What is the general solution for 6sin(θ)=6sin(2θ) ?

The general solution for 6sin(θ)=6sin(2θ) is θ=2pin,θ=pi+2pin,θ= pi/3+2pin,θ=(5pi)/3+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

6sin(θ)=6sin(2θ)

Solution

6 sin (θ) = 6 sin (2 θ)

Solution

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

Degrees

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

Solution steps

6 sin (θ) = 6 sin (2 θ)

Subtract

6 sin (2 θ)

from both sides

6 sin (θ) - 6 sin (2 θ) = 0

Rewrite using trig identities

- 6 sin (2 θ) + 6 sin (θ)

Use the Double Angle identity:

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

= - 6 \cdot 2 sin (θ) cos (θ) + 6 sin (θ)

Simplify

= - 12 sin (θ) cos (θ) + 6 sin (θ)

6 sin (θ) - 12 cos (θ) sin (θ) = 0

Factor

6 sin (θ) - 12 cos (θ) sin (θ) : - 6 sin (θ) (2 cos (θ) - 1)

6 sin (θ) - 12 cos (θ) sin (θ)

Rewrite

- 12

2 \cdot 6

= 6 sin (θ) + 2 \cdot 6 sin (θ) cos (θ)

Factor out common term

- 6 sin (θ)

= - 6 sin (θ) (- 1 + 2 cos (θ))

- 6 sin (θ) (2 cos (θ) - 1) = 0

Solving each part separately

sin (θ) = 0 or 2 cos (θ) - 1 = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

General solutions for

sin (θ) = 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}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

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

2 cos (θ) - 1 = 0

Move

1

to the right side

2 cos (θ) - 1 = 0

Add

1

to both sides

2 cos (θ) - 1 + 1 = 0 + 1

Simplify

2 cos (θ) = 1

2 cos (θ) = 1

Divide both sides by

2

2 cos (θ) = 1

Divide both sides by

2

\frac{2 cos ( θ )}{2} = \frac{1}{2}

Simplify

cos (θ) = \frac{1}{2}

cos (θ) = \frac{1}{2}

General solutions for

cos (θ) = \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}

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

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

Combine all the solutions

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

Popular Examples

2sin(15)cos(15)=tan(x)

2 sin (1 5^{\circ}) cos (1 5^{\circ}) = tan (x)

6sin^2(x)+5cos(x)=7

6 sin^{2} (x) + 5 cos (x) = 7

sqrt(3)cos(x)+sin(x)=-1

3 cos (x) + sin (x) = - 1

-3=tan^2(θ)-2-2tan(θ)

- 3 = tan^{2} (θ) - 2 - 2 tan (θ)

-2cos^2(x)-9sin(x)-3=0

- 2 cos^{2} (x) - 9 sin (x) - 3 = 0

Frequently Asked Questions (FAQ)

What is the general solution for 6sin(θ)=6sin(2θ) ?
The general solution for 6sin(θ)=6sin(2θ) is θ=2pin,θ=pi+2pin,θ= pi/3+2pin,θ=(5pi)/3+2pin