What is the general solution for tan^2(6x)=3 ?

The general solution for tan^2(6x)=3 is x= pi/(18)+(pin)/6 ,x= pi/9+(pin)/6

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tan^2(6x)=3

Solution

tan^{2} (6 x) = 3

Solution

x = \frac{π}{18} + \frac{πn}{6}, x = \frac{π}{9} + \frac{πn}{6}

Degrees

x = 1 0^{\circ} + 3 0^{\circ} n, x = 2 0^{\circ} + 3 0^{\circ} n

Solution steps

tan^{2} (6 x) = 3

Solve by substitution

tan^{2} (6 x) = 3

Let:

tan (6 x) = u

u^{2} = 3

u^{2} = 3 : u = 3, u = - 3

u^{2} = 3

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 3, u = - 3

Substitute back

u = tan (6 x)

tan (6 x) = 3, tan (6 x) = - 3

tan (6 x) = 3, tan (6 x) = - 3

tan (6 x) = 3 : x = \frac{π}{18} + \frac{πn}{6}

tan (6 x) = 3

General solutions for

tan (6 x) = 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}

6 x = \frac{π}{3} + πn

6 x = \frac{π}{3} + πn

Solve

6 x = \frac{π}{3} + πn : x = \frac{π}{18} + \frac{πn}{6}

6 x = \frac{π}{3} + πn

Divide both sides by

6

6 x = \frac{π}{3} + πn

Divide both sides by

6

\frac{6 x}{6} = \frac{\frac{π}{3}}{6} + \frac{πn}{6}

Simplify

\frac{6 x}{6} = \frac{\frac{π}{3}}{6} + \frac{πn}{6}

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

\frac{\frac{π}{3}}{6} + \frac{πn}{6} : \frac{π}{18} + \frac{πn}{6}

\frac{\frac{π}{3}}{6} + \frac{πn}{6}

\frac{\frac{π}{3}}{6} = \frac{π}{18}

\frac{\frac{π}{3}}{6}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 6}

Multiply the numbers:

3 \cdot 6 = 18

= \frac{π}{18}

= \frac{π}{18} + \frac{πn}{6}

x = \frac{π}{18} + \frac{πn}{6}

x = \frac{π}{18} + \frac{πn}{6}

x = \frac{π}{18} + \frac{πn}{6}

x = \frac{π}{18} + \frac{πn}{6}

tan (6 x) = - 3 : x = \frac{π}{9} + \frac{πn}{6}

tan (6 x) = - 3

General solutions for

tan (6 x) = - 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}

6 x = \frac{2 π}{3} + πn

6 x = \frac{2 π}{3} + πn

Solve

6 x = \frac{2 π}{3} + πn : x = \frac{π}{9} + \frac{πn}{6}

6 x = \frac{2 π}{3} + πn

Divide both sides by

6

6 x = \frac{2 π}{3} + πn

Divide both sides by

6

\frac{6 x}{6} = \frac{\frac{2 π}{3}}{6} + \frac{πn}{6}

Simplify

\frac{6 x}{6} = \frac{\frac{2 π}{3}}{6} + \frac{πn}{6}

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

\frac{\frac{2 π}{3}}{6} + \frac{πn}{6} : \frac{π}{9} + \frac{πn}{6}

\frac{\frac{2 π}{3}}{6} + \frac{πn}{6}

\frac{\frac{2 π}{3}}{6} = \frac{π}{9}

\frac{\frac{2 π}{3}}{6}

Apply the fraction rule:

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

= \frac{2 π}{3 \cdot 6}

Multiply the numbers:

3 \cdot 6 = 18

= \frac{2 π}{18}

Cancel the common factor:

2

= \frac{π}{9}

= \frac{π}{9} + \frac{πn}{6}

x = \frac{π}{9} + \frac{πn}{6}

x = \frac{π}{9} + \frac{πn}{6}

x = \frac{π}{9} + \frac{πn}{6}

x = \frac{π}{9} + \frac{πn}{6}

Combine all the solutions

x = \frac{π}{18} + \frac{πn}{6}, x = \frac{π}{9} + \frac{πn}{6}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for tan^2(6x)=3 ?
The general solution for tan^2(6x)=3 is x= pi/(18)+(pin)/6 ,x= pi/9+(pin)/6