What is the value of csc((13pi)/3) ?

The value of csc((13pi)/3) is (2sqrt(3))/3

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csc((13pi)/3)

csc (\frac{13 π}{3})

\frac{2 3}{3}

Decimal Notation

1.15470 \dots

Solution steps

csc (\frac{13 π}{3})

csc (\frac{13 π}{3}) = csc (\frac{π}{3})

csc (\frac{13 π}{3})

Rewrite

\frac{13 π}{3}

2 π \cdot 2 + \frac{π}{3}

= csc (2 π 2 + \frac{π}{3})

Apply the periodicity of

csc

csc (x + 2 π \cdot k) = csc (x)

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

= csc (\frac{π}{3})

= csc (\frac{π}{3})

Use the following trivial identity:

csc (\frac{π}{3}) = \frac{2 3}{3}

csc (\frac{π}{3})

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

= \frac{2 3}{3}

= \frac{2 3}{3}

4.9 \cdot 1 0^{6} \cdot 1.6 \cdot 1 0^{- 19} \cdot 0.19 \cdot sin (6 4^{\circ})

cot (2 8^{\circ})

arctan (- 0.2)

5 \cdot cos (4 5^{\circ})

- 4 sin (- \frac{π}{4})