Linear momentum ( p ) will be maximally conserved when two particles moving towards one another with a constant velocity ( v ) along a straight line collide:
p1i + p2i = p’1f + p’2f
Things become somewhat more complicated when some measurable entity is maximized or minimized when it passes through some other entity at an angle. Under such circumstances, trigonometric functions can eliminate any ambiguity dealing with the magnitude of the outcome being observed. Prior to delving into this topic, let’s take a quick glance at a unit circle:
In the prior reference to two objects moving with a constant velocity, the consequences associated with such motion will differ greatly if, for example, the angle between these objects is 0o ( moving together along a straight path ). Nonetheless, the unit circle unto itself only has limited usefulness for modeling such interactions. For example, what if an angle in question has a value of 48.2, or 49.6, or any of many countless other possible values??? The unit circle’s radius ( r ) will point towards any angle we choose, but how do we relate these changes to the positive or negative x-axis and y-axis of the unit circle??? We do so by inserting a triangle within the unit circle with lengths that will change in accordance to the position of the hypotenuse within the circle:
The hypotenuse ( hyp ) of the triangle becomes the radius of the unit circle once it is placed inside of it. The opposite ( opp ) side of the triangle will relate any given angle to the positive or negative y-axis of the circle. For example, when the hypotenuse coincides with the 90o or 270o markings on the circle, the opposite side of the triangle will reach the maximum positive and negative y-values. Anywhere between these two extremes, the ( opp ) value will be a fraction of these two extremes. For the sake of simplicity, the length of the hypotenuse is assigned a number 1. Additionally, the adjacent side of the triangle will coincide with the positive and negative x-axis in accordance with the hypotenuse’s position:
We are now ready to relate various ratios of the opposite side, adjacent side, and hypotenuse of the triangle to an angle that the hypotenuse subtends. The sine ( sin ) function, cosine ( cos ) function, and tangent ( tan ) functions tie things together in the following manner:
sin ( θ ) = ( opp / hyp )
cos ( θ ) = ( adj / hyp )
tan ( θ ) = ( opp / adj )
Let’s first deal with the sin ( θ ) function. When the hypotenuse rests on the positive x-axis ( θ = 0o ), the opposite side of the triangle will have a value of 0; and thus, sin ( θ ) will equal 0. When the sin ( θ ) angle is at 90o, the hypotenuse and opposite side will both equal 1. When this is the case, the value of sin ( θ ) will be 1 as well. The value of the sin ( θ ) function will move back to 0 when ( θ ) is at 180o. As the hypotenuse moves further around the circle, the sin ( θ ) function will repeat itself with negative values.
The cos ( θ ) function operates the same way, but it is 90o out of phase with the sin ( θ ) angle. For example, when the hypotenuse rests upon the +x-axis ( θ = 0 ), the cos ( θ ) function will have a 1. When the hypotenuse of the triangle reaches 90o, the cosign ( θ ) function has a value of 0. The sin ( θ ) function will begin to acquire negative values when ( θ ) is greater than 180o, but the cos ( θ ) does so when the ( θ ) is greater than 90o. Since the sin ( θ ) function has a value of 0 at the origin while the cos ( θ ) function has a value of 1, the cos ( θ ) function leads the sin ( θ ) function by 90o:
Finally, the tan ( θ ) function is the ratio of the opposite ( opp ) and adjacent ( adj ) sides of the triangle insert. As an extension of this ratio, the sin ( θ ) and cos ( θ ) functions can be used in place of the ( opp ) and ( adj ) sides when needed:
tan ( θ ) = ( opp / adj )
sin ( θ ) = ( opp / hyp )
cos ( θ ) = ( adj / hyp )
[ sin ( θ ) / cos ( θ ) ] = [ ( opp / hyp ) / ( adj / hyp ) ] = ( opp / adj )
tan ( θ ) = [ sin ( θ ) / cos ( θ ) ]
Let’s now look at how the sin ( θ ) function and cos ( θ ) function give meaning to two physics equations. The first equation regards the force ( FB ) imparted upon a positive charge ( +q ) moving at a speed of ( v ) through a magnetic field ( B ). Unto itself, the equation for ( FB ) is relatively meaningless; its magnitude is maximized when it moves through a magnetic field at a 90o angle as shown:
The addition of the sin ( θ ) function also shows that the force will be at a minimum when the angle between the charged species and B-field is 0o, and it will be somewhere between these two extremes when the angle of travel is between 0o and 90o.
Lastly, we have an equation that relates the area ( A ) of a loop to the magnitude of magnetic flux ( B ) traveling through it. The area in this case has a normal assigned to its cross-section. When a maximum amount of magnetic flux passes through the loop, it will directly coincide with this normal at 0o:
We must use the cos ( θ ) function to appropriately reflect the physics associated with this circumstance:
ɸ = BA cos ( θ )
RENAISSANCE PHYSICS 