High School: Geometry
Similarity, Right Triangles, and Trigonometry HSG-SRT.C.7
7. Explain and use the relationship between the sine and cosine of complementary angles.
Who doesn't love a nice compliment? Love your new hair color! Those mismatched socks are so fashionable! OMG Vinny, your idea to get matching his-and-hers tramp stamps instead of wedding rings is so romantic!
Oops, wrong kind of compliment. We're talking here about complimentary angles. You know, the free ones. Like the how the chocolates they leave on your pillow at the Hilton are complimentary, right?
Wrong again. We're talking about angles that add up to 90°. As in, complement with an e. (Yes, that's it!) Right triangles are chock-full of them. Thanks to the Triangle Angle Sum Theorem, we know that the two acute angles in a right triangle add up to 90°.
Complementary angles belong together, much like sine and cosine. In fact, complementary angles are the peanut butter and jelly of geometry, and sine and cosine are the mate-for-life lobsters of trigonometry. And in math class, complementary angles and sine and cosine go great together. Too bad PB&J doesn't pair so well with lobster. (We sense an interesting Top Chef Quickfire challenge in the works.)
As it turns out, when we're dealing with a pair of complementary angles, the sine of one angle is equal to the cosine of the other angle (and vice versa). This is a very handy fact that students can use to evaluate or otherwise simplify complicated trigonometric equations.
Knowing that sine and cosine are related in this way, we can also find similar relationships between tangent and cotangent, and between secant and cosecant. (While they're not explicitly mentioned in the standard, they are derived from the sine-cosine relationship and may well come up as extensions of the basic content.)
In order to succeed with this standard, as well as its extension into the relationships between other trigonometric identities, students will need to be familiar with the ratios that define them. As long as they're familiar with sine, cosine, and tangent, knowing that cosecant, secant, and cotangent are the respective reciprocals should be enough to point them in the right direction.
Students will also need to be able to compute simple calculations that involve adding and subtracting up to 90, but we're going to hope that, since they're in your geometry class, they've already mastered that skill. Well, "master" might be too strong a word…
And finally, they'll probably need to be able to tell the difference between compliment and complement. (Really, it's not rocket science.)