The document derives the Law of Sines and Law of Cosines, which relate the angles and sides of triangles. It provides examples of using these laws to solve oblique triangles given various combinations of angle and side measures. It also discusses finding the area of triangles using formulas involving side lengths and included angles. Application problems demonstrate using the laws of sines and cosines to solve real-world geometry problems involving distances, bearings, and structures like towers on hills.
The PowerPoint presentation is on the "BASICS OF TRIGONOMETRY".
It includes the --
1) Definition of Trigonometry,
2) History of Trigonometry and its Etymology,
3) Angles of a Right Triangle,
4) About different Trigonometric Ratios,
5) Some useful Mnemonics to remember the Trig. ratios,
6) Theorem, which states that --
"Trigonometric Ratios are same for the same angles"
7) Trigonometric Ratios for some specific/ standard angles.
Siddhar Selvam Missions hindu religions Service get releave from your problems confusions by praying god. commander selvam, Dr commander Selvam, Siddhar Commander Selvam Place for Health,wealth,relationship,Excellence,Yoga,Meditation
The PowerPoint presentation is on the "BASICS OF TRIGONOMETRY".
It includes the --
1) Definition of Trigonometry,
2) History of Trigonometry and its Etymology,
3) Angles of a Right Triangle,
4) About different Trigonometric Ratios,
5) Some useful Mnemonics to remember the Trig. ratios,
6) Theorem, which states that --
"Trigonometric Ratios are same for the same angles"
7) Trigonometric Ratios for some specific/ standard angles.
Siddhar Selvam Missions hindu religions Service get releave from your problems confusions by praying god. commander selvam, Dr commander Selvam, Siddhar Commander Selvam Place for Health,wealth,relationship,Excellence,Yoga,Meditation
Nuorodos apie GTD vienoje vietoje (atsisiųskite failą, tada nuorodos garantuotai veiks). Skirta pradedantiems susipažinti su David Allen susisteminta asmeninio ir organizacinio produktyvumo sistema "Getting Things Done".
Triangles
Introduction
Sum of the angles of a triangle
Types of triangles
Altitude, Median and Angle Bisector
Congruence of triangles
Sides opposite congruent angles
A traverse is a series of connected lines whose lengths and directions are to be measured and the process of surveying to find such measurements is known as traversing. In general, chains are used to measure length and compass or theodolite are used to measure the direction of traverse lines.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
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Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
2. DERIVATION OF LAW OF SINES Let ABC be an oblique triangle with sides a, b, and c opposite their respective angles as shown in the figure below. If an altitude h is drawn to the base, we can write the following relationship: C b a h B A c
3. Equating the two expressions for h gives Dividing both sides of the equation by sin A sin B gives the following relationship: Similarly, if we draw an altitude from angle A to side a, we can derive the following expression: Combining these two results gives the Law of Sines, summarized as follows.
4. LAW OF SINES For any triangle ABC in which a, b and c are the lengths of the sides opposite the angles with measures A, B, and C, respectively, In words, the Law of Sines may be stated as follows: The sides of a triangle are proportional to the sines of the opposite angles.
5. Solution of oblique triangles involves four cases, namely: Two angles and one side are given. Two sides and the angle opposite one of the sides are given. Two sides and the angle between those sides are given. Three sides are given. Note: The Law of Sines is applicable for the first two cases and the Law of Cosines for the last two cases.
6. Case I: Two angles and one side are given. Example: Solve the following triangles. A = 51.30B = 48.70a = 24.5 A = 410B = 570c = 52 B = 1190C = 210b = 59 Case II: Two sides and the angle opposite one of the sides are given. (AMBIGUOUS CASE) When two sides and the angle opposite one of them are given, there may be no, one, or two solutions to the triangle. For this reason, Case II is called the ambiguous case. The following are the summary of the possible cases.
7. If A is an acute angle and a < b, there are three possibilities. C C C b sin A a b b b a=b sin A a a b sin A No solution a < b sin A Two solutions a > b sin A A A B A B c B’ One solution a = b sin A
8. 2. If A is an acute angle and a ≥ b, then there is exactly one solution. C a b A B 2. If A is an obtuse angle, then there are two possibilities. c C C a a b b B A A c No solution a ≤ b One solution a > b
9. EXAMPLE: Determine how many solutions exist. When either one or two solutions exist, solve the triangle or triangles. A = 670 a = 18 b = 20 A = 870 a = 47 b = 50 A = 320 a = 7 b = 10 A = 1130 a = 49 b = 54 A = 108.70 a = 54.3 b = 51.2
10. Application: Two forest ranger stations A and B are 48 miles apart. The bearing from A to B is N700E. A ranger in each tower spots a fire. The fire’s bearing from A and B is N330E and N140W, respectively. Find the distance from the fire to each tower. The Leaning Tower of Pisa was originally approximately 56 m high. If a surveyor trying to calculate the lean of the tower walks 72 m from the center base of the tower, the angle of elevation to the top is 400. find the lean of the tower of Pisa. On a hill inclined at an angle of 15.40 with the horizontal, stands a tower. At a point Q, 61.5 m down the hill from the foot of the tower, the angle of elevation of the top of the tower is 42.60. How tall is the tower?
11. DERIVATION OF LAW OF COSINES Let ABC be an oblique triangle with sides a, b, and c opposite their respective angles as shown in the figure below. The altitude h is drawn perpendicular to the base that divides side AB into two parts: x and x-c. Using the Pythagorean theorem for each triangle gives C b a h c-x x B A c
12. Solving each of these equations for h2 gives Equating the two expressions for h2 gives Solving the equation for b2 gives From the figure:
13. Substituting this expression for x gives one form of the Law of Cosines. Using the same method and drawing altitudes to sides CB and AC gives similar results. The Law of Cosines is summarized as follows.
14. LAW OF COSINES For any triangle ABC, where a, b, and c are the lengths of the sides opposite the angles with measure A, B and C respectively,
15. Case III: Two sides and the angle between those sides are given. Example: Solve the following triangle. a = 18.4 c = 26.3 B = 47.90 C = 1150a = 11 b = 21 A = 320b = 23 c = 47 Case IV: Three sides are given. Example: Solve each triangle ABC. a = 11 b = 14 c = 17 a = 23 b = 43 c = 31
16. The Area of a Triangle The area K of any triangle ABC is given by one of these formulas: The above formulas are used to find the area of a triangle when the measures of the two sides and the included angle are known.
17. HERON’S FORMULA: If a, b, and c are the measures of the sides of a triangle, then the area K of the triangle is given by Heron’s Formula is used to find the area of a triangle when three sides are given. EXAMPLE: Find the area of the given triangle. B = 710a = 21 c = 87 a = 31 b = 23 c = 14
18. Application: An airplane flies from city A going west to city B, a distance of 275 miles, and turns through an angle of 430 and flies to city C, a distance of 250 miles. Find the distance from city A to city C. Two ships leave the same port at the same time. One ship sails on a course of 1200 at 16 knots while the other sails on a course of 2200 at 23 knots. Find after 3 hours (a) the distance between the ships and (b) the bearing from the first ship to the second. A 75 ft vertical radio is to be erected on the side of a hill that makes an angle of 90 with the horizontal. Find the length of two guy wires that will be anchored 80 ft uphill and downhill from the base of the tower.