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Understanding the conversion of slope from degrees to radians is crucial when dealing with trigonometric functions, which are a fundamental component of various physics problems, including those involving gravitational potential energy. The slope given in degrees (\(4.3^\text{o}\)) represents the angle of the incline and translating it into radians allows for more straightforward calculations in subsequent steps.

To convert degrees to radians, you utilize the proportion that 180 degrees is equivalent to \(\pi\) radians. The conversion formula is \(\theta_\text{radians} = \theta_\text{degrees} \times (\pi/180)\). Applying this to our problem, the slope \(4.3^\text{o}\) becomes approximately 0.075 radians after the conversion. It's through this mathematical translation that one can then engage with trigonometric functions, which typically operate within the radian system in most scientific calculators.

Trigonometry is indispensable when it comes to analyzing forces and dimensions in physics, which often relate to angles. In the given scenario, we utilize the sine function—one of the basic trigonometric functions—to determine the vertical height reached by the bike racer.

The sine function relates an angle of a right triangle to the ratio of the length of the opposite side (the vertical height, in our case) to the length of the hypotenuse (the length of the road). The formula \(h = l\times\sin(\theta)\) effectively gives us the height when we input the length of the slope (\(l\)) and the angle in radians (\(\theta\)). For our exercise, with a road length of 1200 m and an angle of 0.075 radians, the sine function calculates the vertical height to be 90 meters. This is essential in determining the change in the bike racer's gravitational potential energy.

Moving on to the calculation of potential energy, we now focus on the energy stored due to an object's position within a gravitational field. This is known as gravitational potential energy (PE), and it is dependent on the object's mass, the height it is raised to, and the acceleration due to gravity (\(g\), which is \(9.8 m/s^2\) on Earth).

The formula for calculating gravitational potential energy is \(PE_\text{gravity} = m\times g\times h\), where \(m\) is the mass, \(g\) is the gravitational acceleration, and \(h\) is the height above the reference point. By substituting our determined variables (mass = 72 kg, gravitational acceleration = \(9.8 m/s^2\), and height = 90 m), the gravitational potential energy for the bike racer turns out to be 63504 Joules. This quantifies the racer’s increased potential energy throughout the climb, emphasizing the direct relationship between an object’s change in height and its potential energy in a gravitational field.