What is the value of cos(pi/3+2pi) ?

The value of cos(pi/3+2pi) is 1/2

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cos(pi/3+2pi)

cos (\frac{π}{3} + 2 π)

\frac{1}{2}

Decimal Notation

0.5

Solution steps

cos (\frac{π}{3} + 2 π)

Simplify:

\frac{π}{3} + 2 π = \frac{7 π}{3}

\frac{π}{3} + 2 π

Convert element to fraction:

2 π = \frac{2 π 3}{3}

= \frac{π}{3} + \frac{2 π 3}{3}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π + 2 π 3}{3}

π + 2 π 3 = 7 π

π + 2 π 3

Multiply the numbers:

2 \cdot 3 = 6

= π + 6 π

Add similar elements:

π + 6 π = 7 π

= 7 π

= \frac{7 π}{3}

= cos (\frac{7 π}{3})

cos (\frac{7 π}{3}) = cos (\frac{π}{3})

cos (\frac{7 π}{3})

Rewrite

\frac{7 π}{3}

2 π + \frac{π}{3}

= cos (2 π + \frac{π}{3})

Apply the periodicity of

cos

cos (x + 2 π) = cos (x)

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

= cos (\frac{π}{3})

= cos (\frac{π}{3})

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{π}{3})

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{1}{2}

= \frac{1}{2}

3 \cdot sin (6 0^{\circ})

- ln ∣ cos (5) ∣

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

\frac{3}{8} i cos^{2} (6 0^{\circ})

arctan (\frac{1}{6})