What is the general solution for 4tan^2(x)-12=0 ?

The general solution for 4tan^2(x)-12=0 is x= pi/3+pin,x=(2pi)/3+pin

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4tan^2(x)-12=0

4 tan^{2} (x) - 12 = 0

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

Degrees

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

Solution steps

4 tan^{2} (x) - 12 = 0

Solve by substitution

4 tan^{2} (x) - 12 = 0

Let:

tan (x) = u

4 u^{2} - 12 = 0

4 u^{2} - 12 = 0 : u = 3, u = - 3

4 u^{2} - 12 = 0

Move

12

to the right side

4 u^{2} - 12 = 0

Add

12

to both sides

4 u^{2} - 12 + 12 = 0 + 12

Simplify

4 u^{2} = 12

4 u^{2} = 12

Divide both sides by

4

4 u^{2} = 12

Divide both sides by

4

\frac{4 u ^{2}}{4} = \frac{12}{4}

Simplify

u^{2} = 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 (x)

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

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

tan (x) = 3 : x = \frac{π}{3} + πn

tan (x) = 3

General solutions for

tan (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}

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

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

tan (x) = - 3

General solutions for

tan (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}

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

Combine all the solutions

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

tan (θ) - \frac{1}{3} = 0

tan (\frac{x}{2} + \frac{π}{3}) = 1

- cot (x) + csc (x) = 1

cot (\frac{x}{2}) = 0

sin^{2} (θ) = 2 sin (θ) + 3

What is the general solution for 4tan^2(x)-12=0 ?
The general solution for 4tan^2(x)-12=0 is x= pi/3+pin,x=(2pi)/3+pin