Why cosθ is use for dot product and sinθ is use for vector product

            In vector calculus the magnitude of  two vector is calculated by either scalar product or cross product. we can observe that sinθ is used for cross/vector product and cosθ is used for dot/scalar product, have you ever wonder why is that so, why we don't use tanθ or other trigonometric ratios.

 its because other trigonometric ratios does not agree with the physical nature of the real system for example if we want to rotate or twist a system or object how can we do that we need to applied the force in such a way so that the system can rotates here is a some diagram for illustration 

from the figure (a)  you can observe that if two vector are perpendicular the system will rotate about the center similarly  from figure ( b ) if there is some angle it always make the system rotates but the magnitude of the rotation is affected by the angle and from figure ( c ) the system will move in a straight line, can't rotates about the center. in physics the rotational effect of something is called torque and its defined as the cross product of  applied force times moment arm if we want to calculated the torque or how much thing rotates you need to know how the force is applied to the system. if the direction of the force and moment arm are same mean zero/180 degree in this situation system will not rotates other then zero/180 degree the system can rotate therefore physics is measuremental science we need to represent that physical nature in a mathematical form or express, in order to express the the physical situation in the mathematical form, so that form must satisfied the true nature of the system 

from the give example the true express for the magnitude of the torque can be written as 

                    Torque = Frsinθ 

frsinθ  can satisfied the situation

for some situation cosθ is true