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) = (e^ix + e^-ix) / 2
(e^ix + e^-ix) / 2 = 3
e^ix + e^-ix = 6

Multiply with e^ix

e^2ix + 1 = 6e^ix
y = e^ix

We get quadratic equation:

y² – 6y + 1 = 0
y_1,2 = (6 ± √32)/2
y₁ = 5.828427 = e^ix
y₂ = 0.171573 = e^ix

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

x₁ = ln(5.828427) / i
x₂ = ln(0.171573) / i