What is the inverse cosine of 2?
2 Answers
It does not exist.
Explanation:
The range of the cosine function is only from 1 to -1.
The curve doesn't go past these values in the y-axis (as you've mentioned cosine inverse).
Take a look at the cosine curve.
graph{cosx [-15.8, 15.79, -7.9, 7.9]}
For Real cosine this does not exist.
For Complex cosine:
Explanation:
As a Real valued function of Real angles
The definition of
e^(ix) = cos(x) + i sin(x)
cos(-x) = cos(x)
sin(-x) = -sin(x)
Hence:
cos(x) = (e^(ix)+e^(-ix))/2
Then we can define
cos(z) = (e^(iz)+e^(-iz))/2
Then
(e^(iz)+e^(-iz))/2 = 2
Let
(t+1/t)/2 = 2
Hence:
t^2-4t+1 = 0
Hence:
t = 2+-sqrt(3)
That is:
e^(iz) = 2+-sqrt(3)
So:
iz = ln(e^(iz)) = ln(2+-sqrt(3))
So:
z = ln(2+-sqrt(3))/i = +-i ln(2+sqrt(3))
By convention, the principal value is the solution with positive coefficient of
In fact,