Imagine a bat hitting a baseball head on. The ball-to-bat force pair created upon impact is unbalanced; thus, after impact, the ball will sail outward in the opposite direction. What if, however, the bat-to-ball collision occurred at an angle? How would the magnitude of force imparted to the ball change? Answers to these types of questions necessitate there being a mathematical relationship between angles and lines-of-force in such a circumstance. Fortunately, the unit circle beautifully establishes such a relationship.
Consider the unit circle pictured below:
The circle’s radius has a value of 1. This value enables us to form a valuable relationship between angular displacement ( θ ) from the +x-axis and a right triangle as pictured below:
The side of the triangle that is opposite ( opp ) the angle is red, and the side of the triangle adjacent ( adj ) to the angle is blue. If the radius is rotated counterclockwise, the lengths of these sides will change accordingly. When the radius is at rest on the positive +x-axis, the triangle’s opposite side will have a value of zero, and the adjacent side will have a value of one. Conversely, when the radius rests on the +y-axis, the triangle’s opposite side will have a value of one, and the adjacent side will have a value of zero. Regardless of position, the triangle’s radius, which is the hypotenuse ( hyp ) of the triangle, will have a value of one. These relationships may be summed up as follows:
sin θ = opp/hyp
cos θ = adj/hyp
tan θ = opp/adj
As we will see, the sine ( sin ), cosine ( cos ) and tangent ( tan ) functions enable us to examine force ( and other ) vectors as components when interactions between objects are not collinear.
RENAISSANCE PHYSICS 