What is the general solution for cos(x)-cos(2x)=0,0<= x<360 ?

Q: What is the general solution for cos(x)-cos(2x)=0,0<= x<360 ?

The general solution for cos(x)-cos(2x)=0,0<= x<360 is x=0,x=120,x=240

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

cos(x)-cos(2x)=0,0<= x<360

Solution

cos (x) - cos (2 x) = 0, 0^{\circ} \leq x < 36 0^{\circ}

Solution

x = 0, x = 12 0^{\circ}, x = 24 0^{\circ}

Radians

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

Solution steps

cos (x) - cos (2 x) = 0, 0^{\circ} \leq x < 36 0^{\circ}

Rewrite using trig identities

- cos (2 x) + cos (x)

Use the Sum to Product identity:

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

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

Simplify

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

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

Add similar elements:

x + 2 x = 3 x

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

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

\frac{x - 2 x}{2}

Add similar elements:

x - 2 x = - x

= \frac{- x}{2}

Apply the fraction rule:

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

= - \frac{x}{2}

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

Use the negative angle identity:

sin (- x) = - sin (x)

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

Apply rule

- (- a) = a

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

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

2 sin (\frac{3 x}{2}) sin (\frac{x}{2}) = 0

Solving each part separately

sin (\frac{3 x}{2}) = 0 or sin (\frac{x}{2}) = 0

sin (\frac{3 x}{2}) = 0, 0 \leq x < 36 0^{\circ} : x = 0, x = 12 0^{\circ}, x = 24 0^{\circ}

sin (\frac{3 x}{2}) = 0, 0 \leq x < 36 0^{\circ}

General solutions for

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

\frac{3 x}{2} = 0 + 36 0^{\circ} n, \frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

\frac{3 x}{2} = 0 + 36 0^{\circ} n, \frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Solve

\frac{3 x}{2} = 0 + 36 0^{\circ} n : x = \frac{72 0 ^{\circ} n}{3}

\frac{3 x}{2} = 0 + 36 0^{\circ} n

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

\frac{3 x}{2} = 36 0^{\circ} n

Multiply both sides by

2

\frac{3 x}{2} = 36 0^{\circ} n

Multiply both sides by

2

\frac{2 \cdot 3 x}{2} = 2 \cdot 36 0^{\circ} n

Simplify

3 x = 72 0^{\circ} n

3 x = 72 0^{\circ} n

Divide both sides by

3

3 x = 72 0^{\circ} n

Divide both sides by

3

\frac{3 x}{3} = \frac{72 0 ^{\circ} n}{3}

Simplify

x = \frac{72 0 ^{\circ} n}{3}

x = \frac{72 0 ^{\circ} n}{3}

Solve

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n : x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

2

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

2

\frac{2 \cdot 3 x}{2} = 36 0^{\circ} + 2 \cdot 36 0^{\circ} n

Simplify

3 x = 36 0^{\circ} + 72 0^{\circ} n

3 x = 36 0^{\circ} + 72 0^{\circ} n

Divide both sides by

3

3 x = 36 0^{\circ} + 72 0^{\circ} n

Divide both sides by

3

\frac{3 x}{3} = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

Simplify

x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

x = \frac{72 0 ^{\circ} n}{3}, x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

Solutions for the range

0 \leq x < 36 0^{\circ}

x = 0, x = 12 0^{\circ}, x = 24 0^{\circ}

sin (\frac{x}{2}) = 0, 0 \leq x < 36 0^{\circ} : x = 0

sin (\frac{x}{2}) = 0, 0 \leq x < 36 0^{\circ}

General solutions for

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

\frac{x}{2} = 0 + 36 0^{\circ} n, \frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

\frac{x}{2} = 0 + 36 0^{\circ} n, \frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Solve

\frac{x}{2} = 0 + 36 0^{\circ} n : x = 72 0^{\circ} n

\frac{x}{2} = 0 + 36 0^{\circ} n

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

\frac{x}{2} = 36 0^{\circ} n

Multiply both sides by

2

\frac{x}{2} = 36 0^{\circ} n

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot 36 0^{\circ} n

Simplify

x = 72 0^{\circ} n

x = 72 0^{\circ} n

Solve

\frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} + 72 0^{\circ} n

\frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

2

\frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

2

\frac{2 x}{2} = 36 0^{\circ} + 2 \cdot 36 0^{\circ} n

Simplify

x = 36 0^{\circ} + 72 0^{\circ} n

x = 36 0^{\circ} + 72 0^{\circ} n

x = 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

Solutions for the range

0 \leq x < 36 0^{\circ}

x = 0

Combine all the solutions

x = 0, x = 12 0^{\circ}, x = 24 0^{\circ}

Graph

Popular Examples

3=-3cos(θ)

2cos(θ)-sqrt(3)=0,0<= θ<= 2pi

-4sin(x)=cos^2(x)+1,0<= x<= 2pi

tan(2x)cos(2x)+cot(2x)sin(2x)=1

sin^2(2x)-cos^2(2x)= 1/2

Frequently Asked Questions (FAQ)

What is the general solution for cos(x)-cos(2x)=0,0<= x<360 ?
The general solution for cos(x)-cos(2x)=0,0<= x<360 is x=0,x=120,x=240