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

The general solution for sin(2t)=sin(t) is t=2pin,t=pi+2pin,t= pi/3+2pin,t=(5pi)/3+2pin

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Popular >

sin(2t)=sin(t)

Solution

sin (2 t) = sin (t)

Solution

t = 2 πn, t = π + 2 πn, t = \frac{π}{3} + 2 πn, t = \frac{5 π}{3} + 2 πn

Degrees

t = 0^{\circ} + 36 0^{\circ} n, t = 18 0^{\circ} + 36 0^{\circ} n, t = 6 0^{\circ} + 36 0^{\circ} n, t = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (2 t) = sin (t)

Subtract

sin (t)

from both sides

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

Rewrite using trig identities

sin (2 t) - sin (t)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 2 sin (t) cos (t) - sin (t)

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

Factor

- sin (t) + 2 cos (t) sin (t) : sin (t) (2 cos (t) - 1)

- sin (t) + 2 cos (t) sin (t)

Factor out common term

sin (t)

= sin (t) (- 1 + 2 cos (t))

sin (t) (2 cos (t) - 1) = 0

Solving each part separately

sin (t) = 0 or 2 cos (t) - 1 = 0

sin (t) = 0 : t = 2 πn, t = π + 2 πn

sin (t) = 0

General solutions for

sin (t) = 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}

t = 0 + 2 πn, t = π + 2 πn

t = 0 + 2 πn, t = π + 2 πn

Solve

t = 0 + 2 πn : t = 2 πn

t = 0 + 2 πn

0 + 2 πn = 2 πn

t = 2 πn

t = 2 πn, t = π + 2 πn

2 cos (t) - 1 = 0 : t = \frac{π}{3} + 2 πn, t = \frac{5 π}{3} + 2 πn

2 cos (t) - 1 = 0

Move

1

to the right side

2 cos (t) - 1 = 0

Add

1

to both sides

2 cos (t) - 1 + 1 = 0 + 1

Simplify

2 cos (t) = 1

2 cos (t) = 1

Divide both sides by

2

2 cos (t) = 1

Divide both sides by

2

\frac{2 cos ( t )}{2} = \frac{1}{2}

Simplify

cos (t) = \frac{1}{2}

cos (t) = \frac{1}{2}

General solutions for

cos (t) = \frac{1}{2}

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}

t = \frac{π}{3} + 2 πn, t = \frac{5 π}{3} + 2 πn

t = \frac{π}{3} + 2 πn, t = \frac{5 π}{3} + 2 πn

Combine all the solutions

t = 2 πn, t = π + 2 πn, t = \frac{π}{3} + 2 πn, t = \frac{5 π}{3} + 2 πn

Popular Examples

cos(x)=(17.6)/(26)

cos (x) = \frac{17.6}{26}

tan(2θ)=-2.5

tan (2 θ) = - 2.5

1+4cos(θ)=sqrt(3)sin(θ),0<= θ<= 2pi

1 + 4 cos (θ) = 3 sin (θ), 0 \leq θ \leq 2 π

(150)/(sin(x))=(140.78)/(sin(115))

\frac{150}{sin ( x )} = \frac{140.78}{sin ( 11 5 ^{\circ} )}

tan^2(x)+cot^2(x)=2

tan^{2} (x) + cot^{2} (x) = 2

Frequently Asked Questions (FAQ)

What is the general solution for sin(2t)=sin(t) ?
The general solution for sin(2t)=sin(t) is t=2pin,t=pi+2pin,t= pi/3+2pin,t=(5pi)/3+2pin