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SOLVING COMMON ENGINEERING PROBLEMS
Although we can view engineering as the analysis and design of systems, the root of the profession is using the principles of mathematics and science in order to solve problems. The problem could be humanistic in nature where we are designing a water purification system for a poverty-stricken country; on the other hand, we could design a toaster with the capability to print high-quality pictures onto whole grain bread. Regardless of the project, our intent is to provide a remedy to a problem. Since the idea of a “problem” is broad, we will need a wide range of mathematical ideas. In the next three chapters, we will cover the following topics:
• Coordinate Systems
• Real and Complex Numbers
• Basic Geometry and Trigonometry
• Functions and Their Behavior
• Finding Roots to Functions (both analytically and numerically)
• (An introduction to) Calculus
2.1 CHOICE OF COORDINATE SYSTEMS
The real world can be an amazingly complex place, much more than the mighty pen and paper can capture. In the world of mathematics, abstraction can take the complexity of ideas to higher dimensions, entirely nonphysical situations, and make common knowledge obsolete. Our key abstraction will be how we choose our perspective when solving engineering problems. As long as we are consistent, then the solution will naturally emerge. With this in mind, when confronted with a physical problem, we should ask ourselves: which coordinate system will help us solve the problem in the most practical way?
Definition 2.1: A Coordinate systemuses two or more numbers to describe the position of a point in two- (2-D) or three-dimensional (3-D) space. The concept extends to higher dimensions as well.
Of the various coordinate systems at our disposal, we will focus on two that are the most popular—rectangular and polar.
We use two main coordinate systems in two dimensions. Both systems are equally capable of representing the same object, but the difference lies in how we get to the point. Pictured in Figure 2.1 is a point called P in a 2-D plane. To identify exactly where P lies, we can use rectangular coordinates.
Figure 2.1. Rectangular coordinates
Definition 2.2: Rectangular coordinates are a coordinate system that uses two distances (x and y) with respect to reference lines (called the X-axis and Y-axis) to identify points in two dimensions (or more if needed).
To describe the location of a point in a 2-D space, we first move right (or left) along the X-axis some distance, x, and move up (or down) by some distance, y, until we reach P. The point P is then given by P = (x,y), where (x,y) is called an ordered pair.
Definition 2.3: An ordered pair is two numbers written in a certain order. For example, (1,2), (4,3), and (6,6) are ordered pairs. An ordered triple would be three numbers written in a certain order: (1,4,6). We will use ordered pairs and triples for points, but their use in mathematics extends beyond our purposes.
Take care in which direction you move; moving up and right are positive, and moving down and left are negative. The “2-D space” that point P and any other points of the form (x,y) occupy is called the real plane.
Definition 2.4: The real plane is a collection of points in the form (x,y) where x and y are real numbers.
These coordinates are fundamentally meaningless unless we are clear about how we are obtaining the values. The lines X and Y are called the X-axis and Y-axis, respectively; these are typically marked with units to determine distance—giving us a reference. The X- and Y-axis meet at a special point called the origin. In rectangular coordinates, the origin is the point, O = (0,0).
Definition 2.5: The origin is the mathematical “center” of a coordinate system.
Example 2.1: Placing Points on an Object—A fundamental purpose of points in engineering is to identify critical areas on an object or in space. When using 2-D drafting software like AutoCAD to model a part for production, then the entire space is encoded with a rectangular coordinate system by default (Figure 2.2 ). If we draw a line, the software “understands” this as a straight thin object from one point (represented as an ordered pair) to another point (also an ordered pair). Without this frame of reference, how else would the software know where to place the shape we want? For 3-D parts modeled in AutoCAD or SolidWorks (another platform for 3-D modeling), the concept of knowing where you are becomes even more necessary. As a little exercise to understand why, take a moment to think about how a 3-D printer would work if it could not tell where the printer head was in space.
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2.1.1.1 Moving to a Third Dimension
To place points on 3-D objects, we are going to need another axis, the Z-axis. Imagine taking a piece of paper, which is our usual real plane, and piecing it with a pencil, the Z-axis, straight through the middle of the page—this process provides us with another direction to work with. To identify P in general space, the common method is to use a right-handed coordinate system (as pictured in Figure 2.3 ) and move along the x-axis, followed by the y-axis, and finally on the z-axis. The result is the ordered triple, P = (x,y,z).
Figure 2.2. Screenshot of AutoCAD sketch
Figure 2.3. A 3-D coordinate system
In a right-handed coordinate system, moving forward, to the right, and up are considered positive.
Why do we refer to this orientation as “right hand”? If you were to place your right hand such that it points along the positive X-axis (with your palm up), then close your fingers toward the positive Y-axis, your thumb indicates the positive Z direction.
Example 2.2: Points on a 3-D Object—Pictured in Figure 2.4 is a beam supported by two cables that are bolted into a wall.
Figure 2.4. Beam supported by cables
To analyze the internal forces in the cables, we need to know exactly where the bolts are positioned relative to the beam. If we know our beam is 6 feet long, we can locate the end, E, of the beam in space by moving along the x-axis 6 feet (6) and ignoring the y (0) and z (0) axis, since we already found the point. Therefore, the end of the beam is at E = (6,0,0) (Figure 2.5 ).
Figure 2.5. Finding the end of the beam, E
For the bolt above the beam, we move ignoring the x-axis (0), move to the left on the y-axis (–3), and move up 1 foot to meet the bolt (1). This means the first bolt is at (0,-3,1). The second bolt can be found in a similar manner; we still ignore the x-axis (0), move to the right on the y-axis (1), and move down 0.5 feet to meet the second bolt (–0.5). Therefore, the second bolt is at (0,1,–0.5).
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Figure 2.6. Demonstration of polar coordinates
Despite the popularity of rectangular coordinates, a handful of situations exist where using them to describe crucial locations will cause frustration due to needless complication. If it is painful to assign rectangular coordinates, we may find it useful to try polar coordinates, as shown in Figure 2.6 .
Definition 2.6: Polar coordinates are a coordinate system that uses a radial distance from the origin, r, and an angle with respect to a reference line (X-axis), θ, to identify points on a plane.
In this coordinate system, we use the distance, r, that P is from the origin, O, and the angle (usually given by θ) the line r makes with some reference line, which is the X-axis in the picture. To avoid confusion, we will write a point in polar coordinates as 
to remind ourselves we are dealing with a radius and an angle. Keep in mind that both coordinate systems are equally useful in identifying the same point in space, but one may relieve us from performing laborious calculations in the process of finding the point itself.
Example 2.3: Describing the Friction on a Tire—Pictured in Figure 2.7 is a car tire; if we are interested in the friction between the tire and the pavement, then it would make sense to use polar coordinates since the tire is a circular object. The friction occurs at the point where the tire meets the pavement, and so we will mark this spot with a point F. Say the wheels are 20” in diameter, then the radius of the circle is 10”. If our origin is at the center of the tire, then we need to increase our angle counterclockwise by 270° and extend 10” down to reach the bottom. This means our point 
.
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Figure 2.7. Friction on a tire
If it is desirable to switch between the two coordinate systems, we have a few conversion formulas based on trigonometry to help us. To convert a point (x,y) in rectangular coordinates to a point 
in polar coordinates or vice versa,
then we use the following formulas:
To understand where the conversion formulas come from, the dashed vertical line in Figure 2.6 reveals everything we need. Since it drops directly from the point onto the X-axis, it intersects the axis at a right angle; therefore, it is a triangle with a hypotenuse of r and legs of length x and y, respectively. By implementing the trigonometric functions (to be discussed later), we can translate between the two coordinate systems using the triangle as a bridge! Note that we also used one of the most famous mathematical theorems of all time, the Pythagorean Theorem, within the second set of formulas.
Theorem: Pythagorean Theorem: Given a right triangle with legs x and y with hypotenuse r, then
Example 2.4: Converting from Rectangular to Polar and from Polar to Rectangular—To demonstrate how messy rectangular coordinates can be in certain situations, let us continue the tire example and track the point F as a fixed point on the tire. We were able to identify the point where friction occurs using polar coordinates—excellent—now consider a problem where we want to know how many times the wheel spins in a minute by seeing how many times F crosses the dotted line. At any instant, the point we are considering is probably not going to be at an angle that is going to convert smoothly. Say that F is at 131° (Figure 2.8 ).
Figure 2.8. Point F at 131°
If we want the rectangular equivalent of this point, we need to use the following conversion formulas:
Since our radius is 10”, r = 10, and our angle is 131°, θ = 131°:
Therefore, F = (-6.561,7.547) in rectangular coordinates. Compared to our representation in polar coordinates, we have inadvertently introduced some ugly decimals. As mentioned earlier, one form may be easier to work with but both are equally valid.
ARE THERE ANY ISSUES CONVERTING FROM RECTANGULAR TO POLAR?
Note that converting from rectangular (x, y) to polar 
requires a “common sense” check when finding the angle. For example, if we were to convert our rectangular answer back to polar, our angle could be: 
. However, a calculator may show –49°. Why?
The inverse tangent function gives us the same result for F = (-6.561,7.547) or F = (6.561,-7.547). The first point should result in an angle of 131°, while the second should be –49°. Simply picture which quadrant (which area of the plot) contains the point. In short, the angle on our calculator may be off by 180°.
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When attempting to analyze anything in a physical sense, it is highly unlikely that plain numbers will capture the essence of the problem. Even the simplest situation can require a bit of additional information; failing to incorporate the extra details could make the problem needlessly difficult or impossible to solve.
Example 2.5: More Information Needed—How useful is a global positioning system (GPS) that can tell us only the number of miles we need to drive before our next turn, but cannot tell us where to turn? Well, fundamentally useless! The entire point of a GPS is to guide us to our destination, we have to turn at some point! When leading us through the maze of unfamiliar streets, the GPS offers us an invaluable piece of information—direction—along with how far we need to drive—magnitude. When solving physical problems, neglecting the direction of movement is equivalent to driving to a new destination with our useless GPS—not a smart decision.
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The mathematical idea that helps us describe objects in terms of a magnitude and a direction is called a vector.
Definition 2.7: Vectors are quantities that have both a magnitude and a direction. One way we can denote vectors is using boldface letters (e.g., B is a vector).
To contrast with vectors, we deem so-called “regular” numbers—quantities without any additional sense of direction—scalars.
Definition 2.8: Scalars are quantities that have only a magnitude.
To distinguish between a vector and a scalar, we will use bold print for vectors like so: A is a vector, whereas A is a scalar.
Example 2.6: A Tensile Test—Say we are testing a piece of steel’s strength by performing a tensile test (Figure 2.9 ). In this test, the material is subjected to an automated “tug of war” between two mechanical grips on either end, but the grips are equally strong. To illustrate the action in terms of vectors, we need to mathematically describe only the force the grips are exerting on the steel. Since the force has a magnitude and a direction, we will call the force, F. If we want to know what the value of F is at any point during the test, we are referring to F’s magnitude. For example, we will assume the magnitude of F is around 20 kN (note that kN is shorthand for kilonewtons, where newton is the unit for force and kilo means 1000—meaning we have 20 * 1000 newtons of force). We have a few different notations for the “magnitude of F,” but we will simply use the nonbold version of the letter or symbol. Therefore, we can make the following distinction:
One mechanical grip is pulling upward, but we have one more mechanical arm pulling in the opposite direction with the same magnitude. Multiplying the vector, F, by 1 will reverse the direction, but it will not change the magnitude. Thus,
We can see these two forces at play in Figure 2.9 . To abstract them, we opt to represent vectors as arrows pointing in the appropriate directions.
Countless methods for representing vectors exist, but the “approved” notation depends on the field. Thankfully, the names for special vectors remain relatively untouched, and so we will use a fairly neutral notation to carry on; in fact, we will borrow from rectangular and polar coordinates to do so.
Figure 2.9. Visualizing the tensile test
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Example 2.7: Two Ways of Writing Down a Vector—Example 2.6 included vectors pointing in standard cardinal directions, north and south, but we need better ways to denote direction instead of saying “up” or “down.” To demonstrate, consider a vector describing the acceleration of an object in Figure 2.10 .
Figure 2.10. Acceleration vector A
Describing this vector does not require much effort on our part. We know this vector begins at (2, 4) and ends at (8, 2); therefore, we can write down the expression for A using position vectors.
Definition 2.9 Position vectors are vectors with their tails beginning at the origin.
We can think of these types of vectors as spotlights at the theater pointing out important characters on stage—although the light can point in all directions, the light still starts in the same place. In this situation, the position vectors P1 and P2 want to highlight the starting and ending points of A as the most important characters on stage (Figure 2.11 ).
Figure 2.11. Position vectors pointing out the end points of A
If we know the end points of a vector, then we express it mathematically by taking the coordinates of the vector’s tip and subtracting the coordinates of the vector’s tail like so:
All we did was take the difference between the x coordinates and y coordinates; in fact, we just performed vector subtraction. To illustrate, we can find the representation of P1 and P2 using the same logic.
Notice that P1 = Tail of A and P2 = Tail of A, so let us perform a substitution:
Just like scalars (numbers without a direction), we can use algebra with addition and subtraction to manipulate the equality. For instance, we could add P1 to both sides to obtain:
A graphical interpretation of vector addition and subtraction can be seen in Figure 2.11 ; the resulting vector is just another side of the triangle completing the tail to tip chain!
Alternatively, a vector can be expressed using polar coordinates. The same notation holds as before, but this time we are using the coordinates to identify a vector. Using the conversion formula...
It is customary to avoid using negative angles, so we add 360° to counter the negative sign. Then, vector A can also be written as: 
.
Example 2.8: Unit vectors—The notion of “direction” is not arbitrarily constructed; vectors have a natural physical directional component. Whether we recognize them or not, unit vectors are the component that makes our vectors have a direction in the first place.
Definition 2.10 A unit vector is any vector with a magnitude of 1.
To indicate we are considering a unit vector as opposed to a vector with any magnitude, we place a “hat” or “cap” (^) on the vector. To demonstrate, say we have a cable in tension with the following vector describing the force, F = (–2,5,4) newtons. Our vector has three coordinates, that means we know it is a 3-D vector and is given in Cartesian Coordinates. The unit vector is written as 
and can be found by using the following formula:
All we do is divide the vector, F, by its magnitude, F. Note: the magnitude is found by squaring each component of the vector, adding them together, then taking the square root.
This process is often called normalizing the vector. We could manipulate the equation a bit and isolate the force vector like so,
which gives us a mathematical formulation of our idea. The unit vector, , acts like our compass or GPS telling us which way to go, then the magnitude, F, tells us how far!
Note that F = F = (magnitude)·(direction) is similar to saying: The store is 5 miles west, or (5 miles) (1 mile west).
In engineering, we use specific unit vectors in the x, y, or z direction called versors. These tiny vectors occupy a single direction in the Cartesian coordinate system and are often written as 
, 
and 
. Each of the three is a unit vector with a specific representation:
We could rewrite the vector F in terms of , , and by remembering our basic rules of addition and factoring. First, treat the vector as a sum of its pieces in each direction,
then factor out the components of t2, 5, and 4 from each vector, respectively.
Look carefully, our versors just appeared! Let us replace them with the shorthand version:
This representation has a name, the Cartesian vector form.
Definition 2.11: A vector V is said to be in Cartesian vector form if V is written as
for some numbers x, y, and z. (Note: The notation extends to higher dimensions, but we will not need more than three dimensions for our purposes.)
Regardless of whether we write the vector as coordinates or in the Cartesian form, the underlying essence of the vector will not change.
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2.1.3.1 Vector Addition and Subtraction
We have seen vector addition and subtraction: in our previous example, we found:
When our vectors are given using rectangular coordinates, we simply add (or subtract) the X, Y, and Z values to add (or subtract) vectors themselves. If our vectors are given in polar coordinates, the method for addition/subtraction amounts to converting to rectangular form first, performing the addition component wise like before, and then converting the result back to polar coordinates.
When we are multiplying a vector by a scalar, we can visualize the process as making a vector longer or shorter, but not changing the direction. Therefore, we are “scaling” the vector by a constant value when a vector is multiplied by a scalar—we often call this the simple product.
Definition 2.12: The simple product is the multiplication of a scalar a by a vector V, written as aV. If V = (x,y), then aV = (ax,ay).
Example 2.9: The Simple Product—From our previous example, we have a vector
If we multiply P2 by a scalar, say 4.1, we multiply each component (x and y) by the scalar value:
Multiplying a scalar by a vector in polar form is even more straightforward: just multiply the magnitude of the vector by the scalar value. First, we will convert to polar form since our vector was given in rectangular form:
Therefore, we now have 
. We then perform the multiplication:
And, just to double check, let us convert our polar result back to rectangular:
Verified! Our additional work confirmed our claim about multiplication in polar form.
Another form of multiplication with vectors is the dot product, which involves two vectors that produce a scalar.
Definition 2.13: The dot product is the multiplication of two vectors, A and B, that produces a scalar—written as, A⋅B = (Ax,Ay)⋅(Bx,By) = AxBx + AyBy.
The formula may look strange, but this type of multiplication is quick and easy to do. In other words, we just multiply each of the components together (x with x and y with y) then add all of the products together. What exactly is the point of the dot product? Well, the above formula does not quite do the dot product justice. There is one more way to write it:
Imagine that, the dot product allows us to calculate the angle formed by two vectors if we solve for θ!
Example 2.10: The Dot Product—The dot product is used to calculate work (W, measured in Joules, J), the amount of energy transferred through an applied force. A classic physics example involves a person pulling a block along a frictionless surface at an angle θ we don’t care about. If we know the vector characterizing the force F and the displacement d, then
Say the person is pulling the block with a force F = (2,2)N, and the block is displaced along d = (5,2)m.
What do these values mean? To provide some context, imagine a person pushing a block in the horizontal direction (x) with a force of 2 N: we would expect the block to move along the x-axis. Instead, the person is pushing in the x-direction with a force of 2 N and in the y-direction with a force of 2 N; therefore, the force is at a 45° angle (Figure 2.12 ).
Figure 2.12. Visualizing the scenario
We are given two vectors: one describes the force applied to the block, F = (2,2)N, and the other is the distance upon which the block was pushed, d = (5,2)m. With these vectors, we know we can directly apply the definition of work:
To find the angle between the two vectors, we simply use the alternate formula for the dot product:
Taking the inverse cosine of both sides allows us to solve for θF,d, which turns out to be 23.2° [cos-1(0.9191) = 23.2°]. Therefore, the incline of the ramp in this problem is 23.2°.
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The last common form of vector multiplication is called the cross product.
Definition 2.14: The cross product is the multiplication of two vectors A and B, which produces another vector.
Performing the calculations by hand can be a bit laborious considering the cross product is defined as a determinant (which we will not cover in this text, but it can be found in any linear algebra textbook). We will present it in the expanded form after performing the necessary operations, so the rightmost expression is the most meaningful for us. Unlike the dot product, crossing a vector with another vector yields yet another vector; in fact, the resulting vector is perpendicular to the original two vectors (also called normal).
Definition 2.15: The normal to a given line l is a line that is perpendicular to l.
Two interesting facts about the cross product: (1) the cross product is anticommutative, meaning
and (2) any versor crossed with itself is zero.
Example 2.11: The Cross Product—A moment is defined as the tendency of a force to produce motion about an axis. With forces in three dimensions, we can use the cross product to determine this quantity! Here is how it works—say we have a force, F, causing trouble by acting on an object in space. To do anything with that force, we need to locate it. Since the cross product requires two vectors, we will use a position vector, r, to form a bridge from the origin to the force itself (Figure 2.13 ). It is sufficient for the position vector to touch any point along F, everything will work out. With that in mind, we can choose coordinates for our position vector to make calculating the moment easier.
Figure 2.13. Example of a position vector and a force
The calculation is the following cross product:
Let us try a practical example. A force is exerted on the outlet pipe by a crowbar in an attempt to dislodge the unit. Find the moment about the point P where 
(Figure 2.14 ).
Figure 2.14. Outlet pipe with force F
Here we are given a pipe and are asked to find the moment of the force F about the point given by P. For our purposes, P can be our origin (0,0,0). We need to start by finding a position vector that points to the force. To make calculating the moment easier, let us point the vector to the tip of the force. The force is making contact on the pipe 6 cm in the y direction and 2 cm down in the z direction. Therefore, our position vector is:
It will help us quite a bit to write r in Cartesian Vector form and express cm in scientific notation:
Now, let us cross r with F:
The cross product is distributive, so
We know 
and 
, so our result is
We can interpret the moment vector as the tendency of the applied force F to rotate the outlet pipe about an axis, so F will rotate about the x-axis with a turning force of 12 Newton meters. Figure 2.15 demonstrates what happened graphically in the cross product calculation with two simple vectors, A and B.
Figure 2.15. Graphical interpretation of the cross product
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Example 2.12: Using the Cross Product to Find Area—Finding the area of a parallelogram spanned by vectorsa = (1,2,3) and b = (–3,7,11) can also be done using the cross product. In fact, the magnitude (or length) of the cross product a×b gives the area of a parallelogram spanned by a and b. First, we want to find the cross product of the vectors:
Now, the area of the parallelogram is 
=2.87.
Note that the double bars around the cross product of the two vectors is another way to say “magnitude.”
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With such a pervasive title, there must be powerful methods lurking in this section. The strength here lies in our ability to be picky; in other words, we gain a substantial amount of power from our freedom of choice. We are aware that we need to specify a coordinate system in order to solve practical problems; however, the way we introduce the coordinate system into the problem is a different story. Thinking of the real plane as a physical thing that we can drop anywhere and solve the problem, it becomes clear that one person’s (1,1) could be somebody else’s (5,6). For example, if we know the store is 8 miles west of the school, we could say the school sits at 0 on the x-axis and we need to move -8 miles to reach the store: this means the store can be found with the ordered pair (-8,0) and the school has coordinates (0,0). On the other hand, we could say the store sits at 0 and we need to move +8 on the x-axis to find the school; now the store has coordinates (0,0) and the school is described by the ordered pair (8,0).
The reality of this situation is that many choices with coordinate systems are personal—almost political—choices. Consider the physics problem in Figure 2.16 , where is the best place for our coordinate system? To be fair, we cannot say for sure; the ideal coordinate system, its placement, and the orientation of the positive directions will all be a function of the given problem!
Figure 2.16. Where do I place my coordinate system?
Example 2.13: Which Way Is Up?—We have a considerable level of freedom when dealing with coordinate systems, especially in how we define what is positive. For this example, we consider two blocks connected by a rope where block A begins on a platform, while block B is suspended mid-air. For the sake of simplicity, nothing has moved yet, we are holding both blocks in place in order to match the picture (Figure 2.17 ). If we are interested in the velocity of block B if we let go, then we can avoid negative signs by defining the positive directions to be right and down—a bit different than what we would normally do. Can we do this? Of course! The math is self-correcting as long as we are consistent. Remember: a negative magnitude means a positive magnitude in the other direction.
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Figure 2.17. Two-block system
The coordinate plane does not necessarily need to be level either; in fact, it can simplify calculations if we allowed our perspective to “tilt” a bit.
Example 2.14: Tilting the Axis—In addition to defining what is considered positive, we can also adjust the axis to suit the situation. Say we have a block sliding down a rough inclined plane, which introduces friction. Like Example 2.13, if we are interested in the velocity of the block as it slides down, then we can tilt the axis until the block is completely level with the X-axis (Figure 2.18 ). We do need to be sure we reorient everything; otherwise the result we come up with will be wildly incorrect. In other words, decide where to start the analysis and note the change in directions before beginning to solve the problem.
Figure 2.18. Reorientation of the coordinate system
The last subtle idea is the question, “where does the origin go?” Intuitively, we designate a certain point in space to be the “center” or reference. This choice is also arbitrary, and the problem can be simplified depending on where the origin is placed. The placement of the origin boils down to a mathematical game of pin the tail on the donkey at a child’s birthday party: the origin is okay just about anywhere, except there is (usually) a sweet spot where everyone will applaud...and the problem will be solved in the easiest fashion.
Example 2.15: Finding the “Sweet Spot” for the Origin—Suppose we have a pendulum with a ball at the end swinging from left to right, and we are interested in where the ball will be if we took a snapshot of the motion at any time. Equations that describe the motion of a simple pendulum exist, but where do we place the origin in Figure 2.19 to use the equations?
Figure 2.19. A simple pendulum and possible choices for the origin
Surely we can place the origin on the far left or right, below the pendulum, or on the ball itself; as long as we stay consistent with our assumptions, everything will work its way out. The most practical origin in this case is right where the pendulum attaches to the ceiling. This configuration enables us to model the motion of the pendulum in terms of the angle it makes relative to one side of the ceiling.
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In our discussion of coordinate systems, we have been working with real numbers. Although the real numbers may seem to be complete and contain all of the numbers we need to solve practical problems, efforts tend to fall short when finding solutions to an infinite set of problems, which may sound like a huge issue. For instance, we receive electricity as alternating current (AC), which is then converted to direct current (DC) for the majority of our devices. DC can be analyzed using the numbers we know and love, whereas AC needs something ... different. Since AC is based on a wave (Figure 2.20 ), the quantities we would be interested in are now varying; in fact, the quantities will have magnitudes and directions.
Figure 2.20. Alternating current
Sounds eerily familiar, almost like a vector. Except, we need new numbers to fully describe what phenomena we are interested in. Has this happened to us before, where we were simply unable to use the numbers to solve something? When have we needed new numbers before in mathematics? How about the innocent problem: x2 + 1 = 0? Attempting to solve this equation for x yields an odd result: 
. What number when multiplied by itself yields –1? Searching from end to end on the real line is a fruitless endeavor because “no such number exists.” For the community of mathematicians, this mathematical anomaly caused quite the headache until the work of Euler and Gauss—undisputed masters in the field.
To cope, an extension of the real numbers is needed, which can be achieved through abstraction. The extension in question is the introduction of complex numbers (also called imaginary numbers, possibly one of the most unfortunate names in all of mathematics). The complex numbers are built upon the solution to x2 + 1 = 0, namely the solution, 
—this is the imaginary unit.
Definition 2.16: The imaginary unit is the solution to the equation x2 + 1 = 0. We say 
.
Definition 2.17: A complex number z is a number of the form z = a + bi where a and b are real numbers. The term a is called the real part and b is called the imaginary part.
Now, we said the real numbers were extended to include the idea of imaginary numbers, how can we “see” this extension? By taking the real line and the “point” i somewhere not on the real line, the setup begs the question: “where does i belong?” (Figure 2.21 ). Moreover, where does the real line belong relative to the other complex numbers?
Figure 2.21. Attempting to relate i to the real line
We care about how these numbers can be represented and used practically. Although complex numbers seem to exist outside the sphere of “realness” we are accustomed to, representing a complex number mathematically is no trouble at all. Let us think of a complex number, called z, having a real part and an imaginary part. In a sense, we can think of numbers as purely real, purely imaginary, or as a cocktail of real and imaginary like so in Figure 2.22 .
A purely real number, or just “real number,” is precisely the number we are accustomed to; a purely imaginary number is a number that has a factor of 
. We can combine these two numbers together through addition (and subtraction) in order to form a complex number that is not purely real or imaginary. Conveniently, representing a complex number in a plane is just as easy.
Figure 2.22. Purely real versus purely imaginary
Consider the following setup: let the x-axis measure the real part of the complex number and the y-axis measure the imaginary part. In other words, if we have a complex number called z, then z = a + bi, the point in the complex plane corresponding to z is the ordered pair (a,b).
Definition 2.18: The complex plane is a modified real plane containing the numbers z = a + bi where the x-axis corresponds to the real part (a) and the y-axis corresponds to the imaginary part (b).
Representing a complex number as an image that looks nearly identical to a point drawn using a rectangular coordinate system (Figure 2.23 ).
Figure 2.23. The complex plane
Another natural measurement to make is the distance of the complex number from the origin; this is called the modulus or magnitude. We can easily find this distance by applying the Pythagorean Theorem, just like finding the length of a vector.
Definition 2.19: The modulus or magnitude is a complex number’s distance from the origin, (0,0), in the plane, denoted by 
for z = a + bi. Speaking of the modulus, the next measurement to consider is the angle the modulus makes with the real axis, often called the angle or phase.
Definition 2.20: The angle or phase of a complex number is the angle the modulus makes with the positive real axis, denoted by 
for z = a + bi.
The graphical interpretation of the complex plane helped the complex numbers gain acceptance within the mathematical community, particularly due to the intrinsic geometric properties they possess. To observe these properties, we need to perform the familiar operation of addition, subtraction, multiplication, and so on. Treating complex numbers like vectors makes the graphical and algebraic interpretation even simpler; however, one new operation for complex numbers exists—the complex conjugate. The complex conjugate of an imaginary number is found by simply changing the sign in the imaginary portion and is often denoted by an asterisk (*). For instance, the complex conjugate of z = 11+3i is z* = 11-3i.
Example 2.16: The Complex Conjugate—An interesting fact is that the multiplication of an imaginary number and its complex conjugate is always a positive, real number (it has no imaginary component).
Multiplying two complex numbers gives:
If we use a number and its complex conjugate, we find
So, for our complex conjugates above,
Example 2.17: Arithmetic of Complex Numbers with Circuits—When analyzing a circuit where the voltage or current supply is alternating, then the elements of the circuit behave with a certain impedance, which can be thought of as the electrical equivalent of resistance to a force, instead the force is current in this case. When placed in a row like in Figure 2.24 (said to be in series), then impedances will add together.
Figure 2.24. Impedances in series
If we add the impedances together, then we are finding the equivalent impedance Zeq. This means, Zeq = Z1 + Z2 + Z3.
Adding and subtracting complex numbers amount to combining like terms, so we group all of the terms without an i together, group all of the terms with an i, then simplify accordingly. Zeq = (100 + 500) + (-400i-100i + 200i)
Subtraction requires a little more care: say we perform the following operation, Z1-Z3.
Be sure to distribute the negative sign due to the subtraction before grouping like terms:
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2.2 GRAPHICAL RELATIONSHIPS—ANGLES
An engineer’s skill set extends beyond calculations performed by hand or by using a computer to crunch numbers; rather, it includes geometry and other graphical relationships.
When solving physical problems, it can be helpful to apply certain truths from geometry—especially angle theorems. Most of these facts can be used without even knowing the angle measure, but we should at least remind ourselves about how angles are measured.
Plenty of units exist for angles; the most notable two are degree and radian measure. In terms of engineering, the use of degree is more prevalent; however, mathematicians would typically use radians.
Definition 2.21: A degree is a measurement for angles. A full rotation is defined to be 360°.
To see what a radian is, we construct a circle with a radius of r. One radian is numerically defined to be the length of an arc on the circle that is also the length of the radius—hence the name “radian.” Note that we define the angle counterclockwise as shown by the direction of the arrow in Figure 2.25 .
Figure 2.25. Visualizing the radian
Definition 2.22: A radian is a measurement for angles. A full rotation is defined to be 2π radians.
Angle measurements are commonly given as Greek letters; in particular, an arbitrary angle is denoted by the letter theta, θ. Mathematically, we need to make a distinction between the measurement of the angle and the angle itself. Two angles could be equal in measure, but oriented differently, like opening in different directions. Saying two angles are the same means they are exactly the same—that is, no rotations, resizing, and so on. To differentiate between the two concepts, we will use a “
” to emphasize we are talking about the physical angle and drop the “” when we want the measure of the angle.
Since theta can be in radians or degrees, a subscript of “rad” or “deg” will be appended to θ if a distinction is needed. Note that, unfortunately, the subscript is not used in the “real world,” even when it would be helpful.
With this construction, the next logical question would be: “how many radians make up a full rotation?” This is perhaps easily explained in terms of the circumference, the distance traveled on the arc about the circle which makes a full circle. This measure of length is given by 2πr, so the unit circle with radius 1 has a circumference of 2π. This would mean that a full rotation would be 2π radians. From here, it can be deduced that half of a rotation is π, a quarter of a rotation is π/2, and so on.
Now, how do we switch between the two? The conversion is achieved through dimensional analysis, where units are converted through relating equivalent units of measurement. In this case, 360° is equivalent to 2π radians; thus, it is essentially multiplying by “1” in a constructive way. Let θrad be an angle measured in radians and θdeg be an angle measured in degrees. The conversion formulas are as follows:
Example 2.18: How Many Degrees Is One Radian?—Suppose we wanted to find the degree equivalent to 1 radian, then we would use the first formula.
Alright, so the translation between the two systems is a bit messy—who would expect a number like 57.3? Although it is not necessarily a clean correspondence between the radian and degree, we still use both measures.
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A straight line is the simplest angle we can create—a straight angle. These angles have 180° (or π radians) because it is the equivalent of only going halfway around the circle in Figure 2.25 . Splitting the straight angle in half gives us an extremely popular and fundamental angle, a right angle—which is 90°. From here, we can classify angles depending on how they relate to the two basic angles, straight and right (Figure 2.26 ).
Figure 2.26. Angle classifications
If the angle is larger than a straight angle, then we say it is a reflex angle. Once the angle becomes smaller than a straight angle, but bigger than a right angle, it is an obtuse angle (magnitudes between 90° and 180°). Finally, if the angle is smaller than a right angle, it is called an acute angle (between 0° and 90°). The angle types are summarized in Table 2.1.
Table 2.1. Summary of angle types
Example 2.19: Attempting to Determine Angles in a Truss—A truss is a structure assembled in a particular organizational pattern such that each of the pieces, called members, behave as an entire object. Most recognizable from their use in construction and reinforcement for structures, trusses are overflowing with geometry. One type of problem in analysis is to determine the forces in each of the members of the truss, usually to provide a measure of safety. An example of a truss can be seen in Figure 2.27 .
To make the truss function effectively, there needs to be an organizational pattern the members follow. Naturally, we can take advantage of the geometry using some of the common theorems for angles.
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Figure 2.27. Example of a truss
To effectively analyze a truss, we need more terminology. Say we have a right angle with a ray splitting the angle into two pieces, 
and 
(Figure 2.28 ; left). It would make sense that we do not “lose” any degrees or radians, so if we add the angles together, we have θ1 + θ2 = 90°. If two angles satisfy this equality, then and are called complementary angles. Moreover, the figure on the right is a straight angle with a ray forming 
and 
. Using the same idea, θ3 + θ4 = 180°. If two angles sum to a straight angle, then the angles, and , are called supplementary angles. The two angle relationships are shown in Figure 2.28 .
Figure 2.28. Complementary and supplementary angles
Different geometric constructions yield new useful angle properties. Take the intersection of two lines (labeled l1 and l2) as shown in Figure 2.29 . This intersection creates four angles: , , , and .
Figure 2.29. Intersection of two lines
First (moving clockwise) we can see that and , and , and , and and are each supplementary angles. Under this construction, the pair and and the pair and are called vertical angles, which means θ1 = θ2 and θ3 = θ4. Intuitively, we can imagine the equality holds by realizing is an inverted mirror image of .
We can continue our discussion on angle relationships by completing the picture in Figure 2.29 and adding a new line. The new line will be constructed such that l2 is parallel to l3—written as l2/l3. With the introduction of this new line (and the tasteful relocation of l2), we gain four more angles given by 
, 
, 
, and 
(Figure 2.30 ). Based on this construction, we have what are known as corresponding angles.
Figure 2.30. Corresponding angles
The notation is deliberate in alerting which angles are related; for example, and are corresponding angles. This relationship is about equality, so θ1 = α1. A sensible approach to the idea that θ1 = α1 is to realize how Figure 2.30 was constructed. The construction l2/l3 should offer some intuition that l2 and l3, although not the same line exactly, are the same kind of line—only space keeps them apart. The line l1 provides the backbone on the definition of corresponding angles by acting as a transversal—a line that cuts through two other lines.
Using this definition, we can infer the following angles are corresponding (equal) angles:
Again, these relationships imply that the angles are equal in measure. We may recognize the vertical angles lurking in the picture, but two more “special” angle relationships exist, which are worth pointing out.
As an example, let us take advantage of the fact and are vertical angles (along with and ) and that and are corresponding angles (along with and ). Distilling this information into statements of equality, we have the following chain of equal signs: θ1 = θ2 = α1 = α2 in particular, we care about θ2 = α1. Glancing back at Figure 2.30 , we see that this angle relationship is considerably noteworthy. We will call and alternate interior angles.
Through a similar procedure, we can deduce—for instance—that α4 = θ3, yet another notable relationship. Since this equality occurs “outside” of the two parallel lines, we call and alternate exterior angles.
At this stage, it may be tempting to begin abusing vertical and corresponding angles, which greatly reduces the number of symbols we need to use. In fact, it turns out there are only two different angle measures in both Figures 2.29 and 2.30. The new situation below is a simplification of Figure 2.30 without the additional subscripts, and so we now have only two angles, 
and 
.
Figure 2.31. Angle relationships with parallel lines cut by a transversal
For convenience, we will summarize all of the angle relationships in Table 2.2.
Table 2.2. Summary of angle types
Angles were connected to arc length in the formulation of radians, lines were intersected in various ways, and constructions were made to demonstrate the fundamental angle properties. What more can be done? In the picture of an angle in Figure 2.32 , let us continue the construction by introducing a line segment, AC.
Figure 2.32. Constructing the line segment AC
Adding the new line segment forms a popular shape, a triangle, which we denote by ΔABC. The natural question to ask based on the discussion of angles is: “what is the sum of all of the angles in the triangle?” Elementary school tells us that the angle sum is always 180°, but perhaps we can use our discussion of angles to show why this is the case.
Consider Figure 2.33 , let l1/l2 and let l3 and l4 be transversals that intersect l1 at a single point, C. The lines l3 intersects l2 at B, whereas l4 intersects l2 at A. With this construction, we can see the points A, B, and C ΔABC.
Figure 2.33. Angle sum of a triangle
Let , and 
be the angles of ΔABC at A, B, and C, respectively. Notice that and 
are alternate interior angles (if unconvinced, imagine that l3 is not in the picture). The same can be said for and 
This implies that α = α′ and β = β′. Now, we use the “whole is the sum of its parts” idea in order to write α′ + γ + β′ = 180°; however, we just established α = α′ and β = β′. Thus,
Theorem: Angle Sum of a Triangle: Let A, B, and C be the vertices of a triangle ΔABC. Let , , and be the angles of ΔABC at A, B, and C, respectively. Then, α + β + γ = 180°.
Our construction of ΔABC was arbitrarily chosen such that none of the angles were the same in order to preserve generality. Since all of the angles are different, it follows that the side lengths are different as well. Such a triangle is called scalene.
Definition 2.23: Scalene Triangle: Let A, B, and C be the vertices of a triangle ΔABC. ΔABC is scalene if each side length is distinct.
We might also have a case with angles that equal one another. For instance, what if two angles were the same? Suppose α = β. From the Angle Sum of a Triangle Theorem, we will replace β with α.
Assuming we already know α, the angle γ is then given by:
Note that our choice of angles is limited since the angle sum of a triangle must be equal to 180°. In a triangle, all of the angles need to be positive and not equal to zero: α,β,γ>0°. Thus, the upper bound for α will be when γ = 0° (obviously not possible by our inequality).
Thus, α is some angle measure between 0° and 90°, not including the end points. This result brings up the point that a triangle cannot have two right angles otherwise the final angle would have a measure of 0°. When two angles are the same, it will follow that the corresponding side lengths are the same as well. These triangles are called isosceles.
Definition 2.24: Isosceles Triangle: Let A, B, and C be the vertices of a triangle ΔABC. ΔABC is isosceles if two sides are equal.
Finally, assume that all of the angles are equal, α = β = γ. Replacing β and γ by α in the Angle Sum for a Triangle Formula, we have:
Our options have quickly dried up as there is only one possible configuration for this case. Since all of the angles are the same, then all of the side lengths will be equal as well—this triangle is equilateral.
Definition 2.25: Equilateral Triangle: Let A, B, and C be the vertices of a triangle ΔABC. ΔABC is equilateral if each side length is equal.
While the previous few triangles are all important, a considerable amount of work in engineering will involve a triangle that contains a right angle—otherwise known as a right triangle.
Definition 2.26: Right Triangle: Let A, B, and C be the vertices of a triangle ΔABC. ΔABC is right if one of its angles is right.
A fundamental concept surrounding right triangles is how side lengths can be related to the angles in the triangle itself. The relationship will manifest itself as two intimately related curves. Constructing them will be accomplished through the unit circle.
Definition 2.27: The unit circle is a circle with a radius equal to 1.
Figure 2.34. The unit circle
The idea is to begin by relating the side lengths, x and y, to the angle θ makes with respect to the positive x-axis like we do with polar coordinates (Figure 2.34 ). Imagine the point P moving counterclockwise on the unit circle like the end of a clock hand. While P is making its way around the circle, the dotted lines x and y will always form a right triangle as long as we draw the dotted y line straight down to the x-axis. By moving P in this manner, we are gradually increasing θ; the more rotations, the bigger the angle. To make the two curves, we will note the length of x first in order to make a graph plotting the angle size, θ, versus the measurements we take. Through persistence and tedious calculations by ruler and protractor to determine x for each small increment of θ, the result of plotting θ versus x yields a smooth curve (after being tidied up) in Figure 2.35 .
Figure 2.35. Plot of the angle, θ, versus the length of x for one rotation
We picked out a few “nice” points that illustrate the 1>0>–1>0 pattern in the curve. Since there are five points highlighted on the curve, the pattern can be seen beginning to repeat at the final point. In fact, since we can increase θ as much as we please, this curve is going to repeat itself for every rotation around the circle (even if we use negative angles). A convenient result of this curve (Figure 2.36 ) is the ability to find the length of the x leg of the triangle for any θ. We use this curve so often that we give it a name, the cosine curve.
Figure 2.36. Plot of the cosine curve on a wider interval
In fact, there is a similar curve for the length of y called the sine curve, which is obtained using the same process (Figure 2.37 ).
Figure 2.37. Plot of the sine curve
For simplicity, we used the unit circle in order to construct the curves; unfortunately, this restricts us to right triangles with hypotenuses of 1. Although this may seem to be an issue, it can be easily solved. With the knowledge of what the cosine and sine curves represent, the x length and y length, respectively, we can consider a triangle with any hypotenuse by using a circle of radius r instead of the unit circle with radius 1 (r will be the hypotenuse of the triangle). The curves will retain their shape, but will have peak values of r and –r instead of 1 and –1 (Figure 2.38 ).
To refrain from writing “sine curve” and “cosine curve” repeatedly, we will denote the two curves using specific notation such that they become mathematical objects, allowing the use of algebra. Thus, the sine curve will be given by sin(θ) and the cosine curve will be given by cos(θ). To describe Figure 2.38 , we write:
Figure 2.38. The cosine curve scaled for a triangle with a hypotenuse of r
Solving for the curves shows that the relationship sine and cosine create within the triangles is much deeper than just the x and y lengths:
With sin(θ) and cos(θ) loosely defined, we will lay the foundation for trigonometry by writing a formula equivalent to what we already know; that is, cos(θ) gives us the x side of the triangle and sin(θ) gives us the y side of the triangle. Thus, for the unit circle:
The x length and y length correspond to a point, P = (x,y), on the circumference of the circle. Since the hypotenuse of the triangle begins at the origin, O = (0,0), and spans to P, then it is also the radius of the circle.
Note that we can represent our equations for sine and cosine as follows (as shown in Figure 2.39 ):
Figure 2.39. Demonstrating the Pythagorean Theorem for trigonometry
Apply the distance formula (or Pythagorean Theorem),
This result should not be too surprising, and we could easily stop here; however, we have an opportunity to further relate sin(θ) and cos(θ) in the context of the unit circle. Since we are limiting ourselves to the unit circle, r = 1. Now, consider the relationship between the legs of the triangle and the two curves. With x = cos(θ) and y = sin(θ):
Theorem: Pythagorean Theorem for Sine and Cosine: sin2(θ) + cos2(θ) = 1 for all values of θ.
Incredibly, we can use this formula for a variety of calculations.
Example 2.20: Using the Pythagorean Theorem for Sine and Cosine (or Pythagorean Trigonometric Identity)—For instance, suppose we know the y length is 
and want to know what the x length will be. Using the Pythagorean Theorem for Sine and Cosine, the x length can be found with little effort regardless of θ.
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Useful as this may be, the pair 
is a known value; in fact, it is a special value of sin(θ) and cos(θ). For convenience, Table 2.3 contains the special values for specific angles (given in degrees and radians).
Table 2.3. Special values of sin(θ) and cos(θ)
Even though the given angle measurements are restrictive, it is possible to calculate a broader range of angles using a property of the two curves. For example, consider the angle θ = 105°. Upon perusing the table, we find that θ does not appear anywhere on the list; however, θ is the sum of two angles, θ1 and θ2, which do appear! The angles in question are θ1 = 45° and θ2 = 60°. The issue is how we can evaluate the following:
In practice, we do not need the alternative expressions for tedious calculations; instead, they assist us in intermediate steps of derivations.
Since sin(θ) and cos(θ) were born from looking at geometric constructions within the unit circle, the angle sum and difference formulas can be proven using the same ideas—triangles and trigonometry!
Example 2.21: A Trivial Use of the Angle Sum Formulas—Without a calculator, finding the exact value sin(105) can be a chore. As we noted before, 105° = 45° + 60°, both of which are in our table of known values. Applying the formula,
yields the following:
Glancing at this expression, we realize each one of the terms can be found using the table! After a bit of simplification, we have
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Example 2.22: New Identities from the Old—A handful of other identities can be derived from the angle sum identities. For instance, how can we determine expressions for sin(2θ) and cos(2θ)? In both cases, the 2θ can be obtained if we use the angle sum identities and assume θ1 = θ2 and call them θ. By doing so, we obtain
Similarly, we can use the same argument with the cosine equivalent:
which becomes
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2.2.2.1 Laws of Sines and Cosines
When working with triangles, we often find ourselves using right triangles because they frequently appear in engineering problems. Knowing that the sum of the interior angles in a triangle sum to 180 is useful for times when we need to find the value of an angle and the Pythagorean Theorem helps us with determining side lengths; however, when we have partial information or triangles that are not right, we have additional few tools at our disposal to find any values we need.
The Law of Sines is useful in the event that we know the length of one side and the values for two angles (of course, if we know the values for two angles, we should know the value of the third angle). The Law of Sines applies to any triangle (Figure 2.40 , for example) and is:
The Law of Cosines is useful in the event we know two of the lengths and their included angle. For example, if we knew lengths b and c and angle A in Figure 2.40 , we could use the Law of Cosines to find the length a:
or any of the sides:
Figure 2.40. Triangle used to define the Laws of Sines and Cosines
Example 2.23: Design of a Nature Preserve—Suppose we were tasked to design a space for a nature preserve, which included fencing the perimeter. If we were constrained by private property such that side a was 1200 km (Figure 2.41 ), and the largest angles at two vertices can be B = 35° and C = 65°, respectively, then we need to find the length of fence required to span the unknown lengths b and c.
Figure 2.41. Area of nature preserve from Example 2.23
We can find the missing angle easily by using the Triangle Angle Sum Theorem, which means A = 80°. However, since we have one side and (at least) two angles, we can use the Law of Sines to find the length of sides b and c.
Once we have the values of b and c, our job is done! We now know we need 698 km of fencing for side b and 1104 km of fencing for side c. Taking into account the known side a, 1200 km, we can add each of the side lengths together to see we need 3002 km of fencing for the entire nature preserve.