Experience First

In the beginning, students will explore the relationships between sine and cosecant, cosine and secant, and tangent and cotangent as they relate to right triangle trigonometry.  We’d like them to see that the other trig ratios are reciprocals of the ones they already know so well.  They may have trouble with the Unit Circle table in Question 4, so be lenient with them as they try to work through those. Equivalent, unrationalized answers are acceptable; if you want your students to rationalize the denominator, save this conversation for the debrief.  If they have trouble filling it in, ask students to find the sine, cosine, and tangent of those angles and then ask them what they would do to those values to get to the cosecant, secant, and cotangent.

For Question 8, we want to circle back to the right triangle trig relationships by having the students think about when it’s easier to use the reciprocal identities rather than the original ratios.  The hypotenuse is double the adjacent when the secant of the angle is 2, so guide them through that if they get stuck. We want students to be very comfortable with equivalent statements like “the opposite is one-third of the hypotenuse, so the hypotenuse is three times the opposite”. This is a great unit to continue to build on students’ proportional reasoning!

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Formalize Later

The formalization should be quick once they realize the relationship between the ratios.  Feel free to add more specific rules from the Unit Circle, such as sec(Ø) = 1/x, but make sure the students understand the rationale behind it first.  Help them write the proper answers in #4, such as sec(π/4) = √2, since cos(π/4) was originally (1/√2) before being rationalized.