# The Motion of the Moon, Sun, and Stars—Motivating Mathematics

### The Motion of the Moon, Sun, and Stars—Motivating Mathematics

Student Outcomes

• Students explore the historical context of trigonometry as motion of celestial bodies in a presumed circular arc.
• Students describe the position of an object along a line of sight in the context of circular motion.
• Students understand the naming of the quadrants and why counterclockwise motion is deemed the positive direction of turning in mathematics.

### New York State Common Core Math Algebra II, Module 2, Lesson 3

Worksheets for Algebra II, Module 2, Lesson 3

Solutions for Algebra II, Module 2, Lesson 3

Classwork

Opening Why does it look like the sun moves across the sky?
Is the sun moving, or are you moving?
In ancient Greek mythology, the god Helios was the personification of the sun. He rode across the sky every day in his chariot led by four horses. Why do your answers make it believable that in ancient times people imagined the sun was pulled across the sky each day?

Discussion In mathematics, counterclockwise rotation is considered to be the positive direction of rotation, which runs counter to our experience with a very common example of rotation: the rotation of the hands on a clock.

• Is there a connection between counterclockwise motion being considered to be positive and the naming of the quadrants on a standard coordinate system?
• What does the circle’s radius, 𝑟, represent?
• How has the motion of the sun influenced the development of mathematics?
• How is measuring the height of the sun like measuring the Ferris wheel passenger car height in the previous lessons?

Exercises 1–4

1. Calculate jya(7°), jya(11°), jya(15°), and jya(18°) using Aryabhata’s formula, round to the nearest integer, and add your results to the table below. Leave the rightmost column blank for now.
2. Label the angle 𝜃, jya(𝜃°), kojya(𝜃°), and 𝑟 in the diagram shown below.
a. How does this relate to something you have done before?
b. How do jya(𝜃°) and kojya(𝜃°) relate to lengths we already know?
3. Use your calculator to compute 𝑟 sin(𝜃°) for each value of 𝜃 in the table from Exercise 1, where 𝑟 = 3438. Record this in the blank column on the right in Exercise 1, rounding to the nearest integer. How do Aryabhata’s approximated values from around the year 500 C.E. compare to the value we can calculate with our modern technology?
4. We will assume that the sun rises at 6:00 a.m., is directly overhead at 12:00 noon, and sets at 6:00 p.m. We measure the height of the sun by finding its vertical distance from the horizon line; the horizontal line that connects the easternmost point, where the sun rises, to the westernmost point, where the sun sets.
a. Using 𝑟 = 3438, as Aryabhata did, find the height of the sun at the times listed in the following table:
b. Now, find the height of the sun at the times listed in the following table using the actual distance from the earth to the sun, which is 93 million miles.

Lesson Summary
Ancient scholars in Babylon and India conjectured that celestial motion was circular; the sun and other stars orbited the earth in a circular fashion. The earth was presumed to be the center of the sun’s orbit.
The quadrant numbering in a coordinate system is consistent with the counterclockwise motion of the sun, which rises from the east and sets in the west.
The 6th century Indian scholar Aryabhata created the first sine table, using a measurement he called jya. The purpose of his table was to calculate the position of the sun, the stars, and the planets

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