# The Unit Circle: Sine and Cosine Functions

## Angles

A golfer swings to hit a ball over a sand trap and onto the green. An airline pilot maneuvers a plane toward a narrow runway. A dress designer creates the latest fashion. What do they all have in common? They all work with angles, and so do all of us at one time or another. Sometimes we need to measure angles exactly with instruments. Other times we estimate them or judge them by eye. Either way, the proper angle can make the difference between success and failure in many undertakings. In this section, we will examine properties of angles.

### Drawing Angles in Standard Position 

Properly defining an angle first requires that we define a ray. A ray is a directed line segment. It consists of one point on a line and all points extending in one direction from that point. The first point is called the endpoint of the ray. We can refer to a specific ray by stating its endpoint and any other point on it. The ray in  can be named as ray EF, or in symbol form 
 

An angle is the union of two rays having a common endpoint. The endpoint is called the vertex of the angle, and the two rays are the sides of the angle. The angle in  is formed from 
  and 
 . Angles can be named using a point on each ray and the vertex, such as angle DEF, or in symbol form 
  
Greek letters are often used as variables for the measure of an angle.  is a list of Greek letters commonly used to represent angles, and a sample angle is shown in .
Angle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and rotate the other. The fixed ray is the initial side, and the rotated ray is the terminal side. In order to identify the different sides, we indicate the rotation with a small arrow close to the vertex as in .
As we discussed at the beginning of the section, there are many applications for angles, but in order to use them correctly, we must be able to measure them. The measure of an angle is the amount of rotation from the initial side to the terminal side. Probably the most familiar unit of angle measurement is the degree. One degree is 
  of a circular rotation, so a complete circular rotation contains 
  degrees. An angle measured in degrees should always include the unit “degrees” after the number, or include the degree symbol 
  For example, 
 


To formalize our work, we will begin by drawing angles on an x-y coordinate plane. Angles can occur in any position on the coordinate plane, but for the purpose of comparison, the convention is to illustrate them in the same position whenever possible. An angle is in standard position if its vertex is located at the origin, and its initial side extends along the positive x-axis. See .
If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a positive angle. If the angle is measured in a clockwise direction, the angle is said to be a negative angle.
Drawing an angle in standard position always starts the same way—draw the initial side along the positive x-axis. To place the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. We do that by dividing the angle measure in degrees by 
  For example, to draw a 
  angle, we calculate that 
  So, the terminal side will be one-fourth of the way around the circle, moving counterclockwise from the positive x-axis. To draw a 
  angle, we calculate that 
  So the terminal side will be 1 complete rotation around the circle, moving counterclockwise from the positive x-axis. In this case, the initial side and the terminal side overlap. See .
Since we define an angle in standard position by its terminal side, we have a special type of angle whose terminal side lies on an axis, a quadrantal angle. This type of angle can have a measure of 0°, 90°, 180°, 270°, or  360°. See .

### Converting Between Degrees and Radians

Dividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before the circle is completed. The portion that you drew is referred to as an arc. An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the circumference of that circle.
The circumference of a circle is 
  If we divide both sides of this equation by 
  we create the ratio of the circumference, which is always 
  to the radius, regardless of the length of the radius. So the circumference of any circle is 
  times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in .
This brings us to our new angle measure. One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals 
  times the radius, a full circular rotation is 
  radians.
See . Note that when an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel.

### Relating Arc Lengths to Radius

An arc length 
  is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius. 
This ratio, called the radian measure, is the same regardless of the radius of the circle—it depends only on the angle. This property allows us to define a measure of any angle as the ratio of the arc length 
  to the radius r. See .
If 
  then 
 

To elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of a circle is 
  where 
  is the radius. The smaller circle then has circumference 
  and the larger has circumference 
  Now we draw a 
  angle on the two circles, as in .
Notice what happens if we find the ratio of the arc length divided by the radius of the circle.
Since both ratios are 
  the angle measures of both circles are the same, even though the arc length and radius differ.

### Using Radians

Because radian measure is the ratio of two lengths, it is a unitless measure. For example, in , suppose the radius were 2 inches and the distance along the arc were also 2 inches. When we calculate the radian measure of the angle, the “inches” cancel, and we have a result without units. Therefore, it is not necessary to write the label “radians” after a radian measure, and if we see an angle that is not labeled with “degrees” or the degree symbol, we can assume that it is a radian measure.
Considering the most basic case, the unit circle (a circle with radius 1), we know that 1 rotation equals 360 degrees, 
  We can also track one rotation around a circle by finding the circumference, 
  and for the unit circle 
  These two different ways to rotate around a circle give us a way to convert from degrees to radians.

### Identifying Special Angles Measured in Radians

In addition to knowing the measurements in degrees and radians of a quarter revolution, a half  revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown in . Memorizing these angles will be very useful as we study the properties associated with angles.
Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in , which are shown in . Be sure you can verify each of these measures.

### Converting Between Radians and Degrees

Because degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion where 
  is the measure of the angle in degrees and 
  is the measure of the angle in radians.
This proportion shows that the measure of angle 
  in degrees divided by 180 equals the measure of angle 
  in radians divided by 
  Or, phrased another way, degrees is to 180 as radians is to 
 


### Finding Coterminal Angles

Converting between degrees and radians can make working with angles easier in some applications. For other applications, we may need another type of conversion. Negative angles and angles greater than a full revolution are more awkward to work with than those in the range of 
  to 
  or 
  to 
  It would be convenient to replace those out-of-range angles with a corresponding angle within the range of a single revolution.
It is possible for more than one angle to have the same terminal side. Look at . The angle of 
  is a positive angle, measured counterclockwise. The angle of 
  is a negative angle, measured clockwise. But both angles have the same terminal side. If two angles in standard position have the same terminal side, they are coterminal angles. Every angle greater than 
  or less than 
  is coterminal with an angle between 
  and 
  and it is often more convenient to find the coterminal angle within the range of 
  to 
  than to work with an angle that is outside that range.
Any angle has infinitely many coterminal angles because each time we add 
   to that angle—or subtract 
   from it—the resulting value has a terminal side in the same location. For example, 
   and 
   are coterminal for this reason, as is 
  
 
An angle’s reference angle is the measure of the smallest, positive, acute angle 
   formed by the terminal side of the angle 
   and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants. See  for examples of reference angles for angles in different quadrants.

### Finding Coterminal Angles Measured in Radians

We can find coterminal angles measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations.

### Determining the Length of an Arc

Recall that the radian measure 
   of an angle was defined as the ratio of the arc length 
   of a circular arc to the radius 
   of the circle, 
   From this relationship, we can find arc length along a circle, given an angle.

### Finding the Area of a Sector of a Circle

In addition to arc length, we can also use angles to find the area of a sector of a circle. A sector is a region of a circle bounded by two radii and the intercepted arc, like a slice of pizza or pie. Recall that the area of a circle with radius 
   can be found using the formula 
   If the two radii form an angle of 
   measured in radians, then 
   is the ratio of the angle measure to the measure of a full rotation and is also, therefore, the ratio of the area of the sector to the area of the circle. Thus, the area of a sector is the fraction 
   multiplied by the entire area. (Always remember that this formula only applies if 
   is in radians.)

### Use Linear and Angular Speed to Describe Motion on a Circular Path

In addition to finding the area of a sector, we can use angles to describe the speed of a moving object. An object traveling in a circular path has two types of speed. Linear speed is speed along a straight path and can be determined by the distance it moves along (its displacement) in a given time interval. For instance, if a wheel with radius 5 inches rotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or 
   inches, every second. So the linear speed of the point is 
   in./s. The equation for linear speed is as follows where 
   is linear speed, 
   is displacement, and 
   is time.
Angular speed results from circular motion and can be determined by the angle through which a point rotates in a given time interval. In other words, angular speed is angular rotation per unit time. So, for instance, if a gear makes a full rotation every 4 seconds, we can calculate its angular speed as 
   90 degrees per second. Angular speed can be given in radians per second, rotations per minute, or degrees per hour for example. The equation for angular speed is as follows, where 
   (read as omega) is angular speed, 
   is the angle traversed, and 
   is time.
Combining the definition of angular speed with the arc length equation, 
   we can find a relationship between angular and linear speeds. The angular speed equation can be solved for 
   giving 
   Substituting this into the arc length equation gives:
Substituting this into the linear speed equation gives:
Water wheels have been used for thousands of years to transfer the power of flowing water to other devices. The image below depicts the design of the the 3rd century Roman water wheel in Hierapolis, a city in what is now Turkey. Water turned the wheel, which in turn rotated a crank connected to two saws used to cut blocks. These design elements were used in water wheel applications throughout the world, and even provided the underlying principle for the steam engine, invented about 1500 years later.

### Key Equations


### Key Concepts

1. An angle is formed from the union of two rays, by keeping the initial side fixed and rotating the terminal side. The amount of rotation determines the measure of the angle.
2. An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis. A positive angle is measured counterclockwise from the initial side and a negative angle is measured clockwise.
3. To draw an angle in standard position, draw the initial side along the positive x-axis and then place the terminal side according to the fraction of a full rotation the angle represents. See .
4. In addition to degrees, the measure of an angle can be described in radians. See .
5. To convert between degrees and radians, use the proportion 
  See  and .
6. Two angles that have the same terminal side are called coterminal angles.
7. We can find coterminal angles by adding or subtracting 
  or 
  See  and .
8. Coterminal angles can be found using radians just as they are for degrees. See .
9. The length of a circular arc is a fraction of the circumference of the entire circle. See .
10. The area of sector is a fraction of the area of the entire circle. See .
11. An object moving in a circular path has both linear and angular speed.
12. The angular speed of an object traveling in a circular path is the measure of the angle through which it turns in a unit of time. See .
13. The linear speed of an object traveling along a circular path is the distance it travels in a unit of time. See .

### Section Exercises


### Verbal


### Graphical

For the following exercises, draw an angle in standard position with the given measure.
For the following exercises, refer to . Round to two decimal places.
For the following exercises, refer to . Round to two decimal places.

### Algebraic 

For the following exercises, convert angles in radians to degrees.
For the following exercises, convert angles in degrees to radians.
For the following exercises, use the given information to find the length of a circular arc. Round to two decimal places.
For the following exercises, use the given information to find the area of the sector. Round to four decimal places.
For the following exercises, find the angle between 
   and 
   that is coterminal to the given angle.
For the following exercises, find the angle between 0 and 
   in radians that is coterminal to the given angle.

### Real-World Applications 


### Extensions
