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### Ideas from Classroom Teachers for Trigonometry and Triangles

The applications section is the central point to this unit; the last few topics should also be addressed from an applications-motivated perspective. Note that anyone can select a formula and plug numbers in. The real challenge here is to create an appropriate diagram, determine how the given numbers and missing value match up to the variables in an appropriate formula, then substitute and solve (the easy part).

See Mathematics Teacher articles for good applications of this topic.

When working with Law of Sines and Law of Cosines, you will need to review basic properties of triangles such as the angle-sum (180°) and the triangle inequality. These are important since they help determine whether the output of the Law of Sines/Cosines has any validity.

If you choose to explore the ambiguous case: One difficult case conceptually is the ambiguous case of the Law of Sines, in which the given information (certain A-S-S combinations) may allow for 2 different triangle configurations. Allowing students to experience the ambiguity first, by creating triangles from spaghetti pieces, will make it seem more reasonable when the Law of Sines returns two possible angle measures.

Dynamic geometry software can be used to illustrate triangle similarity and trigonometric ratios.

Application topics could include angle of elevation (or depression), distance-rate-time, height of a mountain, etc.

Finding the area of non-right triangles can be considered in the context of proving the Law of Sines.

Students can use the formula for the area of any triangle to find the area of a regular pentagon (or other regular polygon).

A proof of Heron’‘s formula is accessible to students. Historical note: Although named for Heron (c. 1st century), Archimedes (c. 287-212 BC) proved it first.