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Using Trigonometry in our everyday life

Using Trigonometry in our everyday life. BY: BADHON TITHI, SERGIO DIAZ, JIALIN SU. TRIGONOMETRY :.

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Using Trigonometry in our everyday life

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  1. Using Trigonometry in our everyday life BY: BADHON TITHI, SERGIO DIAZ, JIALIN SU

  2. TRIGONOMETRY: Math is always important in our school work but also in our everyday lifestyle. Every day we use math in ways such as counting money, calculating commute time, preparing meals and etc. One of the main topics of math is Trigonometry. Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Trigonometry can be used in a lot of every day applications. Certain job fields require Trigonometry as a mandatory requirement such as Engineering, Criminology, Marine Engineering, Construction, Architecture.

  3. CRIMINOLOGY • Trigonometry can be used in Criminology in a multitude of ways. Investigators can use trigonometry to – • Determine the approximate bullet entry and exit point • Analyze blood splatter to find key information about the criminal such as height, distance, and the estimated location of suspect using the angle of blood spatter. These key information can lead to the catch of the criminal by using the height, distance and angle. We used a principle model to show how one can calculate any distance and angle, solving a crime scene based question where the values are assumable.

  4. MATERIALS: • Measuring Tape • Table • A specific object (book) • Calculator • Pencil • Paper

  5. This is our principle model that shows the central concept of Trigonometry using every day objects such as a book and a ruler on a table. Using different objects, we have proven how we can use trigonometry to find the angle and distance from different perspectives. H Height=9.5 inch Distance=38.75 inch Angle=13.775018° H D D Height=9.5 inch Distance=38.75 inch Angle=13.775018° We used the ruler to measure the height and distance. Once we had those two measurements, we could have found the angle by using the trigonometry function of tan. tan(x)=opposite adjacent tan(x)=9.5 38.75 =13.775018°

  6. PROBLEM: • The police found the blood splatter 5ft away from the victim’s determined location. It was discovered that the blood impacted the surface floor at a 50° angle. The Crime Scene Analyst must determine the approximate height at which the blood originated. To do this, the Crime Scene Analyst must use the tan function to find the height at which the blood originated. This is the trigonometry process the Crime Analyst would work through. The estimated origin of the source of blood is 5’95 ft. This information is key in building a profile in order to catch the criminal. • tan(x)= opposite = height • adjacent distance • tan(50)= h • 5 • h= 5tan(50) • h= 5.95 h But what is the significance of finding the location in which the blood is originated? What if you do not have the angle? Nor distance nor height. How would you solve the problem then? 50° 5 ft

  7. BLOOD SPATTER TRIGONOMETRY How is the angle of impact determined? Blood spatter analysts study blood patterns in order to deduce the events of a crime scene. Using trigonometry functions, analysis can calculate the angle of impact which will aid in solving the events in a crime scene such as the previous problem.

  8. BLOOD SPATTER Low Velocity Spatter • Minimum to no force applied in the effects of blood droplets other than dripping directly to the ground (usually perpendicular to the ground or near 90° angle) • Tends to be larger in size and spherical in shape, the absence of extraneous force may allow blood droplets to reach terminal velocity; where height at which the blood falls does not affect the size of the blood splatter. Medium Velocity Spatter • More energy and force applied to blood droplets compared to low velocity spatter IE. Blunt force, stabbings, slashing. • Blood disperses into smaller droplets varying in sizes due different size of surface area in a blunt object • Blood patterns are relatively linear with less blood deposited due to the smaller focused surface area of weapons in stabbings/slashings. High Velocity Spatter • Large amount of force applied to blood droplets. IE. Gunshot, high speed collision, explosives. • Blood stain patterns resembles a fine spray of mist. The shape of blood droplets affected by velocity when impacting a surface or surfaces can determine the angle of impact, the nature of injury, and the weapon that caused the injury.

  9. CALCULATING THE ANGLE OF IMPACT • Blood droplets that strikes a surface at an acute angle exhibits an elongated elliptical-shaped blood stain. • Dr. Victor Balthazard and Dr. Herbert Leon Macdonell discovered the relationship of the width and length ratio of the ellipse is a sine function of the impact angle Side view Aerial view

  10. AREA OF CONVERGENCE There are many factors that comes into play when determining the point in which the blood is released. Blood may be projected onto target surface(s) in various trajectory paths caused by different weapons or objects. Furthermore, the victim and suspect is not fixed to a specific location in a realistic 3 dimensional scale. Victim may move, twist, turn, struggle, or pushed by blunt force trauma. These variable(s) affect the location in which the blood is released. • By tracing well formed blood stains through the elongated length, blood spatter analysist can determine the distance (represented by d) in which the trajectory of blood stains originated prior to its deposit on different surfaces. • When the area of convergence is formed, an approximate center is then used as the base of measurement in creating a 3 dimensional platform for tracing the blood back to its origin, recreating the approximate location of the victim at the time of the event.

  11. AREA OF ORIGIN After applying sine function to identify the Angle of Impact and tracing the trajectory of blood stains back to an area of convergence, a third measurement is introduced in creating a 3 dimensional scale in the form of height (represented as h). Now that we have the angle (calculating the Angle of Impact using sine function) and the distance (tracing each well formed blood stains back to an area of convergence, creating a center as a base for measuring distance), we can discover the height using the same formula in our 2 dimensional problem from the beginning. • Using this process for every well formed blood stain will eventually create the Area of Origin, indicating an accurate location of the victim during the time of incident and the area of wound the blood originally came from

  12. Using trigonometry, analyst are able to recreate the incident even if the victim has been moved. This will help aid in the investigation of a crime scenes, discovering hints of foul play (possibility of manipulation from the original manner of crime such as making a crime look like a suicide) by finding discrepancies that does not match the blood spatters. Maybe even help catch a suspect over a long distance shooting by retracing the area of origin back to the exit wound of a victim and following pass the point of entry in determining the approximate location of the murderer. Retracing trajectory pass the point of entry Area of Origin l= 10mm W= 7mm 3.8ft 3.9ft 4ft 3.95ft 44.43° 45.89° 45.15° 44.78°

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