What is the arccos of 3 ?

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) = (eix + e-ix) / 2

(eix + e-ix) / 2 = 3

eix + e-ix = 6

Multiply with eix

e2ix + 1 = 6eix

y = eix

We get quadratic equation:

y2 - 6y + 1 = 0

y1,2 = (6 ± √32)/2

y1 = 5.828427 = eix

y2 = 0.171573 = eix

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

x1 = ln(5.828427) / i

x2 = ln(0.171573) / i

 

 


See also

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