What is the value of csc(120) ?

The value of csc(120) is (2sqrt(3))/3

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csc(120)

csc (12 0^{\circ})

\frac{2 3}{3}

Decimal Notation

1.15470 \dots

Solution steps

csc (12 0^{\circ})

Rewrite using trig identities:

\frac{1}{sin ( 12 0 ^{\circ} )}

csc (12 0^{\circ})

Use the basic trigonometric identity:

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

= \frac{1}{sin ( 12 0 ^{\circ} )}

= \frac{1}{sin ( 12 0 ^{\circ} )}

Use the following trivial identity:

sin (12 0^{\circ}) = \frac{3}{2}

sin (12 0^{\circ})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} 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 (θ) - 3 = 0

sin (4 0^{\circ})

2 cos (x + \frac{π}{6}) = 1, 0 < x < 2 π

tan (\frac{13 π}{4})

2 cos (x) + 3 = 0