What is the value of (2tan((5pi)/6))/(1-tan^2((5pi)/6)) ?

The value of (2tan((5pi)/6))/(1-tan^2((5pi)/6)) is -sqrt(3)

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(2tan((5pi)/6))/(1-tan^2((5pi)/6))

Solution

\frac{2 tan ( \frac{5 π}{6} )}{1 - tan ^{2} ( \frac{5 π}{6} )}

Solution

- 3

Decimal Notation

- 1.73205 \dots

Solution steps

\frac{2 tan ( \frac{5 π}{6} )}{1 - tan ^{2} ( \frac{5 π}{6} )}

Rewrite using trig identities:

tan (\frac{5 π}{6}) = - \frac{3}{3}

tan (\frac{5 π}{6})

Rewrite using trig identities:

\frac{sin ( \frac{5 π}{6} )}{cos ( \frac{5 π}{6} )}

tan (\frac{5 π}{6})

Use the basic trigonometric identity:

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

= \frac{sin ( \frac{5 π}{6} )}{cos ( \frac{5 π}{6} )}

= \frac{sin ( \frac{5 π}{6} )}{cos ( \frac{5 π}{6} )}

Use the following trivial identity:

sin (\frac{5 π}{6}) = \frac{1}{2}

sin (\frac{5 π}{6})

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{1}{2}

Use the following trivial identity:

cos (\frac{5 π}{6}) = - \frac{3}{2}

cos (\frac{5 π}{6})

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{3}{2}

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

Simplify

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

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

Apply the fraction rule:

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

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

Divide fractions:

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

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

Refine

= - \frac{2}{2 3}

Cancel the common factor:

2

= - \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}

= - \frac{3}{3}

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

Simplify

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

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

Remove parentheses:

(- a) = - a

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

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

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

Apply exponent rule:

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

n

is even

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

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

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

Apply the fraction rule:

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

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

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

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

Apply exponent rule:

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

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

(3)^{2} : 3

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{3}{3 ^{2}}

Cancel the common factor:

3

= \frac{1}{3}

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

Join

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

1 - \frac{1}{3}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 3 - 1 = 2

1 \cdot 3 - 1

Multiply the numbers:

1 \cdot 3 = 3

= 3 - 1

Subtract the numbers:

3 - 1 = 2

= 2

= \frac{2}{3}

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

Simplify

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

\frac{2 \cdot \frac{3}{3}}{\frac{2}{3}}

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 3 = 6

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

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

Multiply

6 \cdot \frac{3}{3} : 23

6 \cdot \frac{3}{3}

Multiply fractions:

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

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

Divide the numbers:

\frac{6}{3} = 2

= 23

= - \frac{2 3}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 3

= - 3

Popular Examples

cos(-(7pi)/3)

cos (- \frac{7 π}{3})

tan(arccos(-3/5))

tan (arccos (- \frac{3}{5}))

arccos(0.77)

arccos (0.77)

arccos(0.97)

arccos (0.97)

sin(5/2 pi)

sin (\frac{5}{2} π)

Frequently Asked Questions (FAQ)

What is the value of (2tan((5pi)/6))/(1-tan^2((5pi)/6)) ?
The value of (2tan((5pi)/6))/(1-tan^2((5pi)/6)) is -sqrt(3)