Trigonometry, or studying the relationships of the angles and side lengths of triangles, is an fascinating branch of mathematics directly applicable to an endless list of job roles, including engineers, construction workers, architects, archers, shooters, and pilots. Students get trig basics in their high school algebra and geometry classes, and take a deeper dive when they engage in differential and integral calculus.
Visual Pathways
At the very core of solving trigonometry problems is understanding trigonometric relationships. The "unit circle" is a circle with a radius of one, centered at the origin of a Cartesian coordinate plane. Constructing triangles on the unit circle allows students to observe the trigonometric relationships of sine, cosine, and tangent of angles between zero and 360 degrees, or zero to 2pi radians, and every periodic coterminal angle greater than and less than angles in this interval. Many times, students are taught to memorize fifteen or more main angles and values on the unit circle in order to produce the sine, cosine, tangent, cosecant, secant, or cotangent of an angle. Unfortunately, students tend to memorize a table or circle of values for a test, quickly forget them, and then aren't able to solve conceptual trigonometric problems thereafter.
As a teacher of high school mathematics, my experience with geometry, trigonometry, algebra II/precalculus, and calculus has shown me that students need to be able to visualize mathematics to conceptualize it. When students first explored unit circle trigonometry in precalculus, I exposed them to the unit circle, and we developed the values using special right triangles. I then asked them not to memorize the unit circle, but instead to use the 30-60-90 and 45-45-90 triangles, as well as the known side relationships for sine, cosine, and tangent (using opposite and adjacent sides from a given angle and the hypotenuse), to derive the trig relationships each time. When students had to solve for trig ratios of the quadrantal angles, I asked them to find the values by recalling their understanding of the sine and cosine graphs.
Figures 1 and 2. What Unit Circle-Free Trig Looks Like
Memorization Versus Conceptualization
Initially, this process seemed like more work to my students. As they went deeper into their trig studies, however, the visualization and continuous conceptualization of triangle relationships allowed them to solve complicated trig equations, understand trig identities, and even discover domain and range for all six trigonometric functions. One student even said to me, "I just don't understand why anyone would [memorize the unit circle] if you can do it this way, and actually know what you're solving for, every single time!" It became commonplace for students to see a trig problem and immediately set up their triangles on the coordinate plane.
In courses like calculus, where students had previously developed their trig knowledge from memorizing the unit circle, students expressed frustration and anguish at having to attempt trig problems. One student in calculus said to me, "Well, when I see a trig problem, I just know I am getting that one wrong." This year, I have students in my calculus course who are the product of "unit-circle–free trigonometry," where they developed the values and learned trig relationships through visualization and physical sketching of triangles on the coordinate plane. The majority of these students now attack trig problems with confidence. When these students say that they do not know, for example, the sine of 7pi/6, I have the luxury of correcting them that yes, they do know. I instruct them to begin constructing their triangles, and they get the correct value every single time. While this approach takes an additional ten to fifteen seconds longer than knowing the values from the unit circle, the truth is that most students do not remember those values, and so the time it takes a student using the unit circle approach to solve the problem is irrelevant, because they are stuck on the precalculus basics.
Benefits in Differential and Integral Calculus
By creating this visual pathway to understanding trigonometry, students are better able to solve problems involving trigonometric functions in calculus. Even if they cannot recall, for example, the tangent of 4pi/3, they can sketch out a triangle on the coordinate plane, set up the triangle, and find the value for tangent of 4pi/3. This may seem basic, but when a student needs to find the derivative of f(x)=tan(x) at the value x=pi/3, students are unlikely to instantly recall even basic trig values on the unit circle. So even if they figure out the derivative function, they cannot find the instantaneous rate of change at x = pi/3. In contrast, a student who can redevelop the values by creating a visual using the triangles will get an accurate answer, and will likely be more comfortable and confident when they encounter a problem involving trigonometry.
Integrating is even more complicated, and when students begin to integrate using trigonometric substitution in Advanced Placement Calculus BC or in Calculus 2, understanding trigonometry in this way is not only useful, but essential. Students need to be able to recognize integrals that require trig substitution, and later on must back substitute values for trig functions that directly result from the visual development of a triangle with variable side lengths. I have worked with students who lack this visual trigonometric understanding. After redeveloping trigonometry using unit-circle–free trigonometry, most students state that this "makes so much sense" and "brings it all together, finally."
Figures 3 and 4. Benefits to Calculus
Teaching Trig for Understanding
Many students move on in mathematics with a shallow understanding of concepts, and are at a continuous disadvantage as they move on in their mathematical career. Teaching for understanding, and having students engage in mathematics with the intent of developing a deeper meaning of the mathematics, helps students with different learning backgrounds be on the same mathematical playing field. In trigonometry, specifically, developing this visual and conceptual meaning to trigonometry sets students on a path to success.