What is the value of 2csc((5pi)/6) ?

The value of 2csc((5pi)/6) is 4

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2csc((5pi)/6)

2 csc (\frac{5 π}{6})

4

Solution steps

2 csc (\frac{5 π}{6})

Rewrite using trig identities:

csc (\frac{5 π}{6}) = 2

csc (\frac{5 π}{6})

Rewrite using trig identities:

\frac{1}{sin ( \frac{5 π}{6} )}

csc (\frac{5 π}{6})

Use the basic trigonometric identity:

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

= \frac{1}{sin ( \frac{5 π}{6} )}

= \frac{1}{sin ( \frac{5 π}{6} )}

Use the following trivial identity:

sin (\frac{5 π}{6}) = \frac{1}{2}

sin (\frac{5 π}{6})

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{1}{2}

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

Simplify

= 2

= 2 \cdot 2

Simplify

= 4

sin (1) + cos (1 + (1)^{2}) - 1

sin (\frac{3 π}{2}) + cos (\frac{π}{4})

sin (4 0^{\circ}) cos (1 5^{\circ}) - cos (4 0^{\circ}) sin (1 5^{\circ})

cos (2 arctan (\frac{1}{2}))

- 3 cos (\frac{3 π}{2})