How to Graph Sine, Cosine, Tangent by Hand ✍
By: Brett Berry
Tags: Mathematics, Math, Tutorial, Trigonometry, Education

What is a Parent Function?

You’ve probably heard the term Parent Function with relation to graphing. Parent functions are the OGs of functions. They are the unaltered forms of your equations. The archetypes. For example, the equation y=x is the parent linear function; the equation y=2x+1 is still a linear function but it is not the parent function. It is an altered form of the parent where we have changed the slope and y-intercept.

This idea of parent functions applies to all types of functions. In Trigonometry our main parent functions are: y=sin x, y=cos x, y=tan x, y=csc x, y=sec x, and y=cot x.

These are the functions you will be primarily working with in trigonometry (although as you progress you might learn about other interesting functions such the inverse or hyperbolic trig functions). So these are the functions we’ll be learning how to graph today!

The MVPs: Sine, Cosine, and Tangent

These three functions are definitely the all-stars of trig. You probably remember these functions from geometry when you were first introduced to right triangle trigonometry. Although the graphs of Sine, Cosine, and Tangent may seem completely different from what you learned in geometry, we are really just experiencing a new picture of the same geometric concept.

SOH CAH TOA | Right Triangle Trig

Sine, cosine, and tangent still represent SOH CAH TOA, but in trigonmetry, we learn about some new and creative ways we can represent this information.

Essentially what we have done here is scaled our special right triangles so that their hypotenuse is 1 unit in length, organized them around the origin of a xy-coordinate plane, and traded out the concept of degrees for radians (which is just angles measured in respect to a circular rotation).

When we lay out all these triangles around the origin, we can connect the outermost points to form a circle that has a radius of 1 unit. That circle is the Unit Circle and one of the most important topics in trig!

Now the points the lie on the circumference of the circle can be translated into the graphs we’ll be exploring today.

According to SOH CAH TOA, we see that the x-coordinate relates to the cosine value (output) at the given angle rotation (input).

The y-coordinate relates to the sine value (output) at the given angle rotation (input).

And lastly tangent can be defined as cosine divided by sine, so the quotient y/x is the tangent value (output) at the given rotation (input).

Now we can graph a picture of this information that we gleaned from the Unit Circle, by creating a graph where the x-axis represents the rotation in radians, and the y-axis represents the value of our sine, cosine, or tangent functions at the given angle.

As I’m writing this, I’m realizing that this is a complicated set of relationships to understand, so I’m going to put together a separate tutorial to explain how these concepts all relate, but until then bear with me!

In the following video tutorial, I demonstrate how to graph the Sine, Cosine, and Tangent functions using a simple pattern and their periodic nature (ie repetitive) and then I relate it back to the Unit Circle, so you can see where those points come from 🙂