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

The value of csc(-(5pi)/2) is -1

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csc(-(5pi)/2)

csc (- \frac{5 π}{2})

- 1

Solution steps

csc (- \frac{5 π}{2})

Use the following property:

csc (- x) = - csc (x)

csc (- \frac{5 π}{2}) = - csc (\frac{5 π}{2})

= - csc (\frac{5 π}{2})

csc (\frac{5 π}{2}) = csc (\frac{π}{2})

csc (\frac{5 π}{2})

Rewrite

\frac{5 π}{2}

2 π + \frac{π}{2}

= csc (2 π + \frac{π}{2})

Apply the periodicity of

csc

csc (x + 2 π) = csc (x)

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

= csc (\frac{π}{2})

= - csc (\frac{π}{2})

Use the following trivial identity:

csc (\frac{π}{2}) = 1

csc (\frac{π}{2})

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

= 1

= - 1

\frac{11}{cos ( 3 7 ^{\circ} )}

sec (2 7^{\circ})

\frac{8.8}{tan ( 7 2 ^{\circ} )}

sin (7 0^{\circ}) cos (1 0^{\circ}) - cos (7 0^{\circ}) sin (1 0^{\circ})

\frac{1 + cos ( 13 5 ^{\circ} )}{2}