Students should already know the even-odd identities for \(\sin\) and \(\cos.\) Start this lesson by asking students to determine which of the six basic trig functions are even, which are odd, and which are neither, given a graph of each function. After that, ask students how we could prove each case algebraically, using only our knowledge of the reciprocal identities, and the fact that \(\sin\) is odd, and \(\cos\) is even? Next, students will use the even-odd identities to evaluate trig functions and simplify trig expressions.
Concude by giving your students this challenge:
George and Jim want to buy a chocolate bar. George would need 2 cents more to buy it. Jim would need 50 cents more. When they put their money together, it's still not enough to buy the chocolate bar. How much does it cost?
Here's one solution: Let \(c\) be the cost of the chocolate bar, \(g\) the amount of money George has, \(j\) the amount Jim has. Then we have \(g = c - 2\) and \(j = c - 50.\) We know their money combined isn't enough to buy the chocolate bar, so \((c - 2) + (c - 50) \lt c.\) Solving the inequality, we find \(c \lt 52\) cents. But \(g = c - 50\) and \(g \ge 0,\) so \(c \ge 50.\) Thus, the cost of the chocolate bar is either 50, or 51 cents. Another, albeit, less favorable solution, is to use guess and check, as seen here.
Note: The problem comes from here. I have adapted it for Americans, rephrased the question slightly, and succinctly explained the solution.