What is the value of sin(-i) ?

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

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sin(-i)

Solution

sin (- i)

Solution

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

Solution steps

sin (- i)

Rewrite using trig identities:

sin (0) cosh (1) - i cos (0) sinh (1)

sin (- i)

Use the following identity:

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

= sin (0) cosh (- 1) + i cos (0) sinh (- 1)

Use the following property:

sinh (- x) = - sinh (x)

sinh (- 1) = - sinh (1)

= sin (0) cosh (- 1) + i cos (0) (- sinh (1))

Use the following property:

cosh (- x) = cosh (x)

cosh (- 1) = cosh (1)

= sin (0) cosh (1) + i cos (0) (- sinh (1))

Simplify

= sin (0) cosh (1) - i cos (0) sinh (1)

= sin (0) cosh (1) - i cos (0) sinh (1)

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:

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 (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:

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}

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

Simplify

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

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

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

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

Apply rule

0 \cdot a = 0

= 0

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

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

Multiply fractions:

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

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

Multiply:

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

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

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

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

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

Rewrite

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

in standard complex form:

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

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

Expand

i (e^{2} - 1) : e^{2} i - i

i (e^{2} - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = i, b = e^{2}, c = 1

= i e^{2} - i 1

= e^{2} i - 1 i

Multiply:

1 i = i

= e^{2} i - i

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

Apply the fraction rule:

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

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

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

Remove parentheses:

(a) = a, - (- a) = a

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

Cancel

\frac{e ^{2} i}{2 e} : \frac{e i}{2}

\frac{e ^{2} i}{2 e}

Cancel the common factor:

e

= \frac{e i}{2}

= - \frac{e i}{2} + \frac{i}{2 e}

Group the real part and the imaginary part of the complex number

= (- \frac{e}{2} + \frac{1}{2 e}) i

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

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

Least Common Multiplier of

2, 2 e : 2 e

2, 2 e

Lowest Common Multiplier (LCM)

Least Common Multiplier of

2, 2 : 2

2, 2

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

2

2

= 2

Multiply the numbers:

2 = 2

= 2

Compute an expression comprised of factors that appear either in

2

2 e

= 2 e

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

2 e

For

\frac{e}{2} :

multiply the denominator and numerator by

e

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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arctan (- 2.42)

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arccos (\frac{19}{25})

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Frequently Asked Questions (FAQ)

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