What is the general solution for tan(x)-2sin(x)tan(x)=0 ?

The general solution for tan(x)-2sin(x)tan(x)=0 is x=pin,x= pi/6+2pin,x=(5pi)/6+2pin

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

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

x = πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Factor

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

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

Factor out common term

- tan (x)

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

- 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{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

2 sin (x) - 1 = 0

Move

1

to the right side

2 sin (x) - 1 = 0

Add

1

to 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{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

cos (x) = \frac{2}{5}

sin (θ) + cos (θ) = 2

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

2 sin^{2} (x) + cos (x) = 0

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

What is the general solution for tan(x)-2sin(x)tan(x)=0 ?
The general solution for tan(x)-2sin(x)tan(x)=0 is x=pin,x= pi/6+2pin,x=(5pi)/6+2pin