Problem 144 - Algebra and Trigonometry

Question

Describe a natural periodic phenomenon. Give an example of a question that can be answered by a trigonometric equation in the study of this phenomenon.

Step-by-Step Solution

Answer

The tilt of the earth from the sun on the 180th day after the winter solstice can be determined using a trigonometric equation modeled on the earth's annual rotation around the sun.

Step 1: Identify Natural Periodic Phenomenon

A natural periodic phenomenon is an event that repeats at regular intervals. A good example is the movement of the earth around the sun, which takes about 365.25 days to complete a cycle. This cycle influences the pattern of seasons we experience every year.

Step 2: Formulate A Trigonometric Question

A suitable question related to this phenomenon could be: 'Given that it is winter solstice when the tilt of Earth's axis is furthest away from the sun, and it is summer solstice when the tilt is closest to the sun, what is the tilt of the earth from the sun (approximated as a constant 23.5 degrees) on the 180th day after the winter solstice?'

Step 3: Solve The Trigonometric Equation

To solve this problem, one will need to set up a cosine equation. Cosine is used because our reference point, the winter solstice, is a maximum point, and cosine functions begin at a maximum. The equation will look like this: \( Tilt = 23.5 * \cos(\frac {2\pi * (day - 1)}{365.25}) \), where \( Tilt \) is the tilt of the earth in degrees, and \( day \) is the day number after the winter solstice. Substituting 180 for day in the above equation, we get \( Tilt = 23.5 * \cos(\frac {2\pi * (180 - 1)}{365.25}) \). Solving this equation will provide the tilt of the earth from the sun on the 180th day after the winter solstice.