What is the general solution for 3tan^2(x)-3tan(x)=0 ?

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

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3tan^2(x)-3tan(x)=0

Solution

3 tan^{2} (x) - 3 tan (x) = 0

Solution

x = \frac{π}{4} + πn, x = πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n, x = 0^{\circ} + 18 0^{\circ} n

Solution steps

3 tan^{2} (x) - 3 tan (x) = 0

Solve by substitution

3 tan^{2} (x) - 3 tan (x) = 0

Let:

tan (x) = u

3 u^{2} - 3 u = 0

3 u^{2} - 3 u = 0 : u = 1, u = 0

3 u^{2} - 3 u = 0

Solve with the quadratic formula

3 u^{2} - 3 u = 0

Quadratic Equation Formula:

For

a = 3, b = - 3, c = 0

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 3 \cdot 0}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 3 \cdot 0}{2 \cdot 3}

(- 3)^{2} - 4 \cdot 3 \cdot 0 = 3

(- 3)^{2} - 4 \cdot 3 \cdot 0

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 3)^{2} = 3^{2}

= 3^{2} - 4 \cdot 3 \cdot 0

Apply rule

0 \cdot a = 0

= 3^{2} - 0

3^{2} - 0 = 3^{2}

= 3^{2}

Apply radical rule:

assuming

a \geq 0

= 3

u_{1, 2} = \frac{- ( - 3 ) \pm 3}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 3}{2 \cdot 3}, u_{2} = \frac{- ( - 3 ) - 3}{2 \cdot 3}

u = \frac{- ( - 3 ) + 3}{2 \cdot 3} : 1

\frac{- ( - 3 ) + 3}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{3 + 3}{2 \cdot 3}

Add the numbers:

3 + 3 = 6

= \frac{6}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6}{6}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- ( - 3 ) - 3}{2 \cdot 3} : 0

\frac{- ( - 3 ) - 3}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{3 - 3}{2 \cdot 3}

Subtract the numbers:

3 - 3 = 0

= \frac{0}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{0}{6}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

The solutions to the quadratic equation are:

u = 1, u = 0

Substitute back

u = tan (x)

tan (x) = 1, tan (x) = 0

tan (x) = 1, tan (x) = 0

tan (x) = 1 : x = \frac{π}{4} + πn

tan (x) = 1

General solutions for

tan (x) = 1

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}{4} + πn

x = \frac{π}{4} + πn

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

Combine all the solutions

x = \frac{π}{4} + πn, x = πn

Popular Examples

2sin^2(x)+2sin(x)+1=0

2 sin^{2} (x) + 2 sin (x) + 1 = 0

4cos(x)-6sin(x)=3

4 cos (x) - 6 sin (x) = 3

sin(x)=0.64

sin (x) = 0.64

4sin^3(x)-8sin^2(x)+sin(x)+3=0

4 sin^{3} (x) - 8 sin^{2} (x) + sin (x) + 3 = 0

sin(x)=0.71

sin (x) = 0.71

Frequently Asked Questions (FAQ)

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