What is the value of csc((8pi)/3) ?

The value of csc((8pi)/3) is (2sqrt(3))/3

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csc((8pi)/3)

csc (\frac{8 π}{3})

\frac{2 3}{3}

Decimal Notation

1.15470 \dots

Solution steps

csc (\frac{8 π}{3})

csc (\frac{8 π}{3}) = csc (\frac{2 π}{3})

csc (\frac{8 π}{3})

Rewrite

\frac{8 π}{3}

2 π + \frac{2 π}{3}

= csc (2 π + \frac{2 π}{3})

Apply the periodicity of

csc

csc (x + 2 π) = csc (x)

csc (2 π + \frac{2 π}{3}) = csc (\frac{2 π}{3})

= csc (\frac{2 π}{3})

= csc (\frac{2 π}{3})

Rewrite using trig identities:

\frac{1}{sin ( \frac{2 π}{3} )}

csc (\frac{2 π}{3})

Use the basic trigonometric identity:

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

= \frac{1}{sin ( \frac{2 π}{3} )}

= \frac{1}{sin ( \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}

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

Simplify

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

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

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{2}{3}

Rationalize

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

\frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{2 3}{3}

= \frac{2 3}{3}

= \frac{2 3}{3}

cot (\frac{9 π}{4})

tan (arccos (\frac{5}{13}))

arctan (- \frac{1}{0})

- 3 sin (\frac{π}{2})

arcsin (- 3)