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In physics, trigonometry plays an essential role in breaking down complex problems into understandable components, especially in scenarios involving inclined planes or angles. This exercise involving a skier on a lift highlights the usefulness of trigonometric functions. When dealing with angles, the sine, cosine, and tangent functions are key to determining vertical and horizontal components of vectors.

In the given problem, the lift makes an angle of \(14.6^{\circ}\) with the horizontal. To find the vertical height—the crucial factor for calculating gravitational potential energy—we use the sine function. The formula \(h = L \cdot \sin(\theta)\) illustrates this perfectly. Here, \(L\) is the length of the lift, and \(\theta\) is the angle of elevation. This method allows us to efficiently convert the known inclination angle and lift length into a usable height value, offering a clear application of trigonometry to solve physics-related problems.

Energy calculations form the backbone of many physics problems, providing insight into how systems work. In this case, we're focusing on gravitational potential energy (GPE). GPE is the energy an object possesses due to its position relative to the Earth, affected by gravity. The fundamental formula \( \Delta PE = m \cdot g \cdot h \) quantifies it, where \(m\) is mass, \(g\) is the acceleration due to gravity (approximately \(9.8\, \text{m/s}^2\) on Earth), and \(h\) is the height gain.

The lift problem requires determining the skier's change in potential energy as they ascend. After calculating the height using trigonometry, substitute the values into the potential energy formula. With mass (75.0 kg), gravitational acceleration, and computed height (712.76 m), the result of \(525,829.2\) Joules signifies the energy increase, highlighting how physics principles translate into real-world scenarios.

Tackling physics problems often involves a systematic approach, using logical steps and critical thinking. This skier problem exemplifies effective physics problem-solving strategies:

• First, gather all known information. Here, it's the length of the lift, the angle, and the skier's mass.
• Next, identify what you need to find, such as the change in gravitational potential energy.
• Apply the appropriate physics and mathematical principles. Trigonometry helps find the height, while energy formulas calculate the potential energy change.
• Finally, solve in a step-by-step manner, ensuring calculations at each stage are accurate.

This approach—identification, application, solution—simplifies even the most complex problems, providing clarity and precision in physics problem-solving.