What is the value of cos((19pi)/4) ?

The value of cos((19pi)/4) is -(sqrt(2))/2

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cos((19pi)/4)

cos (\frac{19 π}{4})

- \frac{2}{2}

Decimal Notation

- 0.70710 \dots

Solution steps

cos (\frac{19 π}{4})

cos (\frac{19 π}{4}) = cos (\frac{3 π}{4})

cos (\frac{19 π}{4})

Rewrite

\frac{19 π}{4}

2 π \cdot 2 + \frac{3 π}{4}

= cos (2 π 2 + \frac{3 π}{4})

Apply the periodicity of

cos

cos (x + 2 π \cdot k) = cos (x)

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

= cos (\frac{3 π}{4})

= cos (\frac{3 π}{4})

Use the following trivial identity:

cos (\frac{3 π}{4}) = - \frac{2}{2}

cos (\frac{3 π}{4})

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}

= - \frac{2}{2}

= - \frac{2}{2}

cos (17 0^{\circ})

cos (arctan (\frac{4}{3}))

arctan (\frac{12}{- 5})

sin (arcsin (\frac{1}{2}) + arctan (- 3))

- sin (\frac{5 π}{3})