The concept of graphing trigonometric functions contains one of the biggest shifts in mathematical thought for students in secondary mathematics. Students have just been introduced to radian measure after years of working with degree measure. The unit circle is unveiled on the coordinate plane and students are taught that the coordinates of points along its circumference can be represented as (cos ?, sin ?), rather than (x, y). Finally, students are led through the creation of trigonometric graphs that feature a domain of angular measure. With so many new concepts to digest, it’s easy to understand why even some of our most talented math students struggle with the content initially. The key to helping students navigate the tricky trigonometric waters is to help them see these concepts visually.
This is where digital interactives shine. There are a few websites that show trigonometric graphs being created in tandem with a terminal ray being rotated around the unit circle. Each one approaches the concept differently. The best interactive I’ve found to illustrate the concept clearly is the Graphing Sine/Cosine interactive housed in Discovery Education’s Math Techbook. You can play with this interactive yourself if you have access to Math Techbook. And if you don’t, then sign up for a free 60-day trial.
Before reaching the interactive, students are initially prompted to recall the special right triangle relationships (30-60-90 and 45-45-90) they learned in Geometry. Setting the hypotenuse of these triangles to 1, students begin placing these triangles inside the unit circle, aligning the hypotenuse of the triangle to be a radius of the circle. This recall naturally pulls students towards the realization that when the unit circle and their triangles are imposed on a coordinate grid, the lengths of the legs become the coordinates of the point at which the hypotenuse touches the unit circle. This activation of prior knowledge is extremely helpful when students begin the interactive.
The first thing students do in the interactive is “unroll” the unit circle. This helps give a visual connection between angular measure and the domain of trigonometric functions. Students can move the circle back and forth horizontally until they are comfortable with the concept.
Next, student begin with a blank coordinate grid and a unit circle with the terminal ray of the central angle set to zero degrees. Students can grab the point on the unit circle and begin to rotate it around the circle in either direction. As they do, the graph of the sine function (sine is the default function initially) begins to form on the coordinate plane. The y-coordinate of the graph of the sine function is constantly compared to the height of the point on the unit circle above the x-axis. At key positions around the unit circle, both lines become red to illustrate this relationship further. This helps highlight the point that the sine value for any triangle formed within the unit circle will always be equal to the length of the vertical leg, since it is opposite of the central angle. This understanding makes it easier for students to grasp ideas such as why sin ?/2 = 1 and that sin -?/2 = sin 3?/2.
Students can toggle the interactive to display the cosine function as well. This time, the y-coordinate of the graph of the sine function is constantly compared to the lateral distance of the point on the unit circle from the y-axis. At key positions around the unit circle, both lines become blue to illustrate this relationship further. This helps highlight the point that the cosine value for any triangle formed within the unit circle will always be equal to the length of the horizontal leg, since it is adjacent to the central angle. This understanding makes it easier for students to grasp ideas such as why cos ? = 1 and that cos ?/2 = cos -3?/2.
After gaining an initial understanding of the sine and cosine functions, students also have the opportunity to manipulate the radius of the circle, alter the distance multiplier, and view the impacts of those changes on the graph. There is a similar interactive that covers the tangent function.
My students love this interactive and I have witnessed several “light bulb” moments occur as they experimented with its features. Their comment this year was that it made trigonometry “less scary.” The visual context is one that they could reproduce in their minds as they delved deeper into trigonometric content. That proved to be invaluable for us.
Do you have an interesting approach to teaching students how to graph trigonometric functions? Share it with us in the comments below!