What is the general solution for sin(t)=sin(2t) ?

The general solution for sin(t)=sin(2t) is t= pi/3+(4pin)/3 ,t=pi+(4pin)/3 ,t=4pin,t=2pi+4pin

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Popular Trigonometry >

sin(t)=sin(2t)

Solution

sin (t) = sin (2 t)

Solution

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

Degrees

t = 6 0^{\circ} + 24 0^{\circ} n, t = 18 0^{\circ} + 24 0^{\circ} n, t = 0^{\circ} + 72 0^{\circ} n, t = 36 0^{\circ} + 72 0^{\circ} n

Solution steps

sin (t) = sin (2 t)

Subtract

sin (2 t)

from both sides

sin (t) - sin (2 t) = 0

Rewrite using trig identities

- sin (2 t) + sin (t)

Use the Sum to Product identity:

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

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

Simplify

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

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

\frac{t - 2 t}{2} = - \frac{t}{2}

\frac{t - 2 t}{2}

Add similar elements:

t - 2 t = - t

= \frac{- t}{2}

Apply the fraction rule:

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

= - \frac{t}{2}

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

Use the negative angle identity:

sin (- x) = - sin (x)

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

Remove parentheses:

(- a) = - a

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

Add similar elements:

t + 2 t = 3 t

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

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

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

Solving each part separately

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

cos (\frac{3 t}{2}) = 0 : t = \frac{π}{3} + \frac{4 πn}{3}, t = π + \frac{4 πn}{3}

cos (\frac{3 t}{2}) = 0

General solutions for

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

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

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

Solve

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

\frac{3 t}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{3 t}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 \cdot 3 t}{2} : 3 t

\frac{2 \cdot 3 t}{2}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 t}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 t

Simplify

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

3 t = π + 4 πn

3 t = π + 4 πn

3 t = π + 4 πn

Divide both sides by

3

3 t = π + 4 πn

Divide both sides by

3

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

Simplify

t = \frac{π}{3} + \frac{4 πn}{3}

t = \frac{π}{3} + \frac{4 πn}{3}

Solve

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

\frac{3 t}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{3 t}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 \cdot 3 t}{2} : 3 t

\frac{2 \cdot 3 t}{2}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 t}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 t

Simplify

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn : 3 π + 4 πn

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

3 t = 3 π + 4 πn

3 t = 3 π + 4 πn

3 t = 3 π + 4 πn

Divide both sides by

3

3 t = 3 π + 4 πn

Divide both sides by

3

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

Simplify

t = π + \frac{4 πn}{3}

t = π + \frac{4 πn}{3}

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

sin (\frac{t}{2}) = 0 : t = 4 πn, t = 2 π + 4 πn

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

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

Combine all the solutions

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(t)=sin(2t) ?
The general solution for sin(t)=sin(2t) is t= pi/3+(4pin)/3 ,t=pi+(4pin)/3 ,t=4pin,t=2pi+4pin