What is the general solution for 6sin(θ)+sqrt(3)sin(θ)tan(θ)=3sin(θ) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

6sin(θ)+sqrt(3)sin(θ)tan(θ)=3sin(θ)

Solution

6 sin (θ) + 3 sin (θ) tan (θ) = 3 sin (θ)

Solution

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

Degrees

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n, θ = 12 0^{\circ} + 18 0^{\circ} n

Solution steps

6 sin (θ) + 3 sin (θ) tan (θ) = 3 sin (θ)

Subtract

3 sin (θ)

from both sides

3 sin (θ) + 3 sin (θ) tan (θ) = 0

Factor

3 sin (θ) + 3 sin (θ) tan (θ) : sin (θ) (3 + 3 tan (θ))

3 sin (θ) + 3 sin (θ) tan (θ)

Factor out common term

sin (θ)

= sin (θ) (3 + 3 tan (θ))

sin (θ) (3 + 3 tan (θ)) = 0

Solving each part separately

sin (θ) = 0 or 3 + 3 tan (θ) = 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

3 + 3 tan (θ) = 0 : θ = \frac{2 π}{3} + πn

3 + 3 tan (θ) = 0

Move

3

to the right side

3 + 3 tan (θ) = 0

Subtract

3

from both sides

3 + 3 tan (θ) - 3 = 0 - 3

Simplify

3 tan (θ) = - 3

3 tan (θ) = - 3

Divide both sides by

3

3 tan (θ) = - 3

Divide both sides by

3

\frac{3 tan ( θ )}{3} = \frac{- 3}{3}

Simplify

\frac{3 tan ( θ )}{3} = \frac{- 3}{3}

Simplify

\frac{3 tan ( θ )}{3} : tan (θ)

\frac{3 tan ( θ )}{3}

Cancel the common factor:

3

= tan (θ)

Simplify

\frac{- 3}{3} : - 3

\frac{- 3}{3}

Apply the fraction rule:

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

= - \frac{3}{3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

= \frac{3}{3 ^{\frac{1}{2}}}

Apply exponent rule:

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

\frac{3 ^{1}}{3 ^{\frac{1}{2}}} = 3^{1 - \frac{1}{2}}

= 3^{1 - \frac{1}{2}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 3^{\frac{1}{2}}

Apply radical rule:

3^{\frac{1}{2}} = 3

= - 3

tan (θ) = - 3

tan (θ) = - 3

tan (θ) = - 3

General solutions for

tan (θ) = - 3

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

θ = \frac{2 π}{3} + πn

θ = \frac{2 π}{3} + πn

Combine all the solutions

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

Popular Examples

cos(2x)sin(x)=1

cos (2 x) sin (x) = 1

sin(θ)=10

sin (θ) = 10

17.6^2=15^2+13.1^2-2(15)(13.1)*cos(A)

17. 6^{2} = 1 5^{2} + 13. 1^{2} - 2 (15) (13.1) \cdot cos (A)

solvefor x,f=cos(1/(x^2))solve for

x, f = cos (\frac{1}{x ^{2}})

sin(θ)=45

sin (θ) = 45

Frequently Asked Questions (FAQ)

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