What is the value of arccos(cos((5pi)/2)) ?

The value of arccos(cos((5pi)/2)) is pi/2

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arccos(cos((5pi)/2))

arccos (cos (\frac{5 π}{2}))

\frac{π}{2}

Decimal Notation

90

Solution steps

arccos (cos (\frac{5 π}{2}))

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

cos (\frac{5 π}{2})

Rewrite

\frac{5 π}{2}

2 π + \frac{π}{2}

= cos (2 π + \frac{π}{2})

Apply the periodicity of

cos

cos (x + 2 π) = cos (x)

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

= cos (\frac{π}{2})

= arccos (cos (\frac{π}{2}))

Use the following trivial identity:

cos (\frac{π}{2}) = 0

cos (\frac{π}{2})

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= arccos (0)

= arccos (0)

Use the following trivial identity:

arccos (0) = \frac{π}{2}

arccos (0)

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{π}{2}

= \frac{π}{2}

cos (1 5^{\circ}) cos (7 5^{\circ}) + sin (1 5^{\circ}) sin (7 5^{\circ})

arctan (\frac{- 1}{- 3})

sin (3 0^{\circ}) \cdot cos (3 0^{\circ})

tan^{4} (\frac{π}{4})

sin (- 63 0^{\circ})

What is the value of arccos(cos((5pi)/2)) ?
The value of arccos(cos((5pi)/2)) is pi/2