From Trig Identities to My Own by Rachel

From Trig Identities to My Own by Rachel - July 2026 Scholarship Essay

Ironically, math is not a topic for which I have done much “problem-solving”. In fact, I like the straightforwardness of it. There is a correct way to solve everything: a correct formula, a correct method, or even a correct way to punch it into the calculator. You either know it or you don’t. At least, that is what I thought, until being taught about trigonometry identities in AP Precalculus.
In any given example, we were given two different trigonometry formulas and asked to prove that they were equivalent using the known trigonometric identities. The problems started easy, like proving that tangent was equal to sine divided by cosine. Some were much more difficult than others, however, and there was no exact way to solve any of them. One go-to method seemed to work pretty consistently, but it didn’t always work, and that tripped me up. There was no correct way to solve problems like these, as long as the math led to the correct answer. To make it even harder, my classmates often used entirely different processes than me. The comfortable uniformity of mathematics was being broken for the first real time in my experience. Sometimes the same question could be done in two, three, or even five or more steps, depending on the student and the method they chose. Dust collected on my calculator and my eraser wore thin as I spent what felt like eternity working on, getting stuck, and reattempting these specific problems. Even when I felt as if I was getting the hang of the concept, there was always an easier approach that I hadn’t noticed before. My work hardly matched the teacher-led examples projected on the board. It frustrated me that I was spending so much time on questions that didn’t require it. I am always one for a shortcut, but I struggled to recognize them in a sea of seemingly multi-layered problems.
As I practiced, I started to get the hang of those trigonometric identities. Solving was similar to playing an escape game. Each step led me closer to the answer I was seeking. While many of my peers dreaded those types of problems, I grew to enjoy the process of trial, error, and eventual proof of equivalence. Instead of jumping right in and trying different approaches, I analyzed what I was given, trying to recognize any somewhat hidden identities. If I didn’t notice any, then I would begin to solve the problem normally. The work that led to my answers began to match that of my teacher, my peers were asking me for help when it came to the concepts, and finally, on test day, I proved proficient. My eraser, formerly warm from constant use, grew cold on the corner of my desk.
Not only did the practice of proving these equivalences help me memorize my trigonometric identities to later help me in AP Calculus, but it also inadvertently strengthened my critical thinking and ability to problem-solve. I learned to slow down and take in everything I’m given before jumping to a solution. From helping me effectively solve trigonometry identities to allowing me to determine the most efficient order to grocery shop, this skill allows me to be more productive in many different contexts. Many subjects, especially math, aren’t always about trying different methods until something works, but sometimes that is the best way to get something done. I attended tours at colleges I never thought I would attend, reached out to coaches of teams I never thought I would compete for, and attempted workouts that didn’t fit into my normal schedule, all for the sake of trying to find my own process. There is no “right way” to go about all these different parts of my life, and even though there is a general template that many of my peers seem to follow, there are still many other ways to get to my own right answer. Although I am more adept in absorbing possible shortcuts to success, I am still content with the idea that my path may be different from others.
Learning to prove equivalence using trigonometric identities was a seemingly useless task in the grand scheme of calculus, but not when it came to my life. I have a stronger ability to stop and think about what I’m given before immediately taking action, as well as a more powerful mindset when it comes to my future. Mathematical identities may not play a significant role in the lives of many, but personal identity absolutely does, and I would not be as successful finding mine without my struggle and persistence in my sophomore AP Precalculus class.