We are now pretty comfortable with our Unit Circle and our six trigonometric functions.
Would it shock you if I told you that we are now going to take everything we know (triangles stuck in a Unit Circle) and unwrap it so that we can see it graphed out on the coordinate plane like any other function?
Well, that’s exactly what we’re about to do because we’re going to Graph Sine and Cosine!
Because we are now using Radian measure to represent angles, we are able to treat our circular function (i.e., sine, cosine, tangent, etc.) like any other function with a domain or real numbers!
In order to see how this works, we are first going to let our theta become the variable x, so it’s easier for us to apply it to basic graphing techniques.
Then we’re going to take our handy-dandy Unit Circle and create a table of values from all the radian measures and ordered pairs. In doing so, we will plot these point on an xy-plane and see what it creates.
Guess what we discover …. All trigonometric functions are periodic!
Periodic? What does that mean?
Well, a function that is indeed periodic, is one that repeats forever at regular intervals – just like the waves/tide in the ocean.
Visualizing the Amplitude of the Sine Function
When we look at the graph that is generated by unwrapping our unit circle, we will see that it generates a wave-like graph, where the domain is all real numbers and the range is between -1 and 1.
This is due to the fact that as we move around the unit circle in a counterclockwise fashion, we notice that the largest value for sine or cosine is 1, therefore it’s amplitude (height) is also 1, as nicely shown by Khan Academy.
That’s great, but why do we care?
Well, it’s no fun to be stuck running around a circle forever – it’s much easier for us to stretch our legs and graph each function along the Cartesian Plane (x-y-axes). Besides, we would miss out on being able to visualize what a sine or cosine function actually does as it travels around the unit circle.
In this video we will discover how to identify all the major component of a sine and cosine graph, and learn an awesome pattern or technique for graphing these periodic functions to make is quick and easy!
We will see that the same techniques we used to graph functions in algebra, like parabolas and absolute value functions, can be applied to how we transform and translate trigonometric functions, as nicely stated by Purple Math.
How to Graph Trig Functions
- Graphing Sin(x) and Cos(x) Worksheet: Practice your skills by graphing the most fundamental trigonometry functions, sine and cosine. This handout includes 4 worked out examples.
How to Graph Sine and Cosine – Video
Get access to all the courses and over 450 HD videos with your subscription
Monthly and Yearly Plans Available
Still wondering if CalcWorkshop is right for you?
Take a Tour and find out how a membership can take the struggle out of learning math.