What is the value of csc(-420) ?

The value of csc(-420) is -(2sqrt(3))/3

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csc(-420)

csc (- 42 0^{\circ})

- \frac{2 3}{3}

Decimal Notation

- 1.15470 \dots

Solution steps

csc (- 42 0^{\circ})

Use the following property:

csc (- x) = - csc (x)

csc (- 42 0^{\circ}) = - csc (42 0^{\circ})

= - csc (42 0^{\circ})

csc (42 0^{\circ}) = csc (6 0^{\circ})

csc (42 0^{\circ})

Rewrite

42 0^{\circ}

36 0^{\circ} + 6 0^{\circ}

= csc (36 0^{\circ} + 6 0^{\circ})

Apply the periodicity of

csc

csc (x + 36 0^{\circ}) = csc (x)

csc (36 0^{\circ} + 6 0^{\circ}) = csc (6 0^{\circ})

= csc (6 0^{\circ})

= - csc (6 0^{\circ})

Use the following trivial identity:

csc (6 0^{\circ}) = \frac{2 3}{3}

csc (6 0^{\circ})

csc (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

= \frac{2 3}{3}

= - \frac{2 3}{3}

arccos (π)

- sin (6 0^{\circ})

cot (- 1)

arccos (sin (\frac{5 π}{6}))

tan (5 7^{\circ})