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

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

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tan(2pi 1/3)

tan (2 π \frac{1}{3})

- 3

Decimal Notation

- 1.73205 \dots

Solution steps

tan (2 π \frac{1}{3})

Rewrite using trig identities:

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

tan (2 π \frac{1}{3})

Use the Double Angle identity:

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

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

Simplify:

π \frac{1}{3} = \frac{π}{3}

π \frac{1}{3}

Multiply fractions:

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

= \frac{1 π}{3}

Multiply:

1 π = π

= \frac{π}{3}

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

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

Use the following trivial identity:

tan (\frac{π}{3}) = 3

tan (\frac{π}{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}

= 3

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

Simplify

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

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

(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}{1 - 3}

Subtract the numbers:

1 - 3 = - 2

= \frac{2 3}{- 2}

Apply the fraction rule:

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

= - \frac{2 3}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 3

= - 3

arcsech (\frac{2}{3})

sin (\frac{1}{\frac{2}{11 π}})

sec (- 1 4^{\circ})

60 \cdot sin (4 5^{\circ})

2 cos^{2} (\frac{3 π}{4}) - 1