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

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

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Popular >

sin(2t)=2sin(t)

Solution

sin (2 t) = 2 sin (t)

Solution

t = 2 πn, t = π + 2 πn

Degrees

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

Solution steps

sin (2 t) = 2 sin (t)

Subtract

2 sin (t)

from both sides

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

Rewrite using trig identities

sin (2 t) - 2 sin (t)

Use the Double Angle identity:

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

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

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

Factor

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

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

Factor out common term

2 sin (t)

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

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

Solving each part separately

sin (t) = 0 or 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

cos (t) - 1 = 0 : t = 2 πn

cos (t) - 1 = 0

Move

1

to the right side

cos (t) - 1 = 0

Add

1

to both sides

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

Simplify

cos (t) = 1

cos (t) = 1

General solutions for

cos (t) = 1

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 = 0 + 2 πn

t = 0 + 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

Combine all the solutions

t = 2 πn, t = π + 2 πn

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Frequently Asked Questions (FAQ)

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