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Trigonometry: The Unit Circle [in Plain English]

Updated on October 26, 2017
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I am a 3rd year engineering student with five years experience tutoring students from Algebra I to multivariable Calculus.

The Idea:

The unit circle lets us visualize the coordinates of a circle on a graph. Of course, there are lots more things the unit circle is used for, but we will get into them later. The important thing to realize is that the unit circle is just a picture of a circle with a radius of one! This helps us to see the connection between the Pythagorean Theorem (A2 + B 2 = C2 ) and sines, cosines, and tangent.

In this article, we will learn how to

  • Construct a unit circle
  • Find the sine or cosine of any angle
  • Use angles in degrees and radians

The Unit Circle

Building a Unit Circle
Building a Unit Circle

Constructing a Unit Circle

For now, we will only focus on the first quadrant, which is the upper-right part of the graph. Notice that there is a line going up at an angle, from the center of the circle (the origin) to the edge of a circle. It is going up at 30o, touching the circle at the point (√3/2, 1/2). These two numbers are the cosine(30) and the sine(30), respectively. So how does sin(30) = 1/2?

Let's draw a picture.

Sin(30): In a Picture

Let's Break it Down

Here are some important things to remember:

  • Sine = the ratio of the opposite side of a triangle to its hypotenuse, or longest side
  • Cosine = the ratio of the adjacent side of a triangle to its hypotenuse
  • When we say opposite or adjacent, we mean with respect to the angle we're measuring

When we draw a line from the origin to a point on the circle, it creates a little triangle with the side lengths given by the coordinates of where it touches. Since the hypotenuse is always 1 on the unit circle, the value of the sine and cosine are simply whatever the opposite and adjacent side lengths are. That's it!


Note: If we choose the other angle, 600, to be what we find the sine of, the value of the sine and cosine would just be reversed.

Also Note: No matter what point we choose on the circle, the sum of its squares will always equal 1. This is where the trig identity sin2(x) + cos2(x) = 1 comes from: an alternate form of the Pythagorean Theorem. Test the answers we found above to confirm the theorem!


Now that we know that sin(x) = opposite/hypotenuse and cos(x) = adjacent/hypotenuse (x represents any angle our line makes with the X-axis), we can find all the points where our line touches the circle. All we need to know is the angle the line is making with the X-axis.

Notice that the values of cosine and sine switched from our previous example! In fact, the value of sine and cosine alternate between just a few values for the common angles used on the unit circle. Here is the complete circle:

Why can I have a positive cos(x) with a negative angle?

The Complete Unit Circle
The Complete Unit Circle

Using Radians

At some point, you may encounter a strange looking unit called a radian that's used to measure an angle, usually expressed as some form of π. You might need to convert from one unit to another, and take the sine or cosine of a radian measurement. It's actually quite simple!

Steps:

  1. First, note that 2π = 360o. This means that for every rotation around the circle, we go 2π, or about 6.28, radians. (We try to keep all our radians in terms of π).
  2. To convert degrees to radians, multiply by 2π/360.
  3. To convert radians to degrees, multiply by 360/2π.

This works because the ratio of radians to degrees remains the same, so we can just use unit math with fractions to get the degrees or radians to drop out - leaving us with our desired unit! This approach of cancelling units works for many, many types of problems from physics to chemistry, and is well worth mastering.

Converting from degrees to radians (and vice versa)
Converting from degrees to radians (and vice versa)

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      Loan Nguyen 4 weeks ago

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