This lesson is students’ first encounter with trigonometry, although they won’t encounter the word “trigonometry” yet. In this lesson, students are focused on looking for patterns and estimating, which allows them to move back and forth between predicting angle measures from ratios and ratios from angle measures. Taking time to build students’ intuition helps them make sense of trigonometry so that in subsequent lessons they will be able to ask themselves, “Is that a reasonable answer?” (MP1).

Students start by measuring side lengths and calculating ratios in several triangles of various sizes but with the same angle measures. This reinforces students’ understanding of similarity. Do not name these ratios yet; the long descriptions are important for students to build understanding. The decision to put the columns in the order that will eventually be named “cosine, sine, tangent” is purposeful. Because cosine represents the -coordinate in the unit circle and sine represents the -coordinate, tables with cosine first correctly correspond to the coordinates that students will see later. 

As students measure side lengths and compute ratios, there is an opportunity to discuss measurement error and the relationships between precision in measurement and precision in values calculated with those measurements. In this unit, students should generally round side lengths to the nearest tenth and angle measures to the nearest degree. Students are instructed to calculate the ratios of side lengths using measured lengths rounded to the hundredths place, and, when using digital tools, to use calculated ratios rounded to the thousandths place.

Students begin looking for patterns, including the relationship between cosine and sine of complementary angles. These observations will be topics in subsequent lessons, so students need not justify their conjectures at this point.

Note on language:  In previous courses, students may have learned that a ratio is an association between two or more quantities. However, in more advanced work, such as the work in this course, “ratio” is commonly used as a synonym for “quotient.” This expanded use of the word “ratio” first came into play in a previous unit. This usage continues in this unit.