What is the value of sin((pi+2i)/2) ?

The value of sin((pi+2i)/2) is (e^2+1)/(2e)

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sin((pi+2i)/2)

Solution

sin (\frac{π + 2 i}{2})

Solution

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

Decimal Notation

1.54308 \dots

Solution steps

sin (\frac{π + 2 i}{2})

Rewrite using trig identities:

sin (\frac{π}{2}) cosh (1) + i cos (\frac{π}{2}) sinh (1)

sin (\frac{π + 2 i}{2})

Use the following identity:

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

= sin (\frac{π}{2}) cosh (1) + i cos (\frac{π}{2}) sinh (1)

= sin (\frac{π}{2}) cosh (1) + i cos (\frac{π}{2}) sinh (1)

Use the following trivial identity:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

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}

= 1

Rewrite using trig identities:

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

cosh (1)

Use the Hyperbolic identity:

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

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

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

\frac{e ^{1} + e ^{- 1}}{2}

Apply rule

a^{1} = a

e^{1} = e

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

Apply exponent rule:

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

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

Join

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

e + \frac{1}{e}

Convert element to fraction:

e = \frac{ee}{e}

= \frac{ee}{e} + \frac{1}{e}

Since the denominators are equal, combine the fractions:

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

= \frac{ee + 1}{e}

ee + 1 = e^{2} + 1

ee + 1

ee = e^{2}

ee

Apply exponent rule:

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

ee = e^{1 + 1}

= e^{1 + 1}

Add the numbers:

1 + 1 = 2

= e^{2}

= e^{2} + 1

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

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

Apply the fraction rule:

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

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

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

Use the following trivial identity:

cos (\frac{π}{2}) = 0

cos (\frac{π}{2})

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}

= 0

Rewrite using trig identities:

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

sinh (1)

Use the Hyperbolic identity:

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

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

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

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

Apply rule

a^{1} = a

e^{1} = e

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

Apply exponent rule:

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

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

Join

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

e - \frac{1}{e}

Convert element to fraction:

e = \frac{ee}{e}

= \frac{ee}{e} - \frac{1}{e}

Since the denominators are equal, combine the fractions:

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

= \frac{ee - 1}{e}

ee - 1 = e^{2} - 1

ee - 1

ee = e^{2}

ee

Apply exponent rule:

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

ee = e^{1 + 1}

= e^{1 + 1}

Add the numbers:

1 + 1 = 2

= e^{2}

= e^{2} - 1

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

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

Apply the fraction rule:

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

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

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

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

Simplify

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

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

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

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

Multiply:

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

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

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

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

Multiply fractions:

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

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

Apply rule

0 \cdot a = 0

= 0

= \frac{e ^{2} + 1}{2 e} + 0

\frac{e ^{2} + 1}{2 e} + 0 = \frac{e ^{2} + 1}{2 e}

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

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

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Frequently Asked Questions (FAQ)

What is the value of sin((pi+2i)/2) ?
The value of sin((pi+2i)/2) is (e^2+1)/(2e)