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

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### Complex arccos function

*x* = arccos(3)

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

cos(*x*) = 3

From Euler’s formula

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

*e*) / 2

^{-ix}(

*e*+

^{ix}*e*) / 2 = 3

^{-ix}*e*+

^{ix}*e*= 6

^{-ix}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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