What is the value of csc(405) ?

The value of csc(405) is square root of 2

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csc(405)

csc (40 5^{\circ})

2

Decimal Notation

1.41421 \dots

Solution steps

csc (40 5^{\circ})

csc (40 5^{\circ}) = csc (4 5^{\circ})

csc (40 5^{\circ})

Rewrite

40 5^{\circ}

36 0^{\circ} + 4 5^{\circ}

= csc (36 0^{\circ} + 4 5^{\circ})

Apply the periodicity of

csc

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

csc (36 0^{\circ} + 4 5^{\circ}) = csc (4 5^{\circ})

= csc (4 5^{\circ})

= csc (4 5^{\circ})

Use the following trivial identity:

csc (4 5^{\circ}) = 2

csc (4 5^{\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

= 2

= 2

sin (\frac{1}{8})

5 cos (17 0^{\circ})

cos (\frac{1}{3})

3 cos (3)

cos (5) (0)