Cannot understand how angle between two vectors is calculated On the picture below I am not getting why we calculate $\cos^{-1}(\frac{1}{3})$ instead of $\cos(\frac{1}{3})$. 

Sorry if the question is dumb.
 A: You have that  
$\cos(\theta)=\frac{1}{3}$
However, you are not looking for $\cos(\theta)$, you are looking for $\theta$ itself, 
so that is, $\arccos(\frac{1}{3})=\theta$
A: Notice, in general an inverse trigonometric function gives the value of an angle. 
$\cos^{-1}\left(\frac{1}{3}\right)=1.23\ \text{rad}$ is the angle $(\theta)$ between $v$ & $w$ which you are looking for. While $\cos\left(\frac{1}{3}\right)$ is a value (not an angle )i.e. cosine of angle $\frac 13$ rad. 
A: The angle formed between the two vectors is denoted by θ. Hence, to find the angle, we solve for/find the value for θ.
You ended up with the equation cos θ = 1⁄3.
Finding the value for θ from here is simple. All we have to do is remove the cos function from the left side of the equal sign and we will get the value for θ.
How we do that:


*

*Take the inverse of cos on both sides.

*Cancel cos and cos−1 since, they are opposites. This means that taking cos of a number and taking cos−1 of the result will give you the original value thus, we can ignore the two operations.

*We are left with θ = cos−1(1⁄3) 
Cos θ = 1⁄3
Cos-1(Cos θ) = Cos-1(1⁄3)
Cos-1(Cos θ) = Cos-1(1⁄3)       (Notice θ is the only value on the left side of equal sign)
θ = Cos-1(1⁄3)
θ = 1.23 rad
Another way to think of it:

Whenever we take something from one side of the equal sign to the other, it becomes the opposite of itself (for + => -, * => /, sin => sin-1, etc. and vice versa).

To find θ, we simply take the cos function to the other side therefore, making it cos-1 resulting in the equation:
θ = Cos-1(1⁄3)
Hope that helped.
