Explore relationships between all six trigonometric functions, including the Pythagorean identities.
Use identities to establish and verify other trigonometric relationships and solve trigonometric equations.
Quick Lesson Plan
In these final two lessons we will look at some trigonometric identities. While this may feel like a departure from the heavy emphasis on functions, trigonometric identities do support Mathematical Practice 2, which is to construct equivalent representations of functions. Students will see some applications where being able to rewrite an expression can reveal a relationship, simplify an expression, or make solving an equation easier.
In today’s lesson “Identity Crisis” students use the same diagram from yesterday but think about the relationship between the side lengths of the various right triangles, applying the Pythagorean Theorem. As you are monitoring students, make sure students are using the side lengths to write the ratios (aka the six trig functions) rather than just the segment names (AB, OA, KO, etc.) Some students may find it helpful to write both.
In questions 5-8, students use similar triangles to write different but equivalent ratios for sine, cosine, and tangent. We wanted students to have a visual proof of some of these identities rather than just verifying them by writing each function in terms of sine and cosine and seeing algebraically how terms cancel. We suggest making a big class list of additional statements students come up with in question 8 (on poster paper preferably to be preserved as a class artifact). There are so many that can be seen in the diagram and highlighting the fact that students can generate them on their own contributes to a space that fosters positive mathematical identities.
Which sides are the legs in this right triangle? Which side is the hypotenuse?
What does it mean if triangles have three sets of congruent angles?
What can you say about the ratios of sides in similar triangles?
Why are the three expressions for sin theta (cos theta, tan theta) equivalent if they are based on different triangles?
You may wish to have students verify some of the identities they wrote in the table using an algebraic proof, rather than the visual one of the diagram. Emphasize that both are proofs and ask whether there is an advantage of one over the other. In questions 2-4 of the Check Your Understanding, students get more practice applying the identities to simplify expressions. You may want to remind students that rewriting expressions and producing equivalent forms is actually a creative process that requires some trial and error! If the goal was just to memorize facts, we would simply have them restate the identities. Figuring out what would make a good substitution, or a clever substitution, is what can make a fairly dry concept more mathematically rich.