# 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,