What is the value of cos(2pii) ?

The value of cos(2pii) is (e^{4pi}+1)/(2e^{2pi)}

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Popular Trigonometry >

cos(2pii)

Solution

cos (2 πi)

Solution

\frac{e ^{4 π} + 1}{2 e ^{2 π}}

Decimal

267.74676 \dots

Solution steps

cos (2 πi)

Rewrite using trig identities:

cos (0) cosh (2 π) - i sin (0) sinh (2 π)

cos (2 πi)

Use the following identity:

cos (a + bi) = cos (a) cosh (b) - i sin (a) sinh (b)

= cos (0) cosh (2 π) - i sin (0) sinh (2 π)

= cos (0) cosh (2 π) - i sin (0) sinh (2 π)

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

Rewrite using trig identities:

cosh (2 π) = \frac{e ^{4 π} + 1}{2 e ^{2 π}}

cosh (2 π)

Use the Hyperbolic identity:

cosh (x) = \frac{e ^{x} + e ^{- x}}{2}

= \frac{e ^{2 π} + e ^{- 2 π}}{2}

\frac{e ^{2 π} + e ^{- 2 π}}{2} = \frac{e ^{4 π} + 1}{2 e ^{2 π}}

\frac{e ^{2 π} + e ^{- 2 π}}{2}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= \frac{e ^{2 π} + \frac{1}{e ^{2 π}}}{2}

Join

e^{2 π} + \frac{1}{e ^{2 π}} : \frac{e ^{4 π} + 1}{e ^{2 π}}

e^{2 π} + \frac{1}{e ^{2 π}}

Convert element to fraction:

e^{2 π} = \frac{e ^{2 π} e ^{2 π}}{e ^{2 π}}

= \frac{e ^{2 π} e ^{2 π}}{e ^{2 π}} + \frac{1}{e ^{2 π}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{2 π} e ^{2 π} + 1}{e ^{2 π}}

e^{2 π} e^{2 π} + 1 = e^{4 π} + 1

e^{2 π} e^{2 π} + 1

e^{2 π} e^{2 π} = e^{4 π}

e^{2 π} e^{2 π}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{2 π} e^{2 π} = e^{2 π + 2 π}

= e^{2 π + 2 π}

Add similar elements:

2 π + 2 π = 4 π

= e^{4 π}

= e^{4 π} + 1

= \frac{e ^{4 π} + 1}{e ^{2 π}}

= \frac{\frac{e ^{4 π} + 1}{e ^{2 π}}}{2}

Apply the fraction rule:

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

= \frac{e ^{4 π} + 1}{e ^{2 π} \cdot 2}

= \frac{e ^{4 π} + 1}{2 e ^{2 π}}

Use the following trivial identity:

sin (0) = 0

sin (0)

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

= 0

Rewrite using trig identities:

sinh (2 π) = \frac{e ^{4 π} - 1}{2 e ^{2 π}}

sinh (2 π)

Use the Hyperbolic identity:

sinh (x) = \frac{e ^{x} - e ^{- x}}{2}

= \frac{e ^{2 π} - e ^{- 2 π}}{2}

\frac{e ^{2 π} - e ^{- 2 π}}{2} = \frac{e ^{4 π} - 1}{2 e ^{2 π}}

\frac{e ^{2 π} - e ^{- 2 π}}{2}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= \frac{e ^{2 π} - \frac{1}{e ^{2 π}}}{2}

Join

e^{2 π} - \frac{1}{e ^{2 π}} : \frac{e ^{4 π} - 1}{e ^{2 π}}

e^{2 π} - \frac{1}{e ^{2 π}}

Convert element to fraction:

e^{2 π} = \frac{e ^{2 π} e ^{2 π}}{e ^{2 π}}

= \frac{e ^{2 π} e ^{2 π}}{e ^{2 π}} - \frac{1}{e ^{2 π}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{2 π} e ^{2 π} - 1}{e ^{2 π}}

e^{2 π} e^{2 π} - 1 = e^{4 π} - 1

e^{2 π} e^{2 π} - 1

e^{2 π} e^{2 π} = e^{4 π}

e^{2 π} e^{2 π}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{2 π} e^{2 π} = e^{2 π + 2 π}

= e^{2 π + 2 π}

Add similar elements:

2 π + 2 π = 4 π

= e^{4 π}

= e^{4 π} - 1

= \frac{e ^{4 π} - 1}{e ^{2 π}}

= \frac{\frac{e ^{4 π} - 1}{e ^{2 π}}}{2}

Apply the fraction rule:

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

= \frac{e ^{4 π} - 1}{e ^{2 π} \cdot 2}

= \frac{e ^{4 π} - 1}{2 e ^{2 π}}

= 1 \cdot \frac{e ^{4 π} + 1}{2 e ^{2 π}} - i 0 \cdot \frac{e ^{4 π} - 1}{2 e ^{2 π}}

Simplify

1 \cdot \frac{e ^{4 π} + 1}{2 e ^{2 π}} - i 0 \cdot \frac{e ^{4 π} - 1}{2 e ^{2 π}} : \frac{e ^{4 π} + 1}{2 e ^{2 π}}

1 \cdot \frac{e ^{4 π} + 1}{2 e ^{2 π}} - i 0 \cdot \frac{e ^{4 π} - 1}{2 e ^{2 π}}

1 \cdot \frac{e ^{4 π} + 1}{2 e ^{2 π}} = \frac{e ^{4 π} + 1}{2 e ^{2 π}}

1 \cdot \frac{e ^{4 π} + 1}{2 e ^{2 π}}

Multiply:

1 \cdot \frac{e ^{4 π} + 1}{2 e ^{2 π}} = \frac{e ^{4 π} + 1}{2 e ^{2 π}}

= \frac{e ^{4 π} + 1}{2 e ^{2 π}}

i 0 \cdot \frac{e ^{4 π} - 1}{2 e ^{2 π}} = 0

i 0 \cdot \frac{e ^{4 π} - 1}{2 e ^{2 π}}

Multiply fractions:

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

= 0 \cdot \frac{i ( e ^{4 π} - 1 )}{2 e ^{2 π}}

Apply rule

0 \cdot a = 0

= 0

= \frac{e ^{4 π} + 1}{2 e ^{2 π}} - 0

\frac{e ^{4 π} + 1}{2 e ^{2 π}} - 0 = \frac{e ^{4 π} + 1}{2 e ^{2 π}}

= \frac{e ^{4 π} + 1}{2 e ^{2 π}}

= \frac{e ^{4 π} + 1}{2 e ^{2 π}}

Popular Examples

cos(-1(5/7))

tan(3/2 pi)

e^{pi/2}-e^{arctan(1+(\sqrt[4]{2/3})^2)}

cot(47)

(tan(50)-tan(20))/(1+tan(50)tan(20))

Frequently Asked Questions (FAQ)

What is the value of cos(2pii) ?
The value of cos(2pii) is (e^{4pi}+1)/(2e^{2pi)}