Graphing Sine, Cosine and Tangent
Graphing sine, cosine and tangent are fundamental skills and concepts required to analyze and graph more complicated trigonometric functions, and even to understand the short cuts for transformations of trigonometric functions.
If you take your time and really understand this material, what follows later, which might seem more complicated, will be accessible and you’ll learn more quickly.
Go through each tab below, watch the videos, try the practice problems. Take your time and be patient. When you’re done, you’ll be a better math student for your efforts!
Q: Is the Unit Circle a graph?
A great way to think of a graph is as a picture of all of the solutions to an equation. Each point on a line or curve satisfies an equation, makes it true. For example, if we had the equation
y = 3x + 1, the coordinate (1, 4) would be on the graph. However, the point (1, 8) would not be.
Another thing to keep in mind is that this is a function, each input has exactly one output. Sine, cosine and tangent are also functions, each input has exactly one output.
The Unit Circle is a diagram that helps us understand how Trigonometry is defined and also a reference for finding common Trigonometric values. But, it is NOT a graph. It is not a picture of the solutions to an equation, though can you can write an equation that is a circle. But that circle will not have anything to do with Trigonometry.
The point is this: When you see a 30º angle on the unit circle, and you see the coordinate that coordinate is NOT a solution to a trigonometric function. A coordinate, (x, y), is an input and an output, as in f(x) = y. What we have with the coordinates on the Unit Circle are a reference like a multiplication chart. The x and y on the Unit Circle are values for cosine (x) and sine (y), not an input and an output. The input of a Trigonometric function is an angle and the output is a ratio of sides. Since the Unit Circle is defined as having a radius of 1, we can take the x or y coordinate as the adjacent (x) or opposite (y) sides, which gives us the information needed for Trigonometric values.
In this section we will discuss how to graph sine, cosine and tangent, and learn the basic features of those graphs. This will provide us a common, basic understanding of the graphs of the three fundamental Trigonometric functions.
Graphing sine, cosine and tangent is nuanced. There are a lot of little pieces to keep in mind. No single piece of information itself is difficult to grasp, but combined it can be overwhelming. As such, pay careful attention and give plenty of time to your first few graphs. This will help you to understand the basics of the graphs, as well as the mechanical skill of creating the graphs. This will help you to be efficient and accurate when it counts most (on assessments).
Idea #1: Curves are continuous, without gaps or spaces between consecutive points. The curves are made from points, with an input (x), which is an angle’s measure in degrees or radians, and an output (y), which is a ratio of sides of a right triangle. Because degrees are measured in Real Numbers, and Real Numbers are continuous, the graphs of Trigonometric functions are curves.
We will use the angles on the Unit Circle to create the outline of our curve, but we can connect those points because there are infinitely many angles between 30º and 45º, for example.
Idea #2: We will use a t-chart to find the coordinates.
Idea #3: Many Trigonometric values are irrational, like We will need to approximate these to find their position on the y – axis. There are only two values we need.
Idea #4: There are angles with negative measures and angles whose measure are more than 360º. We will limit our graphs for sine and cosine, initially, to 0º ≤ x ≤ 360º. Tangent will be limited to -90º ≤ x ≤ 90º. This will provide us with a graph that is one period. A period is one cycle of Trigonometric values. (Notice how the sine of 30º is the same as the sine of 390º.)
Idea #5: Setting up the coordinate plane is a bit tricky. The scale for the x and y axes are very different. The y – axis will have a range of -1 to 1. However, the x – axis will have a domain of either 0 to 360 (degrees) or 0 to 2π (radians).
To set up your x – axis, draw your horizontal line and place a mark where you’d like 360º (or 2π) to be. Divide the distance in half, which is 180º. Each of those halves can be cut in half again, providing you with the 90º and 270º locations. Divide each of those segments in half and you have all of the multiples of 45º. From there, you can estimate 15º on either side of the 45º multiples and you’ll have all of your angles from the Unit Circle on the x – axis.
Note: The input for a trigonometric function is an angle, that will be the x – coordinate. On the Unit Circle the x – coordinate is the cosine of an angle.
The y – axis is much simpler. However, give yourself plenty of space because you need to find a consistent and accurate way of finding 0.5, 0.707 and 0.866. As an example, refer to the picture of the hand-drawn coordinate plane that is appropriate for one period of sine or cosine.