Що таке одиничне коло?

The Unit Circle

At first the unit circle can look a little overwhelming but it is very useful once you understand all of the parts. Let's look at some basics.

1. The unit circle is a circle with a radius of $1$ unit (hence the name)
2. Each point along the circumference of the circle relates to a cosine and sine of the angle from the positive $x$-axis, called $\theta$, and can be listed as $(\cos \theta,\sin\theta)$.
3. The listed angles and points are used in relation to special triangles (45-45-90 Triangles and 30-60-90 Triangles)

Sine, Cosine, and Right Triangles

At any given point on the unit circle, a line can be made perpendicular to the $x$-axis from the point to the $x$-axis. This creates a right triangle with side lengths of $1$, $\cos\theta$, and $\sin\theta$. We can relate this back to our special triangles (45-45-90 Triangles and 30-60-90 Triangles).

If $\theta=45^{\circ}$, we get our $45-45-90$ triangle. Our expansive knowledge of this kind of triangle and our good friend soh-cah-toa let us know that both the sine and cosine of $45$ is $\frac{\sqrt{2}}{2}$. This is reflected by the point on the unit circle where $\theta=45^{\circ}$

If $\theta=30^{\circ}$, we get the $30-60-90$ triangle. From this triangle we have calculated that $\cos 30 = \frac{\sqrt{3}}{2}$ and $\sin 30 = \frac{1}{2}$. This same triangle is used for $\theta=60$. Calculating from our triangle we get that $\cos 60 = \frac{1}{2}$ and $\sin 60 =\frac{\sqrt{3}}{2}$. Once again this is reflected on our unit circle.

We can find any point on the circle given the corresponding angle through the same process as above. $$\begin{align} x&= r\cos\theta\\ y&=r\cos\theta \end{align}$$ When using the unit circle, $r=1$, but we can do the same calculations with circles with larger radius by multiplying each function by the radius.

Reference Angles

The unit circle is equally as useful once we leave the first quadrant.

The picture above shows the same triangle rotated to match each of the four possible quadrants. The key to finding the correct sine and cosine when in quadrants $2-4$ is to relate each angle to a reference angle.

A reference angle is the smallest angle that brings you to the closest horizontal line, and is always between $0^{\circ}$ and $90^{\circ}$.

The circle above shows the angle $150^{\circ}$. The closest horizontal line is at $180^{\circ}$, so it's reference angle is $30^{\circ}$. This means that the magnitude of its cosine and sine are the same as for $30^{\circ}$.

Checking with the unit circle, we can see that $150^{\circ}$ has a cosine of $\frac{-\sqrt{3}}{2}$ and a sine of $\frac12$.

This matches our cosine and sine for $30^{\circ}$, but it has a negative sine as it is in the second quadrant.

The reference angle for $225^{\circ}$ is $45^{\circ}$! The cosine and sine will have the same magnitude as $45^{\circ}$ ($\frac{\sqrt{2}}{2}$), but both will be negative as the related triangle would be in the third quadrant. Let's see what this looks like on a graph.

Only sine would be negative. The fourth quadrant has positive $x$-values, and therefore positive values for cosine, but it has negative $y$-values, and therefore negative values for sine.

Wrap Up

The unit circle is a concise way to show common relationships and components of angles. It is a circle with a radius of $1$, but can be expanded to deal with angles and circles of any size. Every point on the circle has a corresponding cosine and sine as the $x$ and $y$ components, and angles greater than $90^{\circ}$ have a corresponding reference angle that shares attributes.

(Video) How to Memorize the Unit Circle

by NancyPi

This video by NancyPi gives you a good way to memorize the unit circle.

Summary

The unit circle has a radius of $1$. When we want to find an angle on the unit circle, we go counterclockwise, starting from the $x$-axis. We're going to mainly talk in radians.

The unit crosses crosses the $x$- and $y$-axes at four points.

The first place the unit circle crosses the $x$-axis has an angle of $0$. If we rotate counterclockwise, the unit circle crosses the $y$-axis at an angle of $\frac{\pi}{2}$. If rotate again counterclockwise, we get half the circle. The angle is $\pi$. Rotating another quarter of a circle, we get an angle of $\frac{3\pi}{2}$. If we move the last quarter of the circle, we've come full circle, so the angle is $2\pi$.

Next, we're going to look at the $\frac{\pi}{4}$ angles. We'll divide each quarter of the unit circle into $2$ equal parts.

In the exact middle between $0$ and $\frac{\pi}{2}$, we will find the angle, $\frac{\pi}{4}$. Similarly, halfway between the angles $\pi$ and $\frac{\pi}{2}$, we will find $\frac{3\pi}{4}$. The same pattern continues around the circle. Note the pattern of $1, 3, 5, 7$.

Next, we'll tackle the $\frac{\pi}{6}$ angles.

Let's first look at the first quadrant. One-third of $\frac{\pi}{2}$ is $\frac{\pi}{6}$. The rest of the $6s$ are $\frac{5\pi}{6}$, $\frac{7\pi}{6}$, and $\frac{11\pi}{6}$.

Lastly, we have the $\frac{\pi}{3}$s.

Our last pattern to look at is: $\frac{\pi}{3}$, $\frac{2\pi}{3}$, $\frac{4\pi}{3}$, and $\frac{5\pi}{3}$.

What We Use the Unit Circle For

We can use the unit circle when we are trying to find the sine, cosine, and tangent of angles. Since each angle intersects the unit circle at one point, we can use trigonometry to solve problems with the unit circle easily.

The Unit Circle

Now that we've learned about trigonometric functions, it's time to learn about the unit circle!

The unit circle can help you easily find the exact values of common sine and cosine functions. Exact values means that the angles are in radians. It might be helpful to memorize the unit circle to help you more quickly solve trig problems!

Your first instinct may be to read the unit circle in a clockwise manner, but it's actually read in a counterclockwise manner. Recall the standard position of an angle!