What is the general solution for 2sin(x)tan(x)=-tan(x) ?

The general solution for 2sin(x)tan(x)=-tan(x) is x=pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin

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2sin(x)tan(x)=-tan(x)

Solution

2 sin (x) tan (x) = - tan (x)

Solution

x = πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (x) tan (x) = - tan (x)

Subtract

- tan (x)

from both sides

2 sin (x) tan (x) + tan (x) = 0

Factor

2 sin (x) tan (x) + tan (x) : tan (x) (2 sin (x) + 1)

2 sin (x) tan (x) + tan (x)

Factor out common term

tan (x)

= tan (x) (2 sin (x) + 1)

tan (x) (2 sin (x) + 1) = 0

Solving each part separately

tan (x) = 0 or 2 sin (x) + 1 = 0

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 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}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

2 sin (x) + 1 = 0 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

2 sin (x) + 1 = 0

Move

1

to the right side

2 sin (x) + 1 = 0

Subtract

1

from both sides

2 sin (x) + 1 - 1 = 0 - 1

Simplify

2 sin (x) = - 1

2 sin (x) = - 1

Divide both sides by

2

2 sin (x) = - 1

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{- 1}{2}

Simplify

sin (x) = - \frac{1}{2}

sin (x) = - \frac{1}{2}

General solutions for

sin (x) = - \frac{1}{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}

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Popular Examples

cos(2x)cos(x)-sin(2x)sin(x)=-(sqrt(3))/2

cos (2 x) cos (x) - sin (2 x) sin (x) = - \frac{3}{2}

sec(θ)=tan(θ)+cot(θ)

sec (θ) = tan (θ) + cot (θ)

sec^2(x)= 4/3 ,0<= x<= 2pi

sec^{2} (x) = \frac{4}{3}, 0 \leq x \leq 2 π

sqrt(3)tan(3θ)-1=0

3 tan (3 θ) - 1 = 0

2tan(x)=sec(x)

2 tan (x) = sec (x)

Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x)tan(x)=-tan(x) ?
The general solution for 2sin(x)tan(x)=-tan(x) is x=pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin