Application of Trigonometry in Data Science and AI
- 1. Team Aryabhata APPLICATION OF TRIGONOMERTY “Mathematicsmaynotteachushowtoaddloveor minushate. Butitgivesuseveryreasontohopethat everyproblemhasasolution.”
- 2. Contributions of Aryabhata in Mathematics: He formulated the tables of sine. He introduced the alphabetical counting system. He was the first to do calculations on the square and cubic roots. He gave the solutions to equations by r = ax + ca by = ax – c He was the first to provide the approximate value of π. He was the one who said that it is an irrational number. He was the first to sum the first n integers. And many more like this…… About Aryabhata Aryabhata was a famous mathematician and astronomer during the Gupta period. He is credited with the discovery of the 'decimal' system. For the first time in Indian history, he analysed astronomy and mathematics as two distinct fields of study.
- 4. INTRODUCTION Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right triangles. The word "trigonometry" is derived from the Greek words "trigonon" (triangle) and "metron" (measure). Ancient astronomers and mathematicians used trigonometric principles to study celestial bodies and navigate the seas.
- 5. Applications ofTrigonometry Trigonometry has been used in a varietyof fields throughout history, including architecture, theoretical physics, and surveying. It can be used for a varietyof things, including: 1. Oceanography, seismology,meteorology,physical sciences, astronomy,acoustics, navigation,electronics, and many other subjects are among them. 2. It may also be used to determine the length of lengthyrivers,measure the heightof a mountain, and so on. 3. Solar, lunar, and stellar locations have all been calculated using spherical trigonometry. Trigonometry is used in measuring the height of a buildingor a mountain. The distance of a building from the viewpoint and the elevationangle can easily determine the height of a building using the trigonometricfunctions.
- 6. Height and distances Trigonometry is used to find the height, distance and depth of anything easily by certain fundamental. Angle of elevation: It the angle made with the horizontal when the observer raises his eyes position . Angle of depression: It the angle made with the horizontal when the observer lowers his eyes position. It's a very easy technique.
- 10. Q 1. The angle of elevation of the top of a tower from a point At the foot of the tower is 30°. And after advancing 150mtrsTowards the foot of the tower, the angle of elevation becomes 60° . Find the height of the tower? Sol: Tan 30° = 1/√3= h/d - - - (1) Tan60°= √3= h/(d – 150) - - - (2) From(1) d = h√3 From(2) √3 (d – 150) = h Substituting the value of ‘d’.... √3(h√3 – 150) = h 3h – 150√3 = h 3h – h = 150√3 2h = 150√3 h= 129.9m Problems h 150m d
- 11. Sol: Let AB be the tower and the angle of elevation from point C (on ground) is 30°. In ΔΑΒC, (AB)/(BC) = tan 30° (AB)/30 = 1/√3 AB = 30/√3 = 10√3 m Therefore, the height of the tower is 10√3 m. Q 2. The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower is 30°. Find the height of the tower. 30m Q 3. A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30°. Sol: It can be observed from the figure that AB is the pole. In AABC, (AB)/(AC) = sin 30° (AB)/20 = ½ AB = 20/2 = 10m Therefore, the height of the pole is 10 m. A B A B
- 12. Trigonometry has diverse applications across various branches of Artificial Intelligence: 1.*Computer Vision:* - Trigonometric functions help calculate angles, distances, and spatial relationships between objects in images or videos. - Rotational transformations, such as rotation matrices using sine and cosine, are essential for image transformations. 2.*Robotics:* - Trigonometry is fundamental for robot kinematics, determining the position and orientation of robotic arms and joints. - In path planning, trigonometric calculations assist in finding optimal trajectories for robot movements. 3.*Signal Processing:* - Fourier Transforms, which heavily involve trigonometry, are used for signal analysis and feature extraction in tasks like speech recognition and Image processing.
- 13. 4.*Image Processing:* - Trigonometry is applied in image transformation and manipulation, such as rotation and scaling. 5. *Machine Learning:* - Trigonometric functions are included in feature engineering, capturing complex patterns in data. - Sine and cosine functions can be used in neural network architectures, aiding in the modeling of periodic behaviors. 6.*Natural Language Processing (NLP):* - Trigonometry is applied in syntactic analysis, aiding in parsing and understanding sentence structures.
- 14. 7.*Reinforcement Learning:* - Trigonometry is used in modeling agent-environment interactions, especially in scenarios where angles and directions are critical. 8.*Optimization Algorithms:* - Trigonometric functions are used in optimization tasks, such as gradient descent, to find the minimum or maximum of a function efficiently. 9.*Simulation and Gaming:* - Trigonometry helps to model realistic movements, angles, and trajectories in simulations and video games.
- 15. 10.*Sensor Fusion:* - In combining data from different sensors, trigonometry is crucial for accurately estimating object positions and orientations. 11.*Physics Simulations:* -Trigonometry plays a role in simulating physical phenomena, especially in areas like physics-based simulations where understanding angles and trajectories is essential. 12.*Activation Function:* -Trigonometry functions, such as the hyperbolic tangent(tanh) and sine, have been used as activation functions in neural networks, influencing how information is passed between Neurons. These applications highlight the versatility of trigonometry in enhancing the capabilities of AI systems across a wide range of domains.
- 16. Trigonometry has many Roles in data science: Data visualization: Trigonometry is used to create and manipulate graphical representations of data, such as charts, graphs, maps, and animations. It is also used to perform geometric transformations, such as scaling, rotating, and translating, on data objects. 📊📊📊 Data analysis: Trigonometry is used to perform various Mathematical operations on data, such as calculating distances, angles, slopes, areas, and volumes. It is also used to apply trigonometric functions, such as sine, cosine, and tangent, to model periodic phenomena, such as waves, cycles, and oscillations. 📊📊📊 Data mining: Trigonometry is used to discover patterns and trends in large and complex data sets, such as text, images, audio, and video. It is also used to measure the similarity and dissimilarity between data points, such as using cosine similarity or Euclidean distance. 📊📊📊
- 17. Spatial Analysis: Trigonometry is used in tasks that Involves spatial data analysis, such as understanding the orientation or relative positions of objects. Time Series Analysis: Trigonometric function are used sometimes employed to model periodic patterns in time series data. These are some of the examples of how trigonometry is used in various aspects of data science. Trigonometry is a useful and essential tool that can help us explore and understand data in different ways. 📊📊📊
- 18. CONCLUSION Trigonometry is a branch of Mathematics with several important and useful applications. Hence it attracts more and more research with several theories published year after year. Trigonometric function are the relationships amongst various sides in right triangles. The enormous number of application of trigonometry include astronomy, geography, optics, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, seismology, land surveying, architecture .
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