Skip to main content

# 1. Theory

## 1.1 Governing Equation

## 1.2 Extended Functions

## 1.3 Identity Rules

# 2. Purpose

# 2. Exercises & Possibilities

## 2.1 Civil Engineering

### 2.1.1 Inclined Bridge

### 2.1.2 Theodolites

## 2.2 Aerospace & Aeronautics

### 2.2.1 Turbine Velocity Diagrams

### 2.2.2 Aircraft Flight Path

### 2.2.3 Satellite Constellation Trajectories

### 2.4.2 Antenna Array Beamforming and Steering

### 2.4.3 Enhancing Data Aerial Transmission

###### Table 1. Coefficients of modulated signal for $\gamma=90$ deg

###### Table 2. Coefficients of modulated signal for $\gamma=120$ deg

## 3.5 Vibration Theory

# Bibliography

Beyond Trigonometry : Applications of Extended Sine and Cosine Functions to Engineering

Published onSep 13, 2022

Beyond Trigonometry : Applications of Extended Sine and Cosine Functions to Engineering

Ever since sailors aligned quadrants with stars to guide themselves through the perils of the oceans (as illustrated in Figure 1), the need to project angles into distances (and vice versa) has been an inevitability for humanity in its endeavor for territorial and technological expansion. In modern times, this need still exists and is again present during the navigation of the SpaceX Dragon “Axiom-1” capsule as it made its final approach (with its four astronauts) into the International Space Station on April 9^{th}, 2022.

Trigonometric functions such as sine *If the right-angle transformed into an angle of or degrees, what would these functions look like (Figure 2)?* The projected lengths of the hypotenuse given by the traditional sine and cosine would change ― this time dependent on both angles and ― or simply put, into and [where the asterisk sign * implies that they are different from the original sine and cosine].

The contemporary approach to dealing with a scalene triangle is to subdivide it into two right triangles and employ trigonometric operations to the two in order to find the lengths of the sides and angles of the scalene triangle. This resorting to orthogonality to solve non-orthogonal problems is not an organic approach, and a more direct trigonometric and mathematical path is required. If an mathematician or scientist is faced with a problem where he/she needs to know the (normalized) sides of a scalene triangle based on its angles, then the functions sine

These extended functions expand the usefulness of application of the already existing trigonometric functions in a variety of scientific fields, ranging from orbital mechanics [1], electronics [2], chemistry [3] and design [4]. Both the Pythagoras’ theorem and trigonometry (in general) form part of most secondary education curricula around the world, including in the American, Canadian and Australian Curriculum [5-7], which makes this paper of interest to both students and professionals.

The Law of Cosines [written below in Eq.(1)] is a broad expression that relates the lengths of the three sides 𝑥, 𝑦 and 𝑧 of any triangle [8,9], which not only covers the particular case of the Pythagoras theorem

The particular case of the Pythagoras theorem is satisfied by replacing

It has been proved [10] that the expressions for the extended sine function

In trigonometry, the angle sum and difference identity rules establish a link between additive or subtractive operations in angles and its impact on the lengths of their respective right triangles, and are commonly defined as

This is only applicable to a triangle with an obtuse angle

For the angle difference identity rule (where

Note that the drawings in Figure 3 and 4 are practically the same, except that the scalene triangle

This article concerns itself in the application of these equations (in Chapter 3); further details on their construction and proofs can be found in the respective publication.

There are several ways to compute problems in trigonometry, and the point of this article is to solve the problems using the explicitly proposed (above) extended sine and cosine functions (section 1.2) and their identity rules (section 1.3), which have a broad application to scalene triangles, much in the same manner as conventional sine and cosine functions are applicable to right triangles (hence enabling a scientist or mathematician the possibility to, when the orthogonal condition fails, to replace the conventional sine and cosine by their extended versions immediately and effortlessly — culminating in a more flexible mathematical solution). Moreover, the new extended trigonometric functions open doors for possibilities of expanding scientific fields in which they (the functions) are the foundations. Some possibilities will be presented and discussed briefly.

In order to assist the confirmation of the solutions of the following trigonometric problems, the open-source program Geogebra [12] can be used to draw the involved scalene triangles, as well as, in determining their lengths and angles.

**Problem. **A suspension bridge has a series of main suspension cables connecting the horizontal road deck (through which the automobiles transit) to the vertical tower, making an angle *How much have the cables stretched due to the inclination?*

**Solution. **The horizontal projection AC is assumed unchanged by the inclination. Hence, this is the starting point, as it is common to both cases straight

However, the inclination changes the right triangle (

Therefore, the inclination of half a degree by the tower has stretched the main suspension cables by 10 meters.

**Problem. **Imagine that civil engineers with theodolites devices (Figure 1 left) are placed at** **either side of a river (located at points A and B), and wish to measure their distance to a location on the bridge (located at point C), which is inaccessible due to constructions. Among other things, Theodolites devices measure vertical and horizontal angles between visual reference points with great accuracy [14]. Point C is located on the bridge and directly over water, so it is not possible to measure the horizontal positions of point A and B to point C, and neither the height. However, the engineers can measure the distance between them, resulting in 850m. Engineer A rotates the theodolite from point B to C and measures a vertical angular distance of 38 degrees, while engineer B does the same from point A to C and measures 24 degrees. *What are the distances of both engineers to point C on the bridge? What is the height from point C to the horizontal line connecting engineer A to B?*

**Solution. **The angle

On the other hand, the distance of the second engineer B to the point C on the bridge is given by direct application of the extended sine function as

Knowing the distance and angle from A to C, it is simply a matter of applying conventional sine function to obtain the vertical projection in the right triangle

**Problem.** Velocity vector diagrams are tools used by aerospace and aeronautics engineers to design and understand the loading of a turbine stage on an aircraft engine (Figure 1 left) [15]. It links the magnitude and direction of air velocity vectors (both absolute *What is the relative velocity ** seen by the rotor as the flow enters it?*

**Solution.** The analysis revolves around the scalene triangle

Finding the angle

which, by replacing the values above, results in

That re-arranged gives

Note that, computing the arctangent of angle γ gives -47.7 degrees. Since we know that the angle γ needs to be obtuse, we add 180 degrees to give

*If the absolute angle of ** to the engine axis ** is in fact 2 degrees more than anticipated, what is the new relative velocity ** (use the angle difference identity rule)?*

The new angle is

As before the resulting obtuse angle is

which expands to

By applying the extended sine and cosine functions given by Eq.(2) and Eq.(3), the end result is

**Problem. **Two engineers placed theodolites (height 1.7m) at a distance of 400m apart over a runway to record the flight path of aircraft taking off. One aircraft departs and climbs directly over them. As it crosses the sky, the engineers measure at either side the vertical angle for different synchronized times. The usage of electro-optical system to track the position of aircraft is common in the world of aviation [16]. The result are a succession of scalene triangles, as illustrated in Figure 1 (with the recorded angles presented on the right). *What is the distance of the aircraft to each of the points A and B in the ground? Sub-sequentially, what is the height of the aircraft at the different points in time?*

**Solution. **Drawing lines from the three elements — engineers A, B and the aircraft — created a series of scalene triangles that change based on the time at which they were taken. While the three ground angles were measured with the theodolites, the third internal angle of each scalene triangle is 116 degrees for T_{1}, 109 degrees for T_{2} and 79 degrees for T_{3}. The solution for each aircraft position with associated time is computed directly from the extended formula. For time T_{1}, the distance from engineer A to the aircraft is

and the distance to engineer B is

and the aircraft is at a vertical distance of

For time T_{2}, the aircraft is at a distance from engineer A of

and a distance from engineer B of

while the aircraft is at a vertical distance of

For time T_{3}, the distance from engineer A to the aircraft is

and the distance from engineer B is

and the aircraft is at a height of

*Now, if the aircraft projects a shadow on the ground at an angle of 30 degrees to the vertical, what is the distance of engineer A to the projection at point O? If the vertical angular distance increases by another 10 deg as the aircraft climbs (here, we assume that the aircraft rotation or flight path is such that its relative distance to point A remains the same), what is the new distance from engineer at point A to the shadow (use the angle sum identity rule)?*

The angle from the horizontal to the projected direction of the shadow is _{2}, the extended cosine function gives directly the answer for the distance from point A to the projected shadow on the ground to be

The fact that the relative distance of the aircraft does not changing to point A means that, as the aircraft rotates in its climb, the distance

The projected direction of the shadow is still

That is, the shadow moved from

**Problem. **Three GPS satellites trailing each other on the same orbit operate best at an optimum normalized relative position to each other as shown in Figure 1. It is worth noting that in reality such a constellation of properly geometrically-spaced GPS satellites orbiting the Earth has typically 24 satellites disposed in a 3D configuration [17] — so this is a very simplified example. Here, each satellite has an optical sensor onboard that measures the satellite´s angle between the others two. *What angles do the sensors of all three need to measure to guarantee that they are flying at the optimum orbital distance to each other?*

**Solution. **Since the sum of the internal angles of a triangle is 180 degrees, all is required is to know two of them. This implies the need for two equations to solve for two variables. The first equation results from the cosine projection of the longest side of the scalene triangle as

which re-arranged gives

Similarly, the cosine projection of the longest side of the scalene triangles gives

By replacing the above result in

that simplifies to

The identity rule gives

Replacing Eq.(10) and Eq.(11) into the identity rule results in

Expanding results in

Placing the denominator to the right side simplifies this to

which reduces further to

resulting in an expression for cos(α) as

Note that, computing the arcsine of angle γ gives 32.39 degrees which is in the first quadrant. Since we know that the angle γ needs to be obtuse, we are interested in the value in the second quadrant of 147.61 degrees which has the same value of sine. The third angle β then becomes

Therefore, in order for the satellites to fly in an optimum distance deployment, the angular sensor in satellite A needs to measure 19.07 degrees (between satellite B and C), the one in satellite B needs to measure 147.61 degrees (between satellite A and C) and satellite C needs to read 13.32 degrees (between satellite A and B) .

**Possibility. ** Imagine an antenna array composed of sensing elements uniformly spaced in a line. Beamforming is a process by which an interference pattern between the radiated or received signals of all the antenna elements allows the array to acquire directivity in reception/emission, by forming a high gain lobe (in which it is most sensitive to transmissions) located at the center of the array, and perpendicular to the array’s axis [18]. This process involves the use of sine functions and trigonometry, and hence we will expand further into its workings.

When two antenna elements are spaced by distance

The gain of an array is typically computed from the summation of all signals, and is used to construct the array radiation pattern, quantifying its directivity.

For these two particular elements (Figure 10), the time delay

where

In reality, this time delay

The new time delay

This formula indicates that

where

**Possibility. **Binary digital modulation techniques (used in modern wireless data transmission) are simple in concept, but are not efficient in terms of their spectral density. Augmentation of spectral efficiency (i.e., boost transmission bit rate without affecting bandwidth requirements) is commonly achieved via the adoption of quaternary signaling schemes, like quadrature phase-shift keying (QPSK) [19]. In QPSK, the number of bits that are combined are 2 so this makes M=4. Quaternary signaling schemes embed information in carrier phase modifications, while at the same time keeping the carrier amplitude and frequency the same.

Digital signals Q and I are used to modulate a carrier wave by altering its phase in four possible ways (i.e., 45, 135, 225 and 315 degrees), each representing a symbol (Figure 12a). The in-phase signal I is along the x-axis in Figure 12a, and is represented in magenta in Figure 12b. The quadrature signal Q is along the y-axis in Figure 12a, and is represented in blue in Figure 12b. In an orthogonal system of axis, the x-axis is at 90 degrees to the y-axis, which means that the in-phase signal I is phase shifted by 90 degrees from the quadrature signal Q. The novelty introduced by the extended sine and cosine functions is in the ability to change the angle between the system’s axis (other than

The digital QPSK modulated signal is given by the equation

where angle

45 - (0)00 | ||

135 - (0)01 | ||

225 - (0)10 | ||

315 - (0)11 |

Until now, the phase distance between the carrier wave Q and I was fixed to 90 degrees, as sine and cosines are governed by the dynamics of a right-angled triangle. With the extended sine and cosines functions, this phase distance can be modified to an arbitrary value given by the angle γ. Since the system axes are now no longer static (frozen in orthogonal mode), this approach is hereby termed the Dynamic-Axis Quadrature Phase Shift Keying Modulation or DA-QPSK Modulation. Replacing the sine and cosine by their extended functions in Eq.(2) and Eq.(3), allows the x-axis to be at virtually any angle

Digital signals Q and I are used to modulate a carrier wave by altering its phase in four possible ways (i.e., 45, 135, 225 and 315 degrees), each representing a symbol (Figure 12a).

Figure 13a shows that by altering the angle between the two axes (in this case, to

The amplitude modulations of the quadrature signal Q and in-phase signal I produce the phase-shifting of the wave produced by their sum (dashed red in Figure 13b for

where

where the extended

For the particular case of an orthogonal axis system (

and the extended function

which re-arranged further simplifies to

For the particular case of

Consider the case of the symbol 100 (in the first quadrant, where

Similarly, the function

Thus, the coefficients that enable the carrier to shift by the above desired phases (i.e., 30, 120, 210 and 300 degrees) can be computed using the same approach as above, resulting in Table 2.

30 - (1)00 | ||

120 - (1)01 | ||

210 - (1)10 | ||

300 - (1)11 |

Figure 1 shows an example of the conceptual block diagram of the transmitter that could produce the Dynamic-Axis QPSK modulation. The key difference between DA-QPSK and QPSK modulation is the ability to alter the phase between the Q and I carrier waves beyond the customary 90 degrees, a choice that is controlled by the binary status of the first of the 3 Bits via an additional switch. Hence, in DA-QPSK, the number of bits that are combined are 3 so this makes M=8 (an improvement from QPSK where M=4).

The reception and interpretation of the signal implies, for each phase change determine what is the magnitude. If it is an angle multiple of 90/2=45 degrees, then the first digit is 0. Likewise, if it is a multiple of 120/2=60 degrees, then the first digit is 1. So, the first digit is determined by the resulting factorization of the phase. Second, the other two digits are determine by the magnitude of the phase, and is located in the axes plus circle diagram in the same manner as for QPSK modulation. The only difference here, is that if the first bit is 1, then the 120 degree system of axis is used, and the location in the diagram will change accordingly to Eq.(17).

**Possibility. **Sine and cosine are functions that known to govern the oscillation of the most fundamental vibratory system — i.e., the pendulum and the mass-spring-damper system [20].

In such a system, the instantaneous conversion of kinetic energy

which, noting that there are two squared terms, it is equivalently represented by the Pythagoras theorem as an area balance

where

Diferentiating Eq.(20) provides the (unforced and undamped) governing equation in its mostly known formation

which has a solution of the type

where A is an arbitrary constant to be defined for particular initial conditions of the system (e.g., holding the spring at a given height, and then releasing it).

Now, consider the possibility that there is a lag in converting kinetic into potential energy. The mass is the same, and the stiffness is the same. Only that kinetic energy does not convert into potential energy immediately, as there is a degree of freedom acting as a buffer in the system. It means that when the mass is at the energetic extremes, all the the energy in the system is converted into kinetic energy half-way in between the oscillations, and all the energy is transformed into potential at maximum displacement. However, in between these extremes part of the energy is stored in a buffer (to which the kinetic and potential energy of the mass-spring system is coupled).

If the Pythagoras theorem governs the energy balance in a conventional mass-spring system, then one that presents a lag is governed by the more general version of such an equation — the Law of Cosines. Prior research has shown that there are other extended version of the Pythagoras theorem using triangles [21] and hexagons [22] that differ from the former by the presence of a coupling element. Figure 16 shows in yellow the coupling area in both extended versions that make them different from the original Pythagoras theorem.

In the case of amodified spring-mass system, the coupling is modeled by replacing the mass block by a spring with distributed mass along its length (Figure 17). This mass distribution has to do the coupling of inertia and stiffness that will be explained better later. The direction of displacement of this “buffer” spring is at an angle γ to the displacement direction of the main mass-spring system. The angle γ defines how much coupling exists between the buffer and the main system. For instance, if

Accounting for this coupling results in the modified energy governing equation as

which can be more readily modeled and understood by replacing the particular case of the Pythagoras theorem by the general case of the Law of Cosines (that reduces to the Pythagoras theorem for

Note that the coupling term here is

whose differentiation leads to the more generalized governing equation [more than Eq.(23)]. Note the coupling term

which was already proven to be true [19], and the extended sine and cosine functions have been shown to be given by Eq.(2) and Eq.(3), respectively. The angle

[1] Curtis, H. D. (2010). *Orbital Mechanics for Engineering Students* 2^{nd} Edition. Butterworth-Heinemann. https://doi.org/10.1016/C2009-0-19374-1

[2] Rawlins, J. C. (2000). *Basic AC Circuits* (2^{nd} Edition). Newnes Publishing. https://doi.org/10.1016/B978-0-7506-7173-6.X5000-7

[3] Howard Mark, H. and Workman Jr. J. (2018). *Chemometrics in Spectroscopy* (2^{nd} Edition). Academic Press. https://doi.org/10.1016/C2015-0-04023-0

[4] Parisher, R. A and Rhea, R. A. (2012). *Pipe Drafting and Design* (3^{rd} Edition) Gulf Professional Publishing. https://doi.org/10.1016/C2011-0-06090-8

[5] Reys, B. J., Dingman, S., Nevels, N. & Teuscher, D. (2007). *High School Mathematics: State-Level Curriculum Standards and Graduation Requirements.* Center for the Study of Mathematics Curriculum. https://files.eric.ed.gov/fulltext/ED535222.pdf

[6] Canadian Ministry of Education (2020). *The Ontario Curriculum, Grades 1–8: Mathematics*. https://www.dcp.edu.gov.on.ca/en/curriculum/elementary-mathematics

[7] Australian Curriculum, Assessment and Reporting Authority. (n.d.). *Australian curriculum: Mathematics F–10*. http://www.australiancurriculum.edu.au

[8] Maor, E. (2007). *The Pythagorean Theorem: A 4,000 Year History*. Princeton University Press. Retrieved from https://www.jstor.org/stable/j.ctvh9w0ks

[9] Pickover, C. A. (2012). *The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics*. Sterling Milestones

[10] Teia, L. (2022). Extended Sine and Cosine Functions for Scalene Triangles. *Journal of Mathematics Research*. (Under Review)

[11] Teia, L. (2022). Extended Angle Sum and Difference Identity Rules for Scalene Triangles. *Journal of Mathematics Research*. (Accepted; Publication scheduled for October 2022)

[12] Feng, G. T. (2013). Introduction to Geogebra – Version 4.4. Retrieved from https://www.academia.edu/34890249/Introduction_to_Introduction_to_GeoGebra

[13] Pipinato, A. (2021). *Innovative Bridge Design Handbook: Construction, Rehabilitation, and Maintenance* (2^{nd} Edition). Elsevier. https://doi.org/10.1016/C2019-0-05398-8

[14] Topcon Corporation. (2016). *DT-200 Series : Advanced Digital Theodolite.* https://content.dicarlotech.com/hubfs/Construction/Topcon%20DT-200/Topcon_DT200_Theodolite_brochure.pdf

[15] Haselbach, F., and Taylor, M., (2013). Axial Flow High Pressure Turbine Aerodynamic Design. *Lecture Series—Von Karman Institute for Fluid Dynamics.* Chap. 7

[16] Wuhan Huazhiyang Technology Corporation (2020). *Security EOS Electro Optical Systems , Radar Tracking System For Vessel / Aircraft*. http://www.eoselectroopticalsystems.com/sale-9967784-security-eos-electro-optical-systems-radar-tracking-system-for-vessel-aircraft.html

[17] Moorefield Jr., F. D. (2020) GPS Standard Positioning Service (SPS) Performance Standard. *Office of the Department of Defense. U.S. Government.* https://www.gps.gov/technical/ps/2020-SPS-performance-standard.pdf

[18] Liu, W. & Weiss, S. (2010). *Wideband Beamforming - Concepts and Techniques*. John Wiley & Sons Inc. New York

[19] Grami, A. (2016). *Introduction to Digital Communications*. Academic Press. https://doi.org/10.1016/C2012-0-06171-6

[20] Timoshenko, S. P., Young, D. H., & Weaver, W. (1974). *Vibration Problems in Engineering* (4^{th} Edition) John Wiley & Sons Inc., New York https://archive.org/details/vibrationproblem031611mbp/mode/2up

[21] Teia, L. (2021). Extended Pythagoras Theorem Using Triangles, and its Applications to Engineering. *The Journal of Open Engineering.* https://doi.org/10.21428/9d720e7a.7b128995

[22] Teia, L. (2022). Extended Pythagoras Theorem Using Hexagons. *Journal of Mathematics Research*. https://doi.org/10.5539/jmr.v14n2p19