What is the general solution for tan(2x)(1-tan^2(x))= 2/(sqrt(3)) ?

The general solution for tan(2x)(1-tan^2(x))= 2/(sqrt(3)) is x= pi/6+pin

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tan(2x)(1-tan^2(x))= 2/(sqrt(3))

Solution

tan (2 x) (1 - tan^{2} (x)) = \frac{2}{3}

Solution

x = \frac{π}{6} + πn

Degrees

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

Solution steps

tan (2 x) (1 - tan^{2} (x)) = \frac{2}{3}

Subtract

\frac{2}{3}

from both sides

tan (2 x) (1 - tan^{2} (x)) - \frac{2}{3} = 0

Simplify

tan (2 x) (1 - tan^{2} (x)) - \frac{2}{3} : \frac{3 tan ( 2 x ) ( 1 - tan ^{2} ( x ) ) - 2}{3}

tan (2 x) (1 - tan^{2} (x)) - \frac{2}{3}

Convert element to fraction:

tan (2 x) (- tan^{2} (x) + 1) = \frac{tan ( 2 x ) ( 1 - tan ^{2} ( x ) ) 3}{3}

= \frac{tan ( 2 x ) ( 1 - tan ^{2} ( x ) ) 3}{3} - \frac{2}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{tan ( 2 x ) ( 1 - tan ^{2} ( x ) ) 3 - 2}{3}

\frac{3 tan ( 2 x ) ( 1 - tan ^{2} ( x ) ) - 2}{3} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 tan (2 x) (1 - tan^{2} (x)) - 2 = 0

Rewrite using trig identities

- 2 + (1 - tan^{2} (x)) 3 tan (2 x)

Use the Double Angle identity:

tan (2 x) = \frac{2 tan ( x )}{1 - tan ^{2} ( x )}

= - 2 + 3 \frac{2 tan ( x )}{1 - tan ^{2} ( x )} (1 - tan^{2} (x))

3 \frac{2 tan ( x )}{1 - tan ^{2} ( x )} (1 - tan^{2} (x)) = 23 tan (x)

3 \frac{2 tan ( x )}{1 - tan ^{2} ( x )} (1 - tan^{2} (x))

Multiply fractions:

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

= \frac{2 tan ( x ) 3 ( 1 - tan ^{2} ( x ) )}{1 - tan ^{2} ( x )}

Cancel the common factor:

1 - tan^{2} (x)

= 2 tan (x) 3

= - 2 + 23 tan (x)

- 2 + 23 tan (x) = 0

Move

2

to the right side

- 2 + 23 tan (x) = 0

Add

2

to both sides

- 2 + 23 tan (x) + 2 = 0 + 2

Simplify

23 tan (x) = 2

23 tan (x) = 2

Divide both sides by

23

23 tan (x) = 2

Divide both sides by

23

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

Simplify

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

Simplify

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

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

Divide the numbers:

\frac{2}{2} = 1

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

Cancel the common factor:

3

= tan (x)

Simplify

\frac{2}{2 3} : \frac{3}{3}

\frac{2}{2 3}

Divide the numbers:

\frac{2}{2} = 1

= \frac{1}{3}

Rationalize

\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}

= \frac{3}{3}

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

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

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

Popular Examples

tan(x)=25762

tan (x) = 25762

sin(8x)=sin(6x)

sin (8 x) = sin (6 x)

1/(sin(x))=sqrt(2)

\frac{1}{sin ( x )} = 2

0=sec((pix)/4)

0 = sec (\frac{π x}{4})

3cos^2(x)+2cos(x)-1=0

3 cos^{2} (x) + 2 cos (x) - 1 = 0

Frequently Asked Questions (FAQ)

What is the general solution for tan(2x)(1-tan^2(x))= 2/(sqrt(3)) ?
The general solution for tan(2x)(1-tan^2(x))= 2/(sqrt(3)) is x= pi/6+pin