Circles and Triangles

The Utility of the Pythagorean Theorem

Our main focus in this course will be trigonometry, or the study of triangles.  We begin however with an exploration of circles.  While this may seem surprising, we will see how circles and triangles are deeply connected and how our understanding of one informs our knowledge of the other.

To make things extra circular (see what I did there?) we will start our discussion of circles with that famous theorem about right triangles, the Pythagorean Theorem.

The Pythagorean Theorem: In a right triangle with side lengths [latex]a[/latex], [latex]b[/latex] and [latex]c[/latex], where [latex]c[/latex] is the hypotenuse, [latex]a^2 + b^2 = c^2[/latex]

Figure 1

It is important to note that the Pythagorean Theorem only holds for right triangles.

We will see that the Pythagorean Theorem pops up throughout this course.  Let’s start by using it to derive what is known as the distance formula.  The distance formula gives us a way of computing the distance between two points on the coordinate plane.  In Figure 2 below, we have two points, [latex](x_1, y_1)[/latex] and [latex](x_2, y_2)[/latex].  We would like a way to compute the distance between these two points.

Figure 2

We can leverage the fact that we are working on the coordinate grid by drawing a right triangle with legs parallel to the [latex]x[/latex] and [latex]y[/latex] axes and hypotenuse connecting the two points as shown in Figure 3 below.

Figure 3

To use the Pythagorean Theorem, we need to know the lengths of two of the three sides of our right triangle.  We can compute the lengths of the two legs of the triangle using the change in the [latex]x[/latex]-values for the horizontal leg and the change in the [latex]y[/latex]-values for the vertical leg as shown in Figure 4 below.

Figure 4

Notice that distances must always be non-negative, so a cautious reader like yourself might be concerned that we might need to take the absolute value of the difference of the values.  Normally, you would be correct in this concern but this will become irrelevant in a minute when we apply the Pythagorean Theorem since it requires that we square the lengths of the sides of the triangle.

Applying the Pythagorean Theorem to our triangle, we have:

[latex]d^2 = (x_2-x_1)^2 + (y_2-y_1)^2[/latex].

For the cautious reader who was worried about negative lengths for the legs of the triangle, notice that when we square the differences in the [latex]x[/latex] or [latex]y[/latex]-values, the negative sign disappears.

Since we are hoping to solve for [latex]d[/latex], we can take the square root of both sides of this equation.  Giving us:

[latex]d = \pm\sqrt{ (x_2-x_1)^2 + (y_2-y_1)^2}[/latex].

Since [latex]d[/latex] is a distance, we can ignore the negative answer.  Thus, we have derived the distance formula as summarized below:

Since we just found the formula for the distance between two points on the coordinate plane, it might also be useful to find the coordinates of the point halfway between any two points on the coordinate plane.  This is often called the midpoint formula.

Figure 5

This formula comes from the fact that the midpoint is located exactly halfway between the [latex]x[/latex] and [latex]y[/latex]-coordinates of the given points.  We can find that halfway point by computing the average (mean) of the two values.  Thus, the halfway point in the [latex]x[/latex]-direction is given by [latex]\frac{x_1 + x_2}{2}[/latex], and a similar process will work for the halfway point in the [latex]y[/latex]-direction.

The Equation of  a Circle

Let’s return to the circle.  You might recall that a

 is a geometric object defined by the set of all points that are a fixed distance (called the radius) away from a given point (called the center).  We would like to find the equation of a circle given its center and radius.

Let’s begin with finding the equation of a circle with radius [latex]r[/latex] centered at the origin.  In Figure 6 below, we have drawn such a circle and labeled an arbitrary point on it [latex](x, y)[/latex].

Figure 6

By dropping a line from the point [latex](x, y)[/latex] perpendicular to the [latex]x[/latex]-axis, we create a right triangle as shown in Figure 7:

Figure 7

We know the lengths of the legs of the right triangle are [latex]x[/latex] and [latex]y[/latex] since those are the distances from the origin to the [latex]x[/latex] and [latex]y[/latex]-coordinates of the point.  Furthermore, we know the hypotenuse of the triangle is [latex]r[/latex] since it is the radius of the circle.  Thus, using the Pythagorean Theorem, we see that

[latex]x^2 + y^2 = r^2[/latex].

This is the equation of the circle centered at the origin and with radius [latex]r[/latex].  Notice that this equation does not have the form [latex]y=f(x)[/latex].  While it is possible to get [latex]y[/latex] by itself on one side of the equation, the format given above is the standard way of writing the equation of a circle.

Recalling that a function must pass the vertical line test, we also note that this circle is most definitely not a function.  Therefore, it is less important to write it in the form [latex]y=f(x)[/latex].  Nevertheless, it is important to note that every point that is on the circle will satisfy the equation above.

What if the circle is not centered at the origin?  We will use a similar process to find the equation of such a circle.  Suppose our circle is centered at the point [latex](h, k)[/latex] and has radius [latex]r[/latex] as shown in Figure 8:

Figure 8

Just as before, we can create a right triangle with legs perpendicular to the [latex]x[/latex] and [latex]y[/latex] axes and hypotenuse given by the radius connecting the center to the point [latex](x,y)[/latex].  We can then find the lengths of the legs by computing the difference of the [latex]x[/latex] and [latex]y[/latex]-coordinates disregarding the sign if it happens to be a negative difference.

Figure 8

Finally, using the Pythagorean Theorem again, we get the following equation:

[latex](x-h)^2 +(y-k)^2 = r^2.[/latex]

This is the desired equation.

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.

Draw angles in standard position

Properly defining an angle first requires that we define a ray. A ray 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 Figure 1 can be named as ray EF, or in symbolic form [latex]\overrightarrow{EF}[/latex].

Figure 9

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 Figure 2 is formed from [latex]\overrightarrow{ED}[/latex] and [latex]\overrightarrow{EF}[/latex]. Angles can be named using a point on each ray and the vertex, such as angle [latex]{DEF}[/latex], or in symbol form [latex]\angle{DEF}[/latex].

Figure 10

Greek letters are often used as variables for the measure of an angle. The table below is a list of Greek letters commonly used to represent angles, and a sample angle is shown in Figure 2.

[latex]\theta[/latex]	[latex]\phi \text{ or }\varphi[/latex]	[latex]\alpha[/latex]	[latex]\beta[/latex]	[latex]\gamma[/latex]
theta	phi	alpha	beta	gamma

 

 

Figure 11. Angle theta, shown as [latex]\angle \theta[/latex]

Figure 12

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 arc and arrow close to the vertex as in Figure 4.

The following video provides an illustration of angles in standard position.

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 [latex]\frac{1}{360}[/latex] of a circular rotation, so a complete circular rotation contains 360 degrees. An angle measured in degrees should always include the unit “degrees” after the number, or include the degree symbol °. For example, 90 degrees = 90°.

Figure 13

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. 

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.

Figure 14

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 360°. For example, to draw a 90° angle, we calculate that [latex]\frac{90^\circ }{360^\circ }=\frac{1}{4}[/latex]. So, the terminal side will be one-fourth of the way around the circle, moving counterclockwise from the positive x-axis. To draw a 360° angle, we calculate that [latex]\frac{360^\circ }{360^\circ }=1[/latex]. 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.

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°.

Figure 15. Quadrantal angles have a terminal side that lies along an axis. Examples are shown.

 

 

A GENERAL NOTE: QUADRANTAL ANGLES

Quadrantal angles are angles whose terminal side lies on an axis, including 0°, 90°, 180°, 270°, or 360°.

 

 

HOW TO: GIVEN AN ANGLE MEASURE IN DEGREES, DRAW THE ANGLE IN STANDARD POSITION.

 

• Express the angle measure as a fraction of 360°.

 

• Reduce the fraction to simplest form.

 

• Draw an angle that contains that same fraction of the circle, beginning on the positive x-axis and moving counterclockwise for positive angles and clockwise for negative angles.

 

 

 

 

EXAMPLE 1: DRAWING AN ANGLE IN STANDARD POSITION MEASURED IN DEGREES

 

1. 1. Sketch an angle of 30° in standard position.

 

1. 1. Sketch an angle of −135° in standard position.

 

Show Solution

1. 1. Divide the angle measure by 360°.
      [latex]\frac{30^\circ }{360^\circ }=\frac{1}{12}[/latex]

      To rewrite the fraction in a more familiar fraction, we can recognize that

      [latex]\frac{1}{12}=\frac{1}{3}\left(\frac{1}{4}\right)[/latex]

      One-twelfth equals one-third of a quarter, so by dividing a quarter rotation into thirds, we can sketch a line at 30° as in Figure 8.
      

      

      Figure 16

 

 

 

• • Divide the angle measure by 360°.
    [latex]\frac{-135^\circ }{360^\circ }=-\frac{3}{8}[/latex]

    In this case, we can recognize that

    [latex]-\frac{3}{8}=-\frac{3}{2}\left(\frac{1}{4}\right)[/latex]

 

• • Negative three-eighths is one and one-half times a quarter, so we place a line by moving clockwise one full quarter and one-half of another quarter, as in Figure 9.
    

    Figure 17

 

 

 

TRY IT 1

Show an angle of 240° on a circle in standard position.

Show Solution

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 [latex]C=2\pi r[/latex]. If we divide both sides of this equation by [latex]r[/latex], we create the ratio of the circumference to the radius, which is always [latex]2\pi[/latex] regardless of the length of the radius. So the circumference of any circle is [latex]2\pi \approx 6.28[/latex] 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 Figure 10.
 

Figure 18

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 [latex]2\pi[/latex] times the radius, a full circular rotation is [latex]2\pi[/latex] radians. So

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 out.

Figure 19. The angle t sweeps out a measure of one radian. Note that the length of the intercepted arc is the same as the length of the radius of the circle.

Relating Arc Lengths to Radius

An arc length [latex]s[/latex] 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 [latex]s[/latex] to the radius [latex]r[/latex].

If [latex]s=r[/latex], then [latex]\theta =\frac{r}{r}=\text{ 1 radian}\text{.}[/latex]

Figure 20. (a) In an angle of 1 radian, the arc length [latex]s[/latex] equals the radius [latex]r[/latex]. (b) An angle of 2 radians has an arc length [latex]s=2r[/latex]. (c) A full revolution is [latex]2\pi[/latex] or about 6.28 radians.

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 [latex]C=2\pi r[/latex], where [latex]r[/latex] is the radius. The smaller circle then has circumference [latex]2\pi \left(2\right)=4\pi[/latex] and the larger has circumference [latex]2\pi \left(3\right)=6\pi[/latex]. Now we draw a 45° angle on the two circles, as in Figure 13.

Figure 21. A 45° angle contains one-eighth of the circumference of a circle, regardless of the radius.
Notice what happens if we find the ratio of the arc length divided by the radius of the circle.

Since both ratios are [latex]\frac{1}{4}\pi[/latex], 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 Figure 12, 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, 360°. We can also track one rotation around a circle by finding the circumference, [latex]C=2\pi r[/latex], and for the unit circle [latex]C=2\pi[/latex]. 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 Figure 14. Memorizing these angles will be very useful as we study the properties associated with angles.

Figure 22. Commonly encountered angles measured in degrees
Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in Figure 14, which are shown in Figure 15. Be sure you can verify each of these measures.
 

[caption id="" align="aligncenter" width="487"] Figure 23. Commonly encountered angles measured in radians

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.

This proportion shows that the measure of angle [latex]\theta[/latex] in degrees divided by 180 equals the measure of angle [latex]\theta[/latex] in radians divided by [latex]\pi .[/latex] Or, phrased another way, degrees is to 180 as radians is to [latex]\pi[/latex].

Recall that the radian measure [latex]\theta[/latex] of an angle was defined as the ratio of the arc length [latex]s[/latex] of a circular arc to the radius [latex]r[/latex] of the circle, [latex]\theta =\frac{s}{r}[/latex]. From this relationship, we can find arc length along a circle, given an angle.

A General Note: Arc Length on a Circle

In a circle of radius r, the length of an arc [latex]s[/latex] subtended by an angle with measure [latex]\theta[/latex] in radians, shown in Figure 20, is

[latex]s=r\theta[/latex]

Figure 29

Finding the Area of a Sector of a Circle

A General Note: Area of a Sector

The area of a sector of a circle with radius [latex]r[/latex] subtended by an angle [latex]\theta[/latex], measured in radians, is

[latex]A=\frac{1}{2}\theta {r}^{2}[/latex]

Figure 30. The area of the sector equals half the square of the radius times the central angle measured in radians.

Example 9: Finding the Area of a Sector

An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown in Figure 22. What is the area of the sector of grass the sprinkler waters?

Figure 31. The sprinkler sprays 20 ft within an arc of 30°.

Show Solution

First, we need to convert the angle measure into radians. Because 30 degrees is one of our special angles, we already know the equivalent radian measure, but we can also convert:

[latex]\begin{align}30\text{ degrees}&=30\cdot \frac{\pi }{180} \\ &=\frac{\pi }{6}\text{ radians} \end{align}[/latex]

The area of the sector is then

[latex]\begin{align}\text{Area} &= \frac{1}{2}\left(\frac{\pi }{6}\right){\left(20\right)}^{2} \\ &\approx 104.72 \end{align}[/latex]

So the area is about [latex]104.72{\text{ ft}}^{2}[/latex].

In the following video you will see how to calculate arc length and area of a sector of a circle.

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 [latex]10\pi[/latex] inches, every second. So the linear speed of the point is [latex]10\pi[/latex] in./s. The equation for linear speed is as follows where [latex]v[/latex] is linear speed, [latex]s[/latex] is displacement, and [latex]t[/latex]
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 [latex]\frac{360\text{ degrees}}{4\text{ seconds}}=[/latex] 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 [latex]\omega[/latex] (read as omega) is angular speed, [latex]\theta[/latex] is the angle traversed, and [latex]t[/latex] is time.

Combining the definition of angular speed with the arc length equation, [latex]s=r\theta[/latex], we can find a relationship between angular and linear speeds. The angular speed equation can be solved for [latex]\theta[/latex], giving [latex]\theta =\omega t[/latex]. Substituting this into the arc length equation gives:

Substituting this into the linear speed equation gives:

Example 10: Finding Angular Speed

A water wheel, shown in Figure 23, completes 1 rotation every 5 seconds. Find the angular speed in radians per second.

Figure 32

Show Solution

The wheel completes 1 rotation, or passes through an angle of [latex]2\pi[/latex] radians in 5 seconds, so the angular speed would be [latex]\omega =\frac{2\pi }{5}\approx 1.257[/latex] radians per second.