Friday, 9 October 2015

Linking Unit Circle and Trigonometric Functions

In my opinion Mathematics is most powerful when the same concept can be demonstrated in multiple representations. Luckily most concepts satisfy this but I have found that Trigonometric functions are a particularly great example of this. We can plot the values on a Cartesian Plane with a reoccurring cycle or on a unit circle (rotating as the functions move through cycles). I also see the set of algebraic rules as another representation of the concept.
Students often hold onto the first representation that they understand, however being able to translate their understanding into a variety of representations truly shows deeper understanding.

Using diagrams such as the following to superimpose the trigonometry function on top of the unti circle shows the piston like function of 'piston' like nature of the Sine Function.
But then this begs the question, what about Cosine? Well we could subtract pi/2 radians (or 90 degrees), or I have found showing the Cosine function on the Y axis (I.E. x = COS(y)) students can visually connect connect the Unit Circle and Cosine Function.
Clearly it is important to highlight that we generally see the Cosine Function moving Horizontally (I.E. y = COS(x)) but this can atleast link representations.

 Once the Unit Circle and Function are linked then we can review where there may be multiple answers for a certain Sine, Cosine or Tangent value.

Providing student an interactive aid can assist in this being explored by students. They can move the 'a' value to extend the angle.
Then additionally they adjust other variables in "y=b f(cx)" to change the size of the circle, or adjust the impact of the angle.
I have found after providing basic description of the below interactive students learn best by playing with it, and trying to use it to answer questions with a set range of x, but with multiple solutions. Click below to explore further.


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