What is the general solution for 3-sqrt(3)tan(2x)=4-2sqrt(3)tan(2x) ?

The general solution for 3-sqrt(3)tan(2x)=4-2sqrt(3)tan(2x) is x= pi/(12)+(pin)/2

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3-sqrt(3)tan(2x)=4-2sqrt(3)tan(2x)

Solution

3 - 3 tan (2 x) = 4 - 23 tan (2 x)

Solution

x = \frac{π}{12} + \frac{πn}{2}

Degrees

x = 1 5^{\circ} + 9 0^{\circ} n

Solution steps

3 - 3 tan (2 x) = 4 - 23 tan (2 x)

Solve by substitution

3 - 3 tan (2 x) = 4 - 23 tan (2 x)

Let:

tan (2 x) = u

3 - 3 u = 4 - 23 u

3 - 3 u = 4 - 23 u : u = \frac{3}{3}

3 - 3 u = 4 - 23 u

Move

3

to the right side

3 - 3 u = 4 - 23 u

Subtract

3

from both sides

3 - 3 u - 3 = 4 - 23 u - 3

Simplify

3 - 3 u - 3 = 4 - 23 u - 3

Simplify

3 - 3 u - 3 : - 3 u

3 - 3 u - 3

Add similar elements:

3 - 3 = 0

= - 3 u

Simplify

4 - 23 u - 3 : - 23 u + 1

4 - 23 u - 3

Group like terms

= - 23 u + 4 - 3

Add/Subtract the numbers:

4 - 3 = 1

= - 23 u + 1

- 3 u = - 23 u + 1

- 3 u = - 23 u + 1

- 3 u = - 23 u + 1

Move

23 u

to the left side

- 3 u = - 23 u + 1

Add

23 u

to both sides

- 3 u + 23 u = - 23 u + 1 + 23 u

Simplify

- 3 u + 23 u = - 23 u + 1 + 23 u

Simplify

- 3 u + 23 u : 3 u

- 3 u + 23 u

Add similar elements:

- 3 u + 23 u = 3 u

= 3 u

Simplify

- 23 u + 1 + 23 u : 1

- 23 u + 1 + 23 u

Add similar elements:

- 23 u + 23 u = 0

= 1

3 u = 1

3 u = 1

3 u = 1

Divide both sides by

3

3 u = 1

Divide both sides by

3

\frac{3 u}{3} = \frac{1}{3}

Simplify

\frac{3 u}{3} = \frac{1}{3}

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

u = \frac{3}{3}

u = \frac{3}{3}

u = \frac{3}{3}

Substitute back

u = tan (2 x)

tan (2 x) = \frac{3}{3}

tan (2 x) = \frac{3}{3}

tan (2 x) = \frac{3}{3} : x = \frac{π}{12} + \frac{πn}{2}

tan (2 x) = \frac{3}{3}

General solutions for

tan (2 x) = \frac{3}{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}

2 x = \frac{π}{6} + πn

2 x = \frac{π}{6} + πn

Solve

2 x = \frac{π}{6} + πn : x = \frac{π}{12} + \frac{πn}{2}

2 x = \frac{π}{6} + πn

Divide both sides by

2

2 x = \frac{π}{6} + πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{6}}{2} + \frac{πn}{2} : \frac{π}{12} + \frac{πn}{2}

\frac{\frac{π}{6}}{2} + \frac{πn}{2}

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{π}{12}

= \frac{π}{12} + \frac{πn}{2}

x = \frac{π}{12} + \frac{πn}{2}

x = \frac{π}{12} + \frac{πn}{2}

x = \frac{π}{12} + \frac{πn}{2}

x = \frac{π}{12} + \frac{πn}{2}

Combine all the solutions

x = \frac{π}{12} + \frac{πn}{2}

Popular Examples

cos(θ)= 3/(sqrt(34))

cos (θ) = \frac{3}{34}

tan(θ)+cot(θ)= 4/(sqrt(3))

tan (θ) + cot (θ) = \frac{4}{3}

sec(x)+4cos(x)=5

sec (x) + 4 cos (x) = 5

-sqrt(3)sec(x)=2

- 3 sec (x) = 2

3cos(x)csc(x)=2sqrt(3)cos(x)

3 cos (x) csc (x) = 23 cos (x)

Frequently Asked Questions (FAQ)

What is the general solution for 3-sqrt(3)tan(2x)=4-2sqrt(3)tan(2x) ?
The general solution for 3-sqrt(3)tan(2x)=4-2sqrt(3)tan(2x) is x= pi/(12)+(pin)/2