The Unit Circle
In this lesson, we find the coordinates of many of the points on the unit circle.
So far, we have evaluated trigonometric functions only for acute angles, those between 0° and 90°. To work with other angles, we use the unit circle. Recall that a unit circle is a circle whose radius is one unit. A unit circle is graphed on the coordinate plane in Figure 10.1.
Every point (x,y) on the unit circle makes an angle with the origin (0,0) and the positive xaxis. This angle (θ is measured counterclockwise, as illustrated in Figure 10.2.
Example 1
What angle does the point (0,1) make?
The point (0,1) makes a right angle, as shown in Figure 10.3, either 90° , or , radians.
Example 2
What angle is defined by the point (0,  1)?
The point (0,1) makes a 270° = radian angle, as shown in Figure 10.4.
The point (1,0) could be associated with either 0° or 360°. In fact, each point could be associated with many different angles if we allow angles greater than 360° (which go all the way around the circle) and negative angles (which go around clockwise, against the usual direction).
Example 3
What point corresponds to an angle of radians?
radians is equivalent to = 450°
This is a full circle 360° plus a quarterturn 90° more, as shown in Figure 10.5.
Thus, the angle corresponds to the point (0,l).
Whenever an angle more than 360° is given, you can subtract 360° (or 2π) from it to remove a full turn. The same point of the unit circle will correspond. In the last example, for instance, we could have used 450° – 360° = 90° and gotten the same point (0,l).
Example 4
What point corresponds to the angle –π?
–π is equivalent to –180°. This corresponds to the point (–1,0), as shown in Figure 10.6. Notice that we go clockwise because the angle is negative.
Multiples of 60°
All of the angles thus far have been multiples of 90°. We can find the point that corresponds to 60° by using the nice 306090 triangle from Lesson 9, shown in Figure 10.7.
Because this has a hypotenuse of 1, it fits exactly into the unit circle of radius 1, as depicted in Figure 10.8.
The point that corresponds to 60° is because it is a unit to the right of the origin and units up.
If we flip this triangle across the yaxis, we can find the point that corresponds to 120°, as shown in Figure 10.9.
The 120° angle corresponds with point . Notice that the xcoordinate is negative because the point is to the left of the origin.
By flipping the triangle below the xaxis, we can see that corresponds to 240° and corresponds to 300°, as shown in Figure 10.10.
Example
What point on the unit circle corresponds to ?
is equivalent to = –240°
If we add a full rotation of 360° to this angle, it will still point in the same direction. Thus, the corresponding point for this angle is the same as that for – 240° + 360° = 120°. Thus, we can find the point either by going clockwise 240° or counterclockwise 120°, as shown in Figure 10.11. In either case, this angle corresponds to the point on the unit circle.

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