right spherical triangle. trigonometry
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This document discusses spherical trigonometry and spherical triangles. It defines key terms like spherical trigonometry, great circles, and spherical triangles. It also outlines Napier's rules for solving right spherical triangles and the six cases for solving oblique spherical triangles using different given parts. Examples are provided to illustrate applications of solving and finding the area of spherical triangles.
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Math12 lesson10
Spherical trigonometry concerns triangles on the surface of a sphere, with formulas to solve for unknown sides and angles of spherical triangles based on relationships between opposite, adjacent, and included parts. Formulas are provided for right, oblique, and general spherical triangles, along with the six cases for solving oblique triangles based on given parts. Examples demonstrate applying the formulas to solve various spherical triangle problems.
spherical triangles
This document provides information on spherical trigonometry. It defines key terms like great circles, small circles, spherical angles, and spherical triangles. It describes properties of these concepts, such as every great circle passing through the center of a sphere. Formulas for solving spherical triangles are presented, including the Sine Formula, Cosine Formula, and Haversine Formula. Examples show how to use these formulas to calculate unknown sides and angles of spherical triangles given other information.
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Since most of the figures are not uploaded I recommend to see this presentation on my website at http://www.solohermelin.com.at Math folder.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
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In the preparation for the Geodetic Engineering Licensure Examination, the BSGE students must memorized the fastest possible solution for the LEAST SQUARES ADJUSTMENT using casio fx-991 es plus calculator technique in order to save time during the said examination. note: lec 2 and above wala akong nilagay na solution para hindi makupya techniques ko. just add me on fb para ituro ko sa inyo solution. Kasi itong solution ko wala sa google, youtube, calc tech books at hindi rin itinuro sa review center.GEODETIC Least Square with fx 991-es plus
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This document is a grid of letters arranged in rows and columns with no spaces or punctuation. It contains 40 rows with 35 letters each for a total of 1,400 letters. The letters are seemingly randomly arranged with no obvious words or patterns.
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Spherical trigonometry concerns triangles on the surface of a sphere, with formulas to solve for unknown sides and angles of spherical triangles based on relationships between opposite, adjacent, and included parts. Formulas are provided for right, oblique, and general spherical triangles, along with the six cases for solving oblique triangles based on given parts. Examples demonstrate applying the formulas to solve various spherical triangle problems.
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This document provides information on spherical trigonometry. It defines key terms like great circles, small circles, spherical angles, and spherical triangles. It describes properties of these concepts, such as every great circle passing through the center of a sphere. Formulas for solving spherical triangles are presented, including the Sine Formula, Cosine Formula, and Haversine Formula. Examples show how to use these formulas to calculate unknown sides and angles of spherical triangles given other information.
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Since most of the figures are not uploaded I recommend to see this presentation on my website at http://www.solohermelin.com.at Math folder.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
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equations, section, and/or paragraph containing the answer has been included together with a
copy of the relevant material from the text. This is provided so that useful feedback and
references to the appropriate parts of the book can be easily provided to the students. Every
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printouts. Since there will be no reprints of the Instructors Manual, corrections will not be made
to these files. However, any erratum that is found is the book will be graciously accepted by the
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The Instructor’s Manual has been prepared as a convenience for instructors who adopt, for use in
their classes, the textbook ELEMENTARY SURVEYING (AN INTRODUCTION TO GEOMATICS), 13th
Edition, by Charles D. Ghilani and Paul R. Wolf. As a benefit to the instructor, each problem
consists of the book question and a derived solution. For most questions a reference to the
equations, section, and/or paragraph containing the answer has been included together with a
copy of the relevant material from the text. This is provided so that useful feedback and
references to the appropriate parts of the book can be easily provided to the students. Every
attempt has been made to provide correct responses to the questions, but most assuredly, a few
mistakes may have occurred. Please accept our apologies for them and correct them in your
printouts. Since there will be no reprints of the Instructors Manual, corrections will not be made
to these files. However, any erratum that is found is the book will be graciously accepted by the
author to update subsequent printings of this and future editions.
Introduction to surveying
Surveying is defined as the art and science of determining positions on or near the Earth's surface through measurements of distance, direction, and elevation. There are two main types of surveying: plane surveying which considers the Earth as flat, and geodetic surveying which accounts for the spherical shape of the Earth over large areas. Common types of surveys include cadastral, city, construction, forestry, hydrographic, industrial, mine, photogrammetric, route, and topographic surveys. Distances are subject to errors from factors like temperature, tension, and sag and are measured using pacing or chaining with a tape.
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This document discusses the strength of figure formula used to determine the reliability of survey networks. It defines the strength of figure factor F as the ratio of the number of stations P to the number of lines L. Several examples are provided of calculating F for different survey networks. It also covers using the distance angle method to solve for unknown sides within a quadrilateral using different combinations of triangles formed by the stations. Formulas are provided for calculating distances based on the respective differences of sines from the distance angles.
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1) Fore and back bearings are the bearings of a line observed from each end, with the fore bearing indicating the direction of progress and the back bearing being 180° different.
2) To find the back bearing from the fore bearing, add or subtract 180° depending on whether the fore bearing is less than or greater than 180°.
3) Local attraction refers to deviations of the magnetic needle from its normal position due to external magnetic forces, while dip is the inclination of the needle from the horizontal plane.
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- Control surveys establish horizontal and vertical reference points and lines that preliminary and construction surveys are referenced to. Horizontal control may be tied to grid monuments, property lines, or baselines while vertical control uses benchmark elevations from leveling surveys.
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The document discusses sets of axioms and finite geometries. It begins by defining geometry as "earth measure" from its Greek roots and discusses some early examples of geometry like Eratosthenes' measurement of the circumference of the Earth. It then discusses the development of geometry through the Greeks and Euclid, including undefined terms, axioms, postulates, and theorems. It provides examples of Euclid's axioms and postulates and discusses modern refinements. Finally, it introduces finite geometries, which have a finite number of elements, and provides an example of a three-point geometry with its axioms and a proof of one of its theorems.
Triangulation and trilateration pdf...
Trilateration and triangulation are surveying methods to establish horizontal control networks. Trilateration involves measuring the lengths of all three sides of triangles without measuring angles, while triangulation measures angles and the length of one base line. Both methods are used to determine coordinate positions through trigonometric computations. Triangulation networks can be classified based on their intended accuracy and purpose, from primary/first order for determining large areas to tertiary/third order for more detailed surveys.
Surveying problem solving
This document provides information about errors in measurements and their propagation. It defines different types of errors like gross errors, systematic errors, and random errors. It discusses concepts like most probable value, standard deviation, variance, standard error of mean, most probable error, confidence limits, and weight. It distinguishes between precision and accuracy. It also describes the propagation of errors, where the error in a computed quantity is evaluated as a function of the errors in the measured quantities used to compute it.
Lesson 12 centroid of an area
Here are the steps to find the centroid of each given plane region:
1. Region bounded by y = 10x - x^2, x-axis, x = 2, x = 5:
- Set up the integral to find the area A: ∫2^5 (10x - x^2) dx
- Evaluate the integral: A = 96
- Set up the integrals to find the x- and y-moments: ∫2^5 x(10x - x^2) dx and ∫2^5 (10x - x^2)x dx
- Evaluate the integrals: Mx = 192, My = 960
- Use the formulas for centroid: C
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1. Angles are usually measured in surveying to determine the positions of points on Earth's surface. Vertical angles are measured between lines of sight in a vertical plane, and can be angles of elevation or depression. Horizontal angles include interior, exterior, and deflection angles.
2. The direction of a line can be described by its azimuth angle or bearing. Azimuths are clockwise angles between the line and a reference direction, usually north. Bearings specify the angle from north or south plus the east or west designation.
3. To determine the back azimuth or bearing of a line, add or subtract 180 degrees to the forward azimuth, or reverse the letters for the bearing. Consistency in designating forward
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s2 5 tutorial traversing - compass rule and transit rule
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The document provides information about the universe and our solar system. It begins by stating that the universe contains celestial bodies like asteroids, comets, stars, and planets. It then discusses that the universe began billions of years ago with an expansion and continues expanding today. Galaxies are described as huge systems of stars, dust and gas held together by gravity. Our solar system is part of the Milky Way galaxy. The document then focuses on our solar system, describing that the Sun is at the center and is important for life on Earth by providing heat and light. It notes there are eight planets that orbit the Sun and reflect its light. Finally, it provides details about the Earth, stating it is the third planet from the Sun and the
Trigonometry
The document provides information about trigonometric identities of the form sin(A+B) and double angle formulae. It includes:
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2) Examples of using the identity formulas to simplify trigonometric expressions and prove identities.
3) Double angle formulas for sin(2A), cos(2A) and their uses in finding exact trig values and solving trig equations.
1 CS - Earth and Universe
The document summarizes key concepts about the universe, solar system, and Earth. It describes the universe as containing many stars and systems, including our Milky Way galaxy. It then discusses our solar system, distinguishing the inner, smaller, rocky planets from the outer, larger gas planets. It notes the Earth has one natural satellite, the Moon, which revolves around us in about 28 days. The document also provides brief descriptions of Earth's rotation, revolution, atmosphere containing nitrogen, oxygen and water vapor, hydrosphere of water, and geosphere of rocks and minerals in the core, mantle and crust.
A1 02 Celestial Sphere
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Right triangle trigonometry
This document provides an overview of right triangle trigonometry including definitions of sides and angles, the sine, cosine, and tangent functions, and how to use them to solve problems. It defines the opposite, adjacent, and hypotenuse sides of a right triangle in relation to the angle θ. Sine is defined as opposite/hypotenuse, cosine as adjacent/hypotenuse, and tangent as opposite/adjacent. Examples are provided to demonstrate using trig functions to find missing side lengths or angle measures.
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In addition, features and terminologies associated with remote sensing, satellite and aerosol optical properties are discussed. This project emphasizes on an interactive material that is best supplemented with lecture video. It is not designed to be conventional lecture slide. Point to note: the question mark appearing in bottom of the slides indicates the author raised a question during the lecture.
This presentation was delivered in coming-of-age lecture style, in contrast to old-school conventional style. This presentation stimulates audiences to think and act than a banal display of abstract data. The lecture videos can be found at:
[1] Part-1/2 (52 minutes): https://youtu.be/-O_mYoeg-us
[2] Part-2/2 (51 minutes): https://youtu.be/IhHHHZYcY0o
This presentation is done as a part of graduate course titled Aerosol Mechanics in Spring 2016. The author was pursuing MS in Environmental Engineering Sciences at University of Florida during the making of this project.
Celestial Sphere SK
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Astronomical surveying
This document provides an overview of astronomical surveying concepts. It discusses the celestial sphere model used to depict celestial objects, important astronomical terms like right ascension and declination, and the motions of celestial bodies like the Sun, stars, and planets. It also covers different coordinate systems used in astronomy like the altitude-azimuth and equatorial systems. Finally, it briefly introduces various time systems used, such as sidereal, mean, and standard times.
FIELD ASTRONOMY SURVEYING
This document provides information about different coordinate systems used in astronomy and mapping. It discusses the celestial sphere projection and altitude-azimuth and equatorial coordinate systems used to describe positions of celestial objects. It also covers horizon coordinates, celestial coordinates including right ascension and declination, and concepts like the celestial equator, ecliptic, and seasons. The document further summarizes coordinate systems used for mapping earth resources, including local geographic coordinates, projected coordinate systems, and specific projections like Lambert conformal conic and transverse Mercator.
Geometry and types of aerial photographs
The document summarizes key properties and types of aerial photographs. It discusses that aerial photographs have angles and scales as important geometric properties. It describes the different types of photographs based on the camera axis direction (vertical, oblique, etc.), angle of coverage, film used (black and white, infrared, color), and scale determined by focal length and height. It provides details on vertical, low and high oblique, convergent, and trimetrogon photographs.
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The document provides information about arithmetic and geometric sequences and series. It defines sequences, terms, arithmetic sequences, geometric sequences, and gives formulas to find specific terms and sums of sequences. For arithmetic sequences, the formula to find the nth term is an = a1 + d(n-1), where a1 is the first term, d is the common difference, and n is the term number. For geometric sequences, the formula is an = a1rn-1, where a1 is the first term, r is the common ratio, and examples are given to demonstrate using the formulas.
Sequences And Series
Gauss solved his teacher's challenge of summing the integers from 1 to 100 immediately by realizing that the normal sum (1 + 2 + ... + 100) and the reverse sum (100 + 99 + ... + 1) would be equal. Adding these sums together results in 101 multiplied by the number of terms (100), giving the solution of 5050. This insight showed that mechanical solving is not as useful as understanding the underlying patterns in mathematical problems.
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2. A triangle has several properties - angles opposite equal sides are equal, sides opposite equal angles are equal, and the sum of any two sides is greater than the third side.
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1. The document discusses different types of triangles based on their sides and angles. It defines triangle congruence and presents several triangle congruence theorems including SAS, ASA, AAS, SSS, and RHS.
2. Properties of triangles such as corresponding angles and sides of congruent triangles being equal are explained. Inequalities in triangles and relationships between sides and angles are also covered.
3. Objectives of the lesson include defining triangle congruence, stating criteria for congruence, and properties of triangles like sum of angles and relationships between sides and angles.
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- 2. SPHERICAL TRIANGLE B AC O b a O c B A C A spherical triangle is that part of the surface of a sphere bounded by three arcs of great circles. The bounding arcs are called the sides of the spherical triangle and the intersection of these arcs are called the vertices. The angle formed by two intersecting arcs is called a spherical angle.
- 3. DEFINITION OF TERMS: • Spherical Trigonometry – is a branch of trigonometry that concerns with triangles extracted from the surface of the sphere. • Great Circle – is a circle obtained by passing a section through the center of the sphere. • Spherical Triangle – is a spherical surface bounded by the area of three great circles. The magnitude of the side of a spherical triangle is the angle subtended by it at the Centre of the sphere and is expressed in degrees and minutes of arc. The maximum value of an angle of a spherical triangle is 1800. The sum of the three sides of a spherical triangle is less than 3600. • Right Spherical Triangle – is a spherical triangle having a right angle. • Quadrantal - spherical triangle is one in which one side equals to 900. • Pole – this axis intersects the sphere at two points, of the great circle. • Axis of the circle – the through the center of the sphere perpendicular to the plane of the circle. • Polar triangle – the vertices of a spherical triangle ABC as poles, construct three great circles.
- 5. Rule 1: The sine of any middle part is equal to the product of the tangents of the adjacent parts. Rule 2: The sine of any middle part is equal to the product of the cosines of the opposite parts. NAPIER’S RULE Illustrate:
- 6. RIGHT SPHERICAL TRIANGLE a b co - A co - c co - B Where: co – A = complement of A co – B = complement of B co – c = complement of c Napier’s Circle
- 7. Solution of oblique spherical triangle involves six cases, namely: Case 1: Two sides and included angle are given. ( SAS ) Case 2: Two angles and the included side. (ASA) Case 3: Two sides and an angle opposite one of them. (SSA ) Case 4: Two angles and a side opposite one of them. (AAS) Case 5: Three sides are given. ( SSS) Case 6: Three angles are given. ( AAA)
- 9. Application: Problem: Solve for the spherical triangle whose parts are a = 73°, b = 62°, and C = 90°.
- 11. Find the area of a spherical triangle of whose angles are 123°, 84°, and 73°. The radius of the sphere is 30 m.