What is the general solution for sin(7x)+sin(x)=0,0<= x<= 2pi ?

Q: What is the general solution for sin(7x)+sin(x)=0,0<= x<= 2pi ?

The general solution for sin(7x)+sin(x)=0,0<= x<= 2pi is

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sin(7x)+sin(x)=0,0<= x<= 2pi

Solution

sin (7 x) + sin (x) = 0, 0 \leq x \leq 2 π

Solution

x = \frac{π}{6}, x = \frac{π}{2}, x = \frac{5 π}{6}, x = \frac{7 π}{6}, x = \frac{3 π}{2}, x = \frac{11 π}{6}, x = 0, x = \frac{π}{4}, x = \frac{3 π}{4}, x = π, x = \frac{5 π}{4}, x = \frac{7 π}{4}, x = 2 π

Degrees

x = 3 0^{\circ}, x = 9 0^{\circ}, x = 15 0^{\circ}, x = 21 0^{\circ}, x = 27 0^{\circ}, x = 33 0^{\circ}, x = 0^{\circ}, x = 4 5^{\circ}, x = 13 5^{\circ}, x = 18 0^{\circ}, x = 22 5^{\circ}, x = 31 5^{\circ}, x = 36 0^{\circ}

Solution steps

sin (7 x) + sin (x) = 0, 0 \leq x \leq 2 π

Rewrite using trig identities

sin (7 x) + sin (x)

Use the Sum to Product identity:

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

= 2 sin (\frac{7 x + x}{2}) cos (\frac{7 x - x}{2})

Simplify

2 sin (\frac{7 x + x}{2}) cos (\frac{7 x - x}{2}) : 2 sin (4 x) cos (3 x)

2 sin (\frac{7 x + x}{2}) cos (\frac{7 x - x}{2})

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

\frac{7 x + x}{2}

Add similar elements:

7 x + x = 8 x

= \frac{8 x}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4 x

= 2 sin (4 x) cos (\frac{7 x - x}{2})

\frac{7 x - x}{2} = 3 x

\frac{7 x - x}{2}

Add similar elements:

7 x - x = 6 x

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

= 2 sin (4 x) cos (3 x)

= 2 sin (4 x) cos (3 x)

2 cos (3 x) sin (4 x) = 0

Solving each part separately

cos (3 x) = 0 or sin (4 x) = 0

cos (3 x) = 0, 0 \leq x \leq 2 π : x = \frac{π}{6}, x = \frac{π}{2}, x = \frac{5 π}{6}, x = \frac{7 π}{6}, x = \frac{3 π}{2}, x = \frac{11 π}{6}

cos (3 x) = 0, 0 \leq x \leq 2 π

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}

Solutions for the range

0 \leq x \leq 2 π

x = \frac{π}{6}, x = \frac{π}{2}, x = \frac{5 π}{6}, x = \frac{7 π}{6}, x = \frac{3 π}{2}, x = \frac{11 π}{6}

sin (4 x) = 0, 0 \leq x \leq 2 π : x = 0, x = \frac{π}{4}, x = \frac{π}{2}, x = \frac{3 π}{4}, x = π, x = \frac{5 π}{4}, x = \frac{3 π}{2}, x = \frac{7 π}{4}, x = 2 π

sin (4 x) = 0, 0 \leq x \leq 2 π

General solutions for

sin (4 x) = 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}

4 x = 0 + 2 πn, 4 x = π + 2 πn

4 x = 0 + 2 πn, 4 x = π + 2 πn

Solve

4 x = 0 + 2 πn : x = \frac{πn}{2}

4 x = 0 + 2 πn

0 + 2 πn = 2 πn

4 x = 2 πn

Divide both sides by

4

4 x = 2 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{2 πn}{4}

Simplify

x = \frac{πn}{2}

x = \frac{πn}{2}

Solve

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

4 x = π + 2 πn

Divide both sides by

4

4 x = π + 2 πn

Divide both sides by

4

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

Simplify

x = \frac{π}{4} + \frac{πn}{2}

x = \frac{π}{4} + \frac{πn}{2}

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

Solutions for the range

0 \leq x \leq 2 π

x = 0, x = \frac{π}{4}, x = \frac{π}{2}, x = \frac{3 π}{4}, x = π, x = \frac{5 π}{4}, x = \frac{3 π}{2}, x = \frac{7 π}{4}, x = 2 π

Combine all the solutions

x = \frac{π}{6}, x = \frac{π}{2}, x = \frac{5 π}{6}, x = \frac{7 π}{6}, x = \frac{3 π}{2}, x = \frac{11 π}{6}, x = 0, x = \frac{π}{4}, x = \frac{3 π}{4}, x = π, x = \frac{5 π}{4}, x = \frac{7 π}{4}, x = 2 π

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(7x)+sin(x)=0,0<= x<= 2pi ?
The general solution for sin(7x)+sin(x)=0,0<= x<= 2pi is