What is the general solution for cos(2t)=cos(t),0<= t<2pi ?

Q: What is the general solution for cos(2t)=cos(t),0<= t<2pi ?

The general solution for cos(2t)=cos(t),0<= t<2pi is t=0,t=(2pi)/3 ,t=(4pi)/3

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cos(2t)=cos(t),0<= t<2pi

Solution

cos (2 t) = cos (t), 0 \leq t < 2 π

Solution

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

Degrees

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

Solution steps

cos (2 t) = cos (t), 0 \leq t < 2 π

Subtract

cos (t)

from both sides

cos (2 t) - cos (t) = 0

Rewrite using trig identities

cos (2 t) - cos (t)

Use the Sum to Product identity:

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

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

Simplify

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

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

Add similar elements:

2 t + t = 3 t

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

Add similar elements:

2 t - t = t

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

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

- 2 sin (\frac{3 t}{2}) sin (\frac{t}{2}) = 0

Solving each part separately

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

sin (\frac{3 t}{2}) = 0, 0 \leq t < 2 π : t = 0, t = \frac{2 π}{3}, t = \frac{4 π}{3}

sin (\frac{3 t}{2}) = 0, 0 \leq t < 2 π

General solutions for

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

\frac{3 t}{2} = 0 + 2 πn, \frac{3 t}{2} = π + 2 πn

\frac{3 t}{2} = 0 + 2 πn, \frac{3 t}{2} = π + 2 πn

Solve

\frac{3 t}{2} = 0 + 2 πn : t = \frac{4 πn}{3}

\frac{3 t}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{3 t}{2} = 2 πn

Multiply both sides by

2

\frac{3 t}{2} = 2 πn

Multiply both sides by

2

\frac{2 \cdot 3 t}{2} = 2 \cdot 2 πn

Simplify

3 t = 4 πn

3 t = 4 πn

Divide both sides by

3

3 t = 4 πn

Divide both sides by

3

\frac{3 t}{3} = \frac{4 πn}{3}

Simplify

t = \frac{4 πn}{3}

t = \frac{4 πn}{3}

Solve

\frac{3 t}{2} = π + 2 πn : t = \frac{2 π}{3} + \frac{4 πn}{3}

\frac{3 t}{2} = π + 2 πn

Multiply both sides by

2

\frac{3 t}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 \cdot 3 t}{2} = 2 π + 2 \cdot 2 πn

Simplify

3 t = 2 π + 4 πn

3 t = 2 π + 4 πn

Divide both sides by

3

3 t = 2 π + 4 πn

Divide both sides by

3

\frac{3 t}{3} = \frac{2 π}{3} + \frac{4 πn}{3}

Simplify

t = \frac{2 π}{3} + \frac{4 πn}{3}

t = \frac{2 π}{3} + \frac{4 πn}{3}

t = \frac{4 πn}{3}, t = \frac{2 π}{3} + \frac{4 πn}{3}

Solutions for the range

0 \leq t < 2 π

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

sin (\frac{t}{2}) = 0, 0 \leq t < 2 π : t = 0

sin (\frac{t}{2}) = 0, 0 \leq t < 2 π

General solutions for

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

\frac{t}{2} = 0 + 2 πn, \frac{t}{2} = π + 2 πn

\frac{t}{2} = 0 + 2 πn, \frac{t}{2} = π + 2 πn

Solve

\frac{t}{2} = 0 + 2 πn : t = 4 πn

\frac{t}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{t}{2} = 2 πn

Multiply both sides by

2

\frac{t}{2} = 2 πn

Multiply both sides by

2

\frac{2 t}{2} = 2 \cdot 2 πn

Simplify

t = 4 πn

t = 4 πn

Solve

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

\frac{t}{2} = π + 2 πn

Multiply both sides by

2

\frac{t}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 t}{2} = 2 π + 2 \cdot 2 πn

Simplify

t = 2 π + 4 πn

t = 2 π + 4 πn

t = 4 πn, t = 2 π + 4 πn

Solutions for the range

0 \leq t < 2 π

t = 0

Combine all the solutions

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

Popular Examples

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sin(θ)=sin(-(2pi)/3)

sin (θ) = sin (- \frac{2 π}{3})

sin(x)=(23/24)

sin (x) = (\frac{23}{24})

tan(θ)=-5/4

tan (θ) = - \frac{5}{4}

5-6sin^2(x)-cos(x)=0

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2t)=cos(t),0<= t<2pi ?
The general solution for cos(2t)=cos(t),0<= t<2pi is t=0,t=(2pi)/3 ,t=(4pi)/3