Definition of an angle in a vector space, law of sines On a lecture in linear algebra we have been given this definition of an angle in vector space with scalar product $\langle , \rangle$:
$\cos \alpha=\frac{\langle u,v\rangle}{||u|||v||}$
Throughout the internet I found the same definition using $\cos$. And I was wondering, whether we can somehow define the angle in vector space by using sinus and scalar product. 
How is it then defined? And why is it usually defined by cosinus? Thanks for answers 
ADDED
I am asking because I have this task: Formulate and prove law of sines for vector space. I am kind of lost, so thanks for help
 A: Scalar product defines orthogonality between two vectors: two vectors are orthogonal if the angle formed between them is $\frac{\pi}{2}$. Then if they are orthogonal and you calculate $\vec{v}\cdot\vec{w}$ you get zero. 
If you define the product with the sine, you lose this property, so "orthogonal" in the scalar product sense means the vectors would be parallel. 
A: The dot product, or scalar product (or sometimes inner product in the context of Euclidean space) between two Euclidean vectors u and v is defined as:
∙ = ‖‖‖‖cos ⁡, where ⁡ is the angle between  and      (1)
(see Wikipedia: dot product)
The scalar projection (orthogonal projection) of  on  is given by:
vu = ‖‖cos ⁡    (2)
The unit vector in the direction of  is given by:
$\hat{}$ = $\frac{}{‖‖}$    (3)
The vector of length vu in the direction of  is then:
 = vu $\hat{}$    (4)
We now have 2 legs ( and v). The third side h is chosen such that 
 =  + h    (5)
Thus,
h =  -     (6)
We now have a right triangle formed by the vectors v,  and h; where v is the hypotenuse and the angle between  and h is 90 degrees.
By definition:
sin ⁡ = $\frac{‖\boldsymbol{h}‖}{‖‖}$    (7)
Using the trigonometric identity
tan ⁡ = $\frac{\text{sin}\  ⁡}{\text{cos}\ ⁡}$ = $\frac{‖\boldsymbol{h}‖}{‖_‖}$    (8)
the sine of the angle between u and v may then be expressed in terms of the dot product as
sin $\theta$ = $\frac{‖\boldsymbol{h}‖}{‖_‖}$ cos $\theta$ = $\frac{‖ - _‖}{‖_‖}$ $\frac{\cdot}{‖‖‖‖}$    (9)
Rearranging (7):
‖‖ = $\frac{‖\boldsymbol{h}‖}{\text{sin}\ ⁡}$    (10)
Similarly, for the other sides and angles:
‖‖ = $\frac{‖‖}{\text{sin}(\pi/2)}$    (11)
since  is the orthogonal projection of v on u, then the angle between h and  is 90 degrees
‖‖ = $\frac{‖_‖}{\text{sin}\ \alpha}$    (12)
where $\alpha$ is the angle between v and h
Equating the hypotenuse for all the sides gives the law of sines:
$\frac{‖\boldsymbol{h}‖}{\text{sin}\ ⁡}$ = $\frac{‖‖}{\text{sin}(\pi/2)}$ = $\frac{‖_‖}{\text{sin}\ \alpha}$    (13)
