x)+3cos(3x)=0 ?
The general solution for sin(5x)-sin(x)+3cos(3x)=0 is x= pi/6+(2pin)/3 ,x= pi/2+(2pin)/3