What is the general solution for sqrt(3)tan(θ)-2sin(θ)tan(θ)=0 ?

The general solution for sqrt(3)tan(θ)-2sin(θ)tan(θ)=0 is θ=pin,θ= pi/3+2pin,θ=(2pi)/3+2pin

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sqrt(3)tan(θ)-2sin(θ)tan(θ)=0

3 tan (θ) - 2 sin (θ) tan (θ) = 0

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

Degrees

θ = 0^{\circ} + 18 0^{\circ} n, θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 12 0^{\circ} + 36 0^{\circ} n

Solution steps

3 tan (θ) - 2 sin (θ) tan (θ) = 0

Factor

3 tan (θ) - 2 sin (θ) tan (θ) : tan (θ) (3 - 2 sin (θ))

3 tan (θ) - 2 sin (θ) tan (θ)

Factor out common term

tan (θ)

= tan (θ) (3 - 2 sin (θ))

tan (θ) (3 - 2 sin (θ)) = 0

Solving each part separately

tan (θ) = 0 or 3 - 2 sin (θ) = 0

tan (θ) = 0 : θ = πn

tan (θ) = 0

General solutions for

tan (θ) = 0

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 = 0 + πn

θ = 0 + πn

Solve

θ = 0 + πn : θ = πn

θ = 0 + πn

0 + πn = πn

θ = πn

θ = πn

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

3 - 2 sin (θ) = 0

Move

3

to the right side

3 - 2 sin (θ) = 0

Subtract

3

from both sides

3 - 2 sin (θ) - 3 = 0 - 3

Simplify

- 2 sin (θ) = - 3

- 2 sin (θ) = - 3

Divide both sides by

- 2

- 2 sin (θ) = - 3

Divide both sides by

- 2

\frac{- 2 sin ( θ )}{- 2} = \frac{- 3}{- 2}

Simplify

sin (θ) = \frac{3}{2}

sin (θ) = \frac{3}{2}

General solutions for

sin (θ) = \frac{3}{2}

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

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

Combine all the solutions

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

\frac{sin ( 2 x )}{sin ( x )} = 1.5

x, yz = sin (x y)

sin^{2} (x) = \frac{( 5 m - 5 )}{2}

2 cot (2 x) = 4

cos (θ) = \frac{4}{3.3 8 ^{2} + 2 ^{2}}

What is the general solution for sqrt(3)tan(θ)-2sin(θ)tan(θ)=0 ?
The general solution for sqrt(3)tan(θ)-2sin(θ)tan(θ)=0 is θ=pin,θ= pi/3+2pin,θ=(2pi)/3+2pin