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The Unit Circle Guide

The circle centered at (0,0) with radius 1 unit is called The Unit Circle.

The line segment from the origin to a point on the circle forms an angle with the line segment from the origin to the point (1,0). We will call this angle $/theta$ (theta). Each point on the unit circle corresponds with a different angle $/theta$.

The angle described by moving one full revolution around the circle and back to (1,0) is called a whole
angle and is the central angle of the circle.

The size of an angle may be quantified by a system of angle measure. There are infinitely many possible angle measurement systems; however, two common angle measurement systems are degrees and radians.

The degree angle measurement system divides a whole angle into 360 equally-sized smaller angles and quantifies the measure of each of these smaller angles as 1 degree. This is commonly written as 1°. A whole angle has measure 360°, $\frac{1}{2}$ of a whole angle has measure 180°, $\frac{3}{4}$ of a whole angle has measure 270°, and $\frac{1}{4}$ of a whole angle has measure 90°.

The radian measurement system takes an entirely different approach quantifying the angle based on the fraction of the circumference of the unit circle corresponding with the angle. Recall the circumference of a circle of radius one unit is 2\pi units. A whole angle corresponds with the entire circumference and has an angle measure of 2$\pi$ radians. One half of a whole angle corresponds with one half of the circumference and has angle measure $\pi$ radians because $\frac{1}{2}\left(2\pi\right)=\pi$ . Three-fourths of a whole angle has measure $\frac{3\pi}{2}$ radians because $\frac{3}{4}\left(2\pi\right)=\frac{3\pi}{2}$ and one-fourth of a whole angle has measure $\frac{\pi}{2}$ radians because $\frac{1}{4}\left(2\pi\right)=\frac{\pi}{2}$.

Observe $\ 360°=2π$ radians. It follows that 1$\ radian=(360°)/2π$. This expression reduces to approximately 57.3°. That is, 1 $\ radian\ \approx57.3°$.

What is the angle measure of the angle that corresponds with $\frac{1}{8}$ of the circumference of the circle? This may be easily determined in any angle measurement system by multiplying the measure of a full angle by $\frac{1}{8}$.

$\frac{1}{8}$(360°)=45° $\frac{1}{8}\left(2\pi\right)=\frac{\pi}{4}$ radians.

What is the angle measure of the angle that corresponds with $\frac{1}{12}$ of the circumference of the circle? This may be easily determined in any angle measurement system be multiplying the measure of a full angle by $\frac{1}{12}$.

$\frac{1}{12}(360°)=30°$ $\frac{1}{12}\left(2\pi\right)=\frac{\pi}{6}$ radians

To create a Unit Circle diagram, follow these steps.

Step 1

Step 2

Divide the circle into eight sectors of equal size. This is most easily accomplished by bisecting each of the right angles at the origin.

Recall that the measure of each angle at the origin will be equal to $\frac{1}{8}$ of the measure of a whole angle. As previously shown, these angles each measure 45° and $\frac{\pi}{4}$ radians.

Step 3

Annotate each point on the unit circle with the corresponding angle measure in degrees and radians. The angle measures will be equal to $\frac{0}{8}$ , $\frac{1}{8},\ \frac{2}{8} ,\ \frac{3}{8} ,\ \frac{4}{8} ,\ \frac{5}{8} ,\ \frac{6}{8} ,\ \frac{7}{8}$ , and $\frac{8}{8}$ of a whole angle. Because $\frac{1}{8}$ of the measure of a whole angle is 45°, the angle measures will be the multiples of 45 between 0 and 360 inclusively. In degree measure, the angles are 0°,45°,90°,135°,180°,225°,270°,315°, and 360°.

Because $\frac{1}{8}$ of the measure of a whole angle is $\frac{\pi}{4}$ radians, the angle measures will be the multiples of $\frac{\pi}{4}$ between 0 and $2\pi$ inclusively. In radian measure, the angles are 0, $\frac{\pi}{4},\ \frac{2\pi}{4},\ \frac{3\pi}{4},\frac{4\pi}{4},$ $\frac{5\pi}{4}$, $\frac{6\pi}{4}$, $\frac{7\pi}{4}$, and $\frac{8\pi}{4}$. Although it is customary to reduce these angle measures to 0, $\frac{\pi}{4}$,$\ \frac{\pi}{2}$, $\frac{3\pi}{4},\pi$, $\frac{5\pi}{4}$, $\frac{3\pi}{2}$, $\frac{7\pi}{4}$, and $2\pi$, it is beneficial to remember that these values are just the multiples of $\frac{\pi}{4}$.

Step 4

Determine the coordinates of the points associated with each of the angles. For the points which lie on the horizontal or vertical axes, determining the coordinates is easy. We know each point is 1 unit from the center. The coordinates are $\left(1,0\right)$, $\left(0,1\right)$, $\left(-1,0\right)$, and $\left(0,-1\right)$.

For certain angles (such as 45°), the Pythagorean Theorem can be used to determine the coordinate values. Observe the right triangle from The Unit Circle associated with 45°. Because the angle measures of a triangle must add up to 180° and we know two of the angle measures (45° and 90°), we can calculate the third angle measure: 80°-90°-45°= 45°. We redraw the triangle and label the angles and side lengths.

We use the Pythagorean Theorem to solve for a.

$a^2+a^2=1^2$
${2a}^2=1$
$a^2=\frac{1}{2}$
$a=\sqrt{\frac{1}{2}}$

We rationalize the denominator to get

a=$\frac{\sqrt2}{2}$

So the vertices of the triangle in the unit circle are $\left(0,0\right),\ \left(\frac{\sqrt2}{2},0\right)$, and $\left(\frac{\sqrt2}{2},\frac{\sqrt2}{2}\right)$. The vertex that lies on The Unit Circle has coordinates $\left(\frac{\sqrt2}{2},\frac{\sqrt2}{2}\right)$. The odd multiples of 45° will have these same coordinates differing only in sign. We update The Unit Circle diagram.

We use the Pythagorean Theorem to solve for a.

$a^2+a^2=1^2$
${2a}^2=1$a^2=\frac{1}{2}a=\sqrt{\frac{1}{2}}$$

We rationalize the denominator to get

\[a=\frac{\sqrt2}{2}\]

So the vertices of the triangle in the unit circle are $\left(0,0\right)$, $\left(\frac{\sqrt2}{2},0\right)$, and $\left(\frac{\sqrt2}{2},\frac{\sqrt2}{2}\right)$. The vertex that lies on The Unit Circle has coordinates $\left(\frac{\sqrt2}{2},\frac{\sqrt2}{2}\right)$. The odd multiples of 45° will have these same coordinates differing only in sign. We update The Unit Circle diagram.

Step 5

Draw another circle of radius 1 unit centered at the origin. This circle will be used to determine additional angles and coordinates which will ultimately be added to the Unit Circle diagram.

Step 6

Divide the circle into twelve sectors of equal size. This is most easily accomplished by trisecting each of the right angles at the origin.

Recall that the measure of each angle at the origin will be equal to $\frac{1}{12}$ of the measure of a whole angle. As previously shown, these angles each measure 30° and $\frac{\pi}{6}$ radians.

Step 7

Annotate each point on the unit circle with the corresponding angle measure in degrees and radians. The angle measures will be equal to $\frac{0}{12}$ , $\frac{1}{12}$, $\frac{2}{12}$, $\frac{3}{12}$, $\frac{4}{12}$, $\frac{5}{12}$, $\frac{6}{12}$, $\frac{7}{12}$, $\frac{8}{12}$, $\frac{9}{12}$, $\frac{10}{12}$, $\frac{11}{12}$, and $\frac{12}{12}$ of a whole angle. Because $\frac{1}{12}$ of the measure of a whole angle is $30°$, the angle measures will be the multiples of 30 between 0 and 360 inclusively. In degree measure, the angles are $0°,30°,60°,90°,120°,150°,180°,210°,240°,270°,300°, 330°$, and $360°$.

Because $\frac{1}{12}$ of the measure of a whole angle is $\frac{\pi}{6}$ radians, the angle measures will be the multiples of $\frac{\pi}{6}$ between 0 and 2$\pi$ inclusively. In radian measure, the angles are 0, $\frac{\pi}{6}$, $\frac{2\pi}{6}$, $\frac{3\pi}{6}$,$\frac{4\pi}{6}$, $\frac{5\pi}{6}$, $\frac{6\pi}{6}$, $\frac{7\pi}{6}$, $\frac{8\pi}{6}$, $\frac{9\pi}{6}$, $\frac{10\pi}{6}$, $\frac{11\pi}{6}$, and $\frac{12\pi}{6}$. Although it is customary to reduce these angle measures to 0, $\frac{\pi}{6}$, $\frac{\pi}{3}$, $\frac{\pi}{2},\frac{2\pi}{3},\frac{5\pi}{6},\pi$, $\frac{7\pi}{6}$, $\frac{4\pi}{3}$, $\frac{3\pi}{2}$, $\frac{5\pi}{3}$, $\frac{11\pi}{6}$, and 2$\pi$, it is beneficial to remember that these values are just the multiples of $\frac{\pi}{6}$.

Step 8

Determine the coordinates of the points associated with each of the angles. We have previously determined the coordinates for the points which lie on the horizontal or vertical axes. The coordinates are $\left(1,0\right)$, $\left(0,1\right)$, $\left(-1,0\right)$, and $\left(0,-1\right)$.

For certain angles (such as 30° and 60°), the Pythagorean Theorem can be used to determine the coordinate values. Observe the right triangle from The Unit Circle associated with 30°. Because the angle measures of a triangle must add up to 180° and we know two of the angle measures (30° and 90°), we can calculate the third angle measure: 180°-30°-90°=60°. We redraw the triangle and label the angles and side lengths.

If we can write the side lengths in terms of a single variable, we will be able to determine the side lengths using The Pythagorean Theorem. We rotate the triangle and create a mirror image. The resultant triangle is an equilateral triangle so all side lengths are equal to 1 unit.

Observe from the figure that 2b=1. So b=$\frac{1}{2}$. We now use the Pythagorean Theorem to solve for a.
$\left(\frac{1}{2}\right)^2+a^2=1^2$
$\frac{1}{4}+a^2=1$
$a^2=\frac{3}{4}$
a=$\sqrt{\frac{3}{4}}$

We rationalize the denominator to get

a=$\frac{\sqrt3}{2}$

So the vertices of the triangle in the unit circle are $\left(0,0\right)$, $\left(\frac{\sqrt3}{2},0\right)$, and $\left(\frac{\sqrt3}{2},\frac{1}{2}\right)$. The vertex that lies on The Unit Circle has coordinates $\left(\frac{\sqrt3}{2},\frac{1}{2}\right)$. The multiples of 30° (except for 90°,180°,270°,and 360°) will have coordinates containing the values of $\frac{1}{2}$ and $\frac{\sqrt3}{2}$ differing only in sign. Observe that $\frac{1}{2}$=0.5 and $\frac{\sqrt3}{2}$approx0.87. When the horizontal distance of the point from the origin is greater than the vertical distance, the value $\frac{\sqrt3}{2}$ will be listed first in the coordinates. Otherwise, it will be listed second. We update The Unit Circle diagram.

Step 9

Combine the results from Step 4 with the results from Step 8 to finalize The Unit Circle diagram.

Summary

The Unit Circle diagram contains a circle of radius 1 unit centered at the origin. The standard angles are the multiples of $\frac{1}{8}$ of a whole angle and the multiples of \frac{1}{12} of a whole angle. The five values used in the coordinates (including their signed variations) include

0, $\frac{1}{2}$, $\frac{\sqrt2}{2}$, $\frac{\sqrt3}{2}$, and 1. One way to remember these values is to observe that they are equivalent to $\frac{\sqrt0}{2},\frac{\sqrt1}{2}$, $\frac{\sqrt2}{2}$, $\frac{\sqrt3}{2}$, and $\frac{\sqrt4}{2}$. In the coordinates, 0 is always paired with 1, $\frac{1}{2}$ is always paired with $\frac{\sqrt3}{2}$, and $\frac{\sqrt2}{2}$ is paired with itself.

The Unit Circle Guide