This spring, our math circle was fortunate to have a special treat: Professor Ivana Alexandrova from UAlbany gave our high school students a guest presentation full of fascinating problems in trigonometry. Professor Alexandrova grew up in Vratsa, Bulgaria, which has a rich and venerable tradition of math circles, and she herself participated in math circles as a student before coming to this country for college and graduate studies. As a graduate student at UC-Berkeley, she met another mathematician from Bulgaria, Professor Zvezdelina Stankova, who leads the Berkeley Math Circle, and she was happy to learn that math circles have started to establish themselves in this country as well.
Professor Alexandrova, whose research focuses on partial differential equations and mathematical physics, chose to discuss trigonometry in her presentation, because it is a beautiful, highly useful, and often misunderstood subject which is unfortunately somewhat neglected in conventional American mathematics education. Many college professors share her wish that their students had stronger backgrounds in trig. Under her guidance scary looking trig problems that might initially make your eyes glaze over sparkled and came to life as she developed beautiful ways to approach them. Her trig problems were indeed very intriguing!
You can see some of her presentation in the whiteboard shots below. She has also created a very nice website for high school students with weekly problems designed to draw less experienced students into the world of intriguing math, with a few algebra problems mixed in among the trig problems. You can find her problems and solutions here. The current problem is very nice: Solve 1 + sin x + cos x < 0.