What is the value of 2+(8cos((pi*3)/3))-(2cos((2pi*3)/3)) ?

The value of 2+(8cos((pi*3)/3))-(2cos((2pi*3)/3)) is -8

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2+(8cos((pi3)/3))-(2cos((2pi3)/3))

2 + (8 cos (\frac{π \cdot 3}{3})) - (2 cos (\frac{2 π \cdot 3}{3}))

- 8

Solution steps

2 + (8 cos (\frac{π 3}{3})) - (2 cos (\frac{2 π 3}{3}))

= 2 + 8 cos (\frac{π 3}{3}) - 2 cos (\frac{2 π 3}{3})

Simplify

= 2 + 8 cos (π) - 2 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)

= 2 + 8 cos (π) - 2 cos (0)

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

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)

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

= 2 + 8 (- 1) - 2 \cdot 1

Simplify

= - 8

What is the value of 2+(8cos((pi*3)/3))-(2cos((2pi*3)/3)) ?
The value of 2+(8cos((pi*3)/3))-(2cos((2pi*3)/3)) is -8