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

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

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cos(3((2pi)/3))

cos (3 (\frac{2 π}{3}))

1

Solution steps

cos (3 (\frac{2 π}{3}))

= cos (3 \cdot \frac{2 π}{3})

Simplify:

3 \cdot \frac{2 π}{3} = 2 π

3 \cdot \frac{2 π}{3}

Multiply fractions:

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

= \frac{2 π 3}{3}

Cancel the common factor:

3

= 2 π

= cos (2 π)

cos (2 π) = cos (0)

cos (2 π)

Rewrite

2 π

2 π + 0

= cos (2 π + 0)

Apply the periodicity of

cos

cos (x + 2 π) = cos (x)

cos (2 π + 0) = cos (0)

= cos (0)

= cos (0)

Use the following trivial identity:

cos (0) = 1

cos (0)

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}

= 1

= 1

cos (2 5^{\circ}) cos (2 0^{\circ}) - sin (2 5^{\circ}) sin (2 0^{\circ})

[2 (cos (\frac{π}{3}) + i sin (\frac{π}{3}))]^{4}

\frac{5}{sin ( 4 0 ^{\circ} )}

sin (arcsin (\frac{5}{13}) + arccos (- \frac{7}{25}))

cos (π) π