What are sin and cos and why are they used so frequently in math?
In: 6
First you draw the unit circle, although it can be a circle of any size if you adjust things slightly. Next draw a line coming out of the origin and it will automatically hit the unit circle at some coordinate (x,y).
This line you draw will also create an angle with the positive x-axis. Different lines you draw create different angles. Each of these angles is in correspondence with a point (x,y) on the circle. That is, you draw a line, this creates an angle, and the line touches the unit circle at a point.
Due to the correspondence between an angle and a point on the unit circle created by this process we use a short hand notation. When someone writes “sin(angle)” they mean the y coordinate of the point corresponding to the angle. Similarly, “cos(angle)” just means the x coordinate of the point corresponding to the angle.
Once you’ve established this meaning for sine and cosine you can start thinking about them as functions that turn angles into numbers between -1 and 1 because again, an angle you plug in corresponds to an x or y coordinate of a point on the unit circle. Since it’s a unit circle the points only go as far up as 1, right as 1, down as -1, and left as -1.
Now that you have functions you can try to do things like graph them as you would any other function. At this point the input angles become x-values for points on the graph, and the output coordinates between -1 and 1 become y-values on the graph.
These two functions have very nice properties because of how they are defined. For one, they are periodic functions, meaning they repeat the same pattern over and over. They also obey nice identities due to the fact that they turn out to be the side lengths of right triangles.
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