What is the value of (2tan((2pi)/3))/(1-tan^2((2pi)/3)) ?

The value of (2tan((2pi)/3))/(1-tan^2((2pi)/3)) is square root of 3

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

(2tan((2pi)/3))/(1-tan^2((2pi)/3))

Solution

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

Solution

3

Decimal Notation

1.73205 \dots

Solution steps

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

Rewrite using trig identities:

tan (\frac{2 π}{3}) = - 3

tan (\frac{2 π}{3})

Rewrite using trig identities:

\frac{sin ( \frac{2 π}{3} )}{cos ( \frac{2 π}{3} )}

tan (\frac{2 π}{3})

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{sin ( \frac{2 π}{3} )}{cos ( \frac{2 π}{3} )}

= \frac{sin ( \frac{2 π}{3} )}{cos ( \frac{2 π}{3} )}

Use the following trivial identity:

sin (\frac{2 π}{3}) = \frac{3}{2}

sin (\frac{2 π}{3})

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

Use the following trivial identity:

cos (\frac{2 π}{3}) = - \frac{1}{2}

cos (\frac{2 π}{3})

cos (x)

periodicity table with

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{1}{2}

= \frac{\frac{3}{2}}{- \frac{1}{2}}

Simplify

\frac{\frac{3}{2}}{- \frac{1}{2}} : - 3

\frac{\frac{3}{2}}{- \frac{1}{2}}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{\frac{3}{2}}{\frac{1}{2}}

Divide fractions:

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

= - \frac{3 \cdot 2}{2 \cdot 1}

Refine

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

Cancel the common factor:

2

= - 3

= - 3

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

Simplify

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

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

Remove parentheses:

(- a) = - a

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

1 - (- 3)^{2} = 1 - (3)^{2}

1 - (- 3)^{2}

Apply exponent rule:

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

n

is even

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

= 1 - (3)^{2}

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

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

1 - (3)^{2} = - ((3)^{2} - 1)

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

(3)^{2} = 3

(3)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

= \frac{2 3}{3 - 1}

Subtract the numbers:

3 - 1 = 2

= \frac{2 3}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3

= 3

Popular Examples

8(cos(30)+isin(30))

8 (cos (3 0^{\circ}) + i sin (3 0^{\circ}))

3*sin((2pi)/3)

3 \cdot sin (\frac{2 π}{3})

18sin(71)

18 sin (7 1^{\circ})

(15)/(sin(68))

\frac{15}{sin ( 6 8 ^{\circ} )}

arccos(cos(-30))

arccos (cos (- 3 0^{\circ}))

Frequently Asked Questions (FAQ)

What is the value of (2tan((2pi)/3))/(1-tan^2((2pi)/3)) ?
The value of (2tan((2pi)/3))/(1-tan^2((2pi)/3)) is square root of 3