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

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

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3tan^2(x)=sqrt(3)tan(x)

Solution

3 tan^{2} (x) = 3 tan (x)

Solution

x = πn, x = \frac{π}{6} + πn

Degrees

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

Solution steps

3 tan^{2} (x) = 3 tan (x)

Solve by substitution

3 tan^{2} (x) = 3 tan (x)

Let:

tan (x) = u

3 u^{2} = 3 u

3 u^{2} = 3 u : u = 0, u = \frac{3}{3}

3 u^{2} = 3 u

Move

3 u

to the left side

3 u^{2} = 3 u

Subtract

3 u

from both sides

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

Simplify

3 u^{2} - 3 u = 0

3 u^{2} - 3 u = 0

Factor

3 u^{2} - 3 u : u (3 u - 3)

3 u^{2} - 3 u

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{2} = uu

= 3 uu - 3 u

Factor out common term

u

= u (3 u - 1 \cdot 3)

Multiply the numbers:

1 \cdot 3 = 3

= u (3 u - 3)

u (3 u - 3) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or 3 u - 3 = 0

Solve

3 u - 3 = 0 : u = \frac{3}{3}

3 u - 3 = 0

Move

3

to the right side

3 u - 3 = 0

Add

3

to both sides

3 u - 3 + 3 = 0 + 3

Simplify

3 u = 3

3 u = 3

Divide both sides by

3

3 u = 3

Divide both sides by

3

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

Simplify

u = \frac{3}{3}

u = \frac{3}{3}

The solutions to the quadratic equation are:

u = 0, u = \frac{3}{3}

Substitute back

u = tan (x)

tan (x) = 0, tan (x) = \frac{3}{3}

tan (x) = 0, tan (x) = \frac{3}{3}

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

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

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

General solutions for

tan (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}

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

Combine all the solutions

x = πn, x = \frac{π}{6} + πn

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Frequently Asked Questions (FAQ)

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