What is the general solution for sin(4x)+sin(2x)=0,0<= x<= 180 ?

Q: What is the general solution for sin(4x)+sin(2x)=0,0<= x<= 180 ?

The general solution for sin(4x)+sin(2x)=0,0<= x<= 180 is x=90,x=0,x=60,x=120,x=180

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Popular Trigonometry >

sin(4x)+sin(2x)=0,0<= x<= 180

Solution

sin (4 x) + sin (2 x) = 0, 0^{\circ} \leq x \leq 18 0^{\circ}

Solution

x = 9 0^{\circ}, x = 0, x = 6 0^{\circ}, x = 12 0^{\circ}, x = 18 0^{\circ}

Radians

x = \frac{π}{2}, x = 0, x = \frac{π}{3}, x = \frac{2 π}{3}, x = π

Solution steps

sin (4 x) + sin (2 x) = 0, 0^{\circ} \leq x \leq 18 0^{\circ}

Rewrite using trig identities

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

Simplify

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

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

\frac{2 x + 4 x}{2} = 3 x

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

Add similar elements:

2 x + 4 x = 6 x

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

= 2 sin (3 x) cos (\frac{2 x - 4 x}{2})

\frac{2 x - 4 x}{2} = - x

\frac{2 x - 4 x}{2}

Add similar elements:

2 x - 4 x = - 2 x

= \frac{- 2 x}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= - x

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

Use the negative angle identity:

cos (- x) = cos (x)

= 2 cos (x) sin (3 x)

= 2 cos (x) sin (3 x)

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

Solving each part separately

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

cos (x) = 0, 0 \leq x \leq 18 0^{\circ} : x = 9 0^{\circ}

cos (x) = 0, 0 \leq x \leq 18 0^{\circ}

General solutions for

cos (x) = 0

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solutions for the range

0 \leq x \leq 18 0^{\circ}

x = 9 0^{\circ}

sin (3 x) = 0, 0 \leq x \leq 18 0^{\circ} : x = 0, x = 6 0^{\circ}, x = 12 0^{\circ}, x = 18 0^{\circ}

sin (3 x) = 0, 0 \leq x \leq 18 0^{\circ}

General solutions for

sin (3 x) = 0

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

3 x = 0 + 36 0^{\circ} n, 3 x = 18 0^{\circ} + 36 0^{\circ} n

3 x = 0 + 36 0^{\circ} n, 3 x = 18 0^{\circ} + 36 0^{\circ} n

Solve

3 x = 0 + 36 0^{\circ} n : x = \frac{36 0 ^{\circ} n}{3}

3 x = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

3 x = 36 0^{\circ} n

Divide both sides by

3

3 x = 36 0^{\circ} n

Divide both sides by

3

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3}

Simplify

x = \frac{36 0 ^{\circ} n}{3}

x = \frac{36 0 ^{\circ} n}{3}

Solve

3 x = 18 0^{\circ} + 36 0^{\circ} n : x = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

3 x = 18 0^{\circ} + 36 0^{\circ} n

Divide both sides by

3

3 x = 18 0^{\circ} + 36 0^{\circ} n

Divide both sides by

3

\frac{3 x}{3} = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

Simplify

x = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

x = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

x = \frac{36 0 ^{\circ} n}{3}, x = 6 0^{\circ} + \frac{36 0 ^{\circ} n}{3}

Solutions for the range

0 \leq x \leq 18 0^{\circ}

x = 0, x = 6 0^{\circ}, x = 12 0^{\circ}, x = 18 0^{\circ}

Combine all the solutions

x = 9 0^{\circ}, x = 0, x = 6 0^{\circ}, x = 12 0^{\circ}, x = 18 0^{\circ}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(4x)+sin(2x)=0,0<= x<= 180 ?
The general solution for sin(4x)+sin(2x)=0,0<= x<= 180 is x=90,x=0,x=60,x=120,x=180