What is the arccosine of 3 ?

arccos 3 = ?

## Real arccos function

The arccosine is the inverse cosine function.

Since the cosine function has output values from -1 to 1,

the arccosine function has input values from -1 to 1.

So arccos x is undefined for x=3.

arccos 3 is undefined

## Complex arccos function

*x* = arccos(3)

cos(*x*) = cos(arccos(3))

cos(*x*) = 3

From Euler's formula

cos(*x*) = (*e ^{ix}* +

*e*

^{-ix}) / 2

(*e ^{ix}* +

*e*

^{-ix}) / 2 = 3

*e ^{ix}* +

*e*

^{-ix}= 6

Multiply with *e ^{ix}*

*e*^{2ix} + 1 = 6*e ^{ix}*

*y* = *e ^{ix}*

We get quadratic equation:

*y*^{2} - 6*y* + 1 = 0

*y*_{1,2} = (6 ± √32)/2

*y*_{1} = 5.828427 = *e ^{ix}*

*y*_{2} = 0.171573 = *e ^{ix}*

Apply ln on both sides gives the solution for arccos(3):

*x*_{1} = ln(5.828427) / *i*

*x*_{2} = ln(0.171573) / *i*

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